HOW TO DECOMPOSE THE BETWEEN AND WITHIN EFFECTS IN CONTEXTUAL MODELS By Siwen Guo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Measurement and Quantitative Methods – Doctor of Philosophy 2019 HOW TO DECOMPOSE THE BETWEEN AND WITHIN EFFECTS IN CONTEXTUAL ABSTRACT MODELS By Siwen Guo In contextual studies, group compositions are often extracted from individual data in the sample, in order to estimate the group compositional effects (e.g., school socioeconomic status effect) controlling for interindividual differences in multilevel models. As the same variable is used at both group level and individual level, an appropriate decomposition of between and within effects is a key to providing a clearer picture of these organizational and individual processes (Zhang, Zyphur, & Preacher, 2009). Two analysis approaches, the manifest and latent aggregation approaches, were proposed in previous studies, which were based on different assumptions about the natures of group-level constructs, populations of research interest, and sampling designs. The current study developed a new approach with within-group finite population correction (fpc) to decompose the between and within effects. Its performances were compared with the manifest and latent aggregation approaches for the two-level random intercept model, 2- 1-1 mediation model, and 1-1-1 mediation model through mathematical derivation and simulation. The simulation conditions included the between-to-within-effect ratio, intraclass correlation coefficient (ICC) of the decomposed predictor or mediator, number of groups in the sample, balanced or unbalanced design, group size in the population, and within-group sampling ratio. When the within-group sampling ratio was moderate (i.e., from 10% to 90%), which is commonly seen in education research, the mathematical derivation and simulation results indicated that, in general, the between effect estimates from the new approach with within-group fpc were of less degrees of biases and higher observed coverage rates compared to those from the manifest and latent aggregation approaches. Although the between effect estimates from the manifest aggregation approach were less variable, the trade-offs were their lower observed coverage rates. For the within effects, there was no significant difference among the three analysis approaches in the relative biases, absolute relative biases, or root mean square errors (RMSE) on average, or across different simulation conditions. However, under the unbalanced design, the standard errors for the within effects from the new approach were underestimated. A real data application was also used to illustrate the three analysis approaches. Copyright by SIWEN GUO 2019 Dedicated to my parents, for their endless love, care, and support. v ACKNOWLEDGMENTS I would like to express my deep thanks to my advisor and committee members, Dr. William H. Schmidt, Dr. Richard Houang, Dr. Kimberly Kelly, and Dr. Joseph Gardiner, for their encouragement, support, and guidance. This dissertation would not be completed without their help. My deep gratitude firstly goes to my advisor, Dr. Schmidt, who always encourages and supports me to pursue the study and career paths I choose, and helps me in different ways. Under his mentorship as an advisee and a research assistant in the Center for the Study of Curriculum Policy, I not only gained relevant knowledge in statistics and research experiences, but also found my icon of the professional researcher, educator, and mentor. I would like to thank Dr. Houang deeply for the time, advice, and mentorship he provided me. His smart ideas, useful suggestions, thoughtful considerations, and kind encouragement helped me to achieve each academic milestone in my doctoral study. He always inspires and enriches me with new ideas and views. I am thankful to Dr. Kelly for her patience and support. She not only helped with my dissertation, but also gave me insightful feedback and comments in the courses I took from her and throughout my doctoral study. My special thanks go to Dr. Gardiner for his kind encouragement, support, and comments on my dissertation. I am grateful for his great help. In addition, thanks especially to Dr. Jeffrey Wooldridge from whom I learned a lot in statistics. My sincere gratitude also goes to Dr. Leland Cogan, Christina DeFouw, Jennifer Cady, and Jacqueline Babcock in the Center for the Study of Curriculum Policy. Thanks for the help, care, and kindness they offered me. I also want to thank my dear friends in China and US for their warm care and support in my academics and social life. I am grateful for their friendship. vi From the bottom of my heart, I would like to thank my family, especially my parents. Without their love and trust, I would not be the person I am right now. Many thanks to all who care, help, and support me. I really do appreciate all that they have done for me. vii TABLE OF CONTENTS LIST OF TABLES .......................................................................................................................... x LIST OF FIGURES ..................................................................................................................... xiii Chapter 1 Introduction .................................................................................................................... 1 Chapter 2 Literature Review ........................................................................................................... 3 2.1 Decomposition of the Between and Within Effects .............................................................. 3 2.1.1 Manifest aggregation approach ...................................................................................... 4 2.1.2 Latent aggregation approach .......................................................................................... 7 2.1.3 Within-group finite population selection ...................................................................... 12 2.2 Estimation Methods............................................................................................................. 12 2.2.1 ML estimation............................................................................................................... 12 2.2.2 Within-group finite population selection and MUML estimation ................................ 13 Chapter 3 The Current Study ........................................................................................................ 17 Chapter 4 The Role of Within-group Sampling Ratio in the Decomposition of the Between and Within Effects ............................................................................................................................... 19 Chapter 5 The Performances of the Manifest, Latent, and New Aggregation Approaches under Different Within-group Sampling Ratios...................................................................................... 24 5.1 Simulation Design ............................................................................................................... 24 5.1.1 Population ..................................................................................................................... 24 5.1.2 Population model .......................................................................................................... 25 5.1.3 Data generation ............................................................................................................. 29 5.1.4 Sample .......................................................................................................................... 30 5.1.5 Estimation method ........................................................................................................ 30 5.1.6 Evaluation criteria ......................................................................................................... 31 5.2 Results ................................................................................................................................. 32 5.2.1 Convergence rate .......................................................................................................... 32 5.2.2 Bias ............................................................................................................................... 35 5.2.3 RMSE ........................................................................................................................... 61 5.2.4 Coverage ....................................................................................................................... 69 5.3 Summary ............................................................................................................................. 78 Chapter 6 The Manifest, Latent, and New Aggregation Approaches: An Application with PISA Dataset........................................................................................................................................... 87 6.1 Background ......................................................................................................................... 87 6.2 Data and Measures .............................................................................................................. 88 6.3 Results ................................................................................................................................. 90 Chapter 7 Discussion .................................................................................................................... 99 viii 7.1 Implications ....................................................................................................................... 104 7.2 Limitations and Future Study ............................................................................................ 106 APPENDICES ............................................................................................................................ 111 APPENDIX A: The Between and Within Effects in the Manifest and Latent Aggregation Approaches under the Assumption of an Infinite Group Size ................................................ 112 APPENDIX B: The Between and Within Effects in the Manifest and Latent Aggregation Approaches Considering the Within-group Finite Population Selection ................................ 134 APPENDIX C: Detailed Results from the Simulation Study ................................................. 151 REFERENCES ........................................................................................................................... 187 ix LIST OF TABLES Table 1. Expectations of between and within effect estimators in the MLM, 2-1-1 mediation, and 1-1-1 mediation models ................................................................................................................ 22 Table 2. The population distributions of error terms in the MLM, 2-1-1 mediation, and 1-1-1 mediation models .......................................................................................................................... 28 Table 3. Proportion of variance in model convergence rate explained by the design factors ...... 34 Table 4. Model convergence rate by analysis approach, ICCX/ICCM, and within-group sampling ratio ............................................................................................................................................... 35 Table 5. Relative bias and absolute relative bias in within effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models........................................................................................ 36 Table 6. Proportion of variance of relative bias in within effect estimates explained by the design factors ............................................................................................................................................ 38 Table 7. Proportion of variance of absolute relative bias in within effect estimates explained by the design factors .......................................................................................................................... 39 Table 8. Absolute relative bias in within effect estimates by number of groups, group size, and within-group sampling ratio .......................................................................................................... 40 Table 9. Relative bias and absolute relative bias in between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models........................................................................................ 42 Table 10. Proportion of variance of relative bias in between effect estimates explained by the design factors ................................................................................................................................ 44 Table 11. Relative bias in between effect estimates by analysis approach and between-to-within- effect ratio ..................................................................................................................................... 45 Table 12. Proportion of variance of absolute relative bias in between effect estimates explained by the design factors ..................................................................................................................... 47 Table 13. Absolute relative bias in between effect estimates by analysis approach, between-to- within-effect ratio, and ICCX/ICCM .............................................................................................. 48 Table 14. Bias and absolute bias in between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models ................................................................................................................ 51 x Table 15. Proportion of variance of bias in between effect estimates explained by the design factors ............................................................................................................................................ 52 Table 16. Bias in between effect estimates by analysis approach, between-to-within-effect ratio, and ICCX/ICCM (or group size) .................................................................................................... 53 Table 17. Proportion of variance of absolute bias in between effect estimates explained by the design factors ................................................................................................................................ 57 Table 18. Absolute bias in between effect estimates by analysis approach, ICCX/ICCM, and group size ................................................................................................................................................ 58 Table 19. RMSE of within effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models ........................................................................................................................................... 61 Table 20. Proportion of variance in RMSE of within effect estimates explained by the design factors ............................................................................................................................................ 62 Table 21. RMSE of within effect estimates by number of groups, group size, and within-group sampling ratio................................................................................................................................ 63 Table 22. RMSE of between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models ........................................................................................................................................... 65 Table 23. Proportion of variance in RMSE of between effect estimates explained by the design factors ............................................................................................................................................ 66 Table 24. RMSE of between effect estimates by analysis approach, ICCX/ICCM, and group size ....................................................................................................................................................... 67 Table 25. Observed coverage rate for within effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models .......................................................................................................................... 70 Table 26. Proportion of variance in observed coverage rate for within effects explained by the design factors ................................................................................................................................ 70 Table 27. Observed coverage rate for within effects by analysis approach, balanced or unbalanced design, and group size................................................................................................ 71 Table 28. Observed coverage rate for between effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models .......................................................................................................................... 73 Table 29. Proportion of variance in observed coverage rate for between effects explained by the design factors ................................................................................................................................ 74 xi Table 30. Observed coverage rate for between effects by analysis approach, RX/RM, and within- group sampling ratio ..................................................................................................................... 76 Table 31. Descriptive statistics of ESCS, OTL, and mathematics performance in each country . 91 Table 32. Within effects from the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc in the 1-1-1 mediation model ................................ 95 Table 33. Between effects from the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc in the 1-1-1 mediation model ................................ 97 Table 34. Model convergence rate using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions ....... 151 Table 35. Relative bias using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions ....................... 155 Table 36. Absolute relative bias using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions ....... 163 Table 37. RMSE using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions .................................... 171 Table 38. Coverage rate using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions ....................... 179 xii LIST OF FIGURES Figure 1. Absolute relative bias in within effect estimates by number of groups, group size, and within-group sampling ratio .......................................................................................................... 41 Figure 2. Relative bias in between effect estimates by analysis approach and between-to-within- effect ratio ..................................................................................................................................... 46 Figure 3. Absolute relative bias in between effect estimates by analysis approach and between- to-within-effect ratio ..................................................................................................................... 49 Figure 4. Absolute relative bias in between effect estimates by between-to-within-effect ratio and ICCX/ICCM .................................................................................................................................... 50 Figure 5. Bias in between effect estimates by between-to-within-effect ratio and ICCX/ICCM ... 54 Figure 6. Bias in between effect estimates by between-to-within-effect ratio and group size ..... 55 Figure 7. Absolute bias in between effect estimates by ICCX/ICCM and group size .................... 59 Figure 8. RMSE of within effect estimates by number of groups, group size, and within-group sampling ratio................................................................................................................................ 64 Figure 9. RMSE of between effect estimates by analysis approach, ICCX/ICCM, and group size 68 Figure 10. Coverage rate for within effects by analysis approach, balanced or unbalanced design, and group size ............................................................................................................................... 72 Figure 11. Coverage rate for between effects by analysis approach, RX/RM, and within-group sampling ratio................................................................................................................................ 77 Figure 12. Simulation results for within effects by analysis approach, within-group sampling ratio, balanced or unbalanced design, number of groups in the sample, and group size .............. 83 Figure 13. Simulation results for between effects by analysis approach, within-group sampling ratio, between-to-within-effect ratio, ICCX/ICCM, and group size ............................................... 85 xiii Chapter 1 Introduction In contextual studies, the same individual-level variables are often used to reflect both individual-level and group-level constructs, and the effects from different levels often have different theoretical interpretations (Lau & Nie, 2008; Marsh et al., 2012). The group-level effects or group compositional effects, and the underlying group-level processes are an area of research interest (Mayer, Nagengast, Fletcher, & Steyer, 2014). The big-fish-little-pond effect is an example in education research, which found student academic self-concept was positively associated with individual achievement, but negatively associated with school average achievement. The school-level effect of achievement on student academic self-concept reflected the way schools were structured and their effects on individuals (Marsh et al., 2009). School compositional effects, or the effects of aggregated student characteristics, like student socioeconomic status (SES), gender, and ethnicity, etc., draw large attention in contextual studies in education. The models that include the same variable at both individual- and group- levels are called contextual models or compositional models (Lüedtke et al., 2008). In these models, the central question is whether the aggregated group compositions have any effect on individual outcomes controlling for interindividual differences (Marsh et al., 2009). If individuals are randomly selected from the entire population with error-free measurements of their own characteristics as well as their group compositions, a single level model would work to separate and describe the effects of group compositions and individual characteristics. Challenges arise, however, in a two-stage cluster sampling design which is often used in data collection in these contextual studies, as individuals are naturally nested in groups. Meanwhile, the group compositions are usually unknown and need to be extracted from 1 individual data in the sample, which generally brings sampling errors into the aggregated group compositions. As the same variable is used at both group level and individual level, an appropriate decomposition of the between-group and within-group effects1 is a key to providing a clearer picture of these organizational and individual processes (Zhang et al., 2009). The current study aims at assessing the performances of different analysis approaches in the decomposition of between and within effects in contextual models. 1 For simplicity, the between-group effect and within-group effect are referred as “between effect” and “within effect” in the current study. 2 Chapter 2 Literature Review 2.1 Decomposition of the Between and Within Effects Previous contextual studies have investigated not only the between effects of group compositions and within effects of individual characteristics on the individual outcomes, but also their roles as mediators as well as their indirect effects through other variables. The study on student and school SES effects is one good example which examines the between effect of group compositions and the within effect of individual characteristics (Lüedtke et al., 2008; Raudenbush & Bryk, 2002). To explore the between and within effects, a two-level random intercept model (referred as MLM model for simplicity) is often used. The study on the school-based tobacco prevention programs which aim at lowering youth initiation of smoking via norms, is an example to explore the mediating effect of a group-level aggregated construct on the relationship between group-level treatment and individual outcomes (Pituch, Stapleton, & Kang, 2006). The group norm is aggregated from individual norms, and its mediating effect is the research focus. This mediating effect is usually modeled in a two-level random intercept 2-1-1 mediation model (i.e., the treatment is measured at group level, and the mediator and outcome are measured at individual level, and it is referred as 2-1-1 mediation model in the current study). Schmidt, Burroughs, Zoido, and Houang (2015) examined the between and within indirect effects of SES on student mathematics achievement through opportunity to learn (OTL). It is an example which explores the indirect effects of aggregated group compositions and individual characteristics. The group-level and individual-level indirect effects are usually modeled in a two-level random intercept 1-1-1 mediation model (i.e., the predictor, mediator and outcome are measured at individual level, and it is referred to 1-1-1 mediation model in the current study). 3 To decompose the between and within effects based on individual data, manifest and latent aggregation approaches were proposed in previous studies. Following the research trend in contextual studies in education, the decompositions of between and within effects in the manifest and latent aggregation approaches are discussed for the MLM, 2-1-1 mediation, and 1-1-1 mediation models in the current study. 2.1.1 Manifest aggregation approach Traditionally, the between and within effects are assessed as the effects of manifest group means and individual deviations from the group means (i.e., group mean centering) in multilevel models (Raudenbush & Bryk, 2002). This approach was referred to the unconflated multilevel model by Preacher, Zyphur, and Zhang (2010). Following the research by Marsh et al. (2009) and Lüedtke, Marsh, Robitzsch, and Trautwein (2011), the term “manifest aggregation approach” is used in the current study as a manifest aggregation is used to construct group means. In the manifest aggregation approach, for a MLM model with only one predictor, Raudenbush and Bryk (2002) proposed the model as: , (1) where Yij is the observed outcome and Xij is the observed predictor of ith individual in jth group; Xj is the manifest group mean of X in jth group; β00 is the intercept, and βxw and βxb are the within and between effects of X on Y; and u0j and eij are group-level and individual-level error terms. For the 2-1-1 mediation model and 1-1-1 mediation model2, Zhang et al. (2009) suggested the models as 2 In the current study, it is assumed that group-level constructs only show group-level effects, and they cannot influence any within-group difference. 4 000()ijxwijjxbjjijYXXXue and , , , , (2) (3) (4) (5) where Mij is the observed mediator of ith individual in jth group; Mj is the manifest group mean of M in jth group; βmw and βmb are the within and between effects of M on Y; α00 is the intercept, and αw and αb are the within and between effects of X on M; and w0j and dij are group-level and individual-level error terms. By decomposing Xij into the uncorrelated group mean Xj and individual deviation from group mean (Xij-Xj), the variance of Xij seems to be separated into the between-group variance in Xj and within-group variance in (Xij-Xj), and the between and within effects seem to be set apart. However, the manifest group mean Xj is not a generally perfect or error-free measurement of the group composition3, and the between effect estimator in the manifest aggregation approach is generally a conflation of between and within effects. To be specific, with a random sample from group j, the sample mean Xj is not the “true” population mean in jth group, and it involves the sampling error (Lüedtke et al., 2011; Lüedtke et al., 2008). Lüedtke et al. (2008) showed the expectation of the between effect estimator of X on Y in the MLM model was 3 The focus in the current study is on the sampling error issue in the aggregation of group compositions, and measurement error is not discussed. Xij is assumed free from measurement error in this study. This assumption is reasonable for some individual characteristics used in this study, like gender and ethnicity. However, for student SES and OTL, this assumption is not generally satisfied. When the measurement error is considered in the model, multiple indicators of Xij are needed. Please see Lüedtke et al. (2011) for further information. 5 000()ijmwijjmbjxbjjijYMMMXue000ijbjjijMXwd000()()ijmwijjmbjxwijjxbjjijYMMMXXXue000()ijwijjbjjijMXXXwd E  (  xb )   xb ICC X  ICC X  1 (1  xw n 1 (1  n  ICC ) X ICC ) X , (6) where  xb is the estimator of the between effect of X on Y in the manifest aggregation approach; ICCX is the intraclass correlation coefficient (ICC) of X; and n is the common group size. Preacher et al. (2010) showed the expectation of the group-level indirect effect estimator of X on Y via M in the 2-1-1 mediation model was E   (    b b  mb ) 2   mb M (  ( 2  M  2  XM 2  X 2  XM 2  X )  )  2   M 1 n mw 1 n 2  M , (7) where  b and  m b are estimators of the between effects of X on M, and of M on Y in the manifest aggregation approach; τM2 and σM2 are the group-level and individual-level variances of M; τX2 is the group-level variance of X; and τXM is the group-level covariance between X and M. Following the logic and assumptions made by Lüedtke et al. (2008) and Preacher et al. (2010), in the 1-1-1 mediation model, the expectation of the group-level indirect effect estimator  2 b X 1 n 2   X  2     X     w 1 n  2 X     MY             E   (   b mb ) is   (  XY  1 n  1 n  MY  XM 2   X 2 X XY )(   1  n 1  n 1 n  2   X  2 X XM  1 n  XM ) 2 )           , (8) (  XM 2   M 2 M  1 n  6 where σX 2 is the individual-level variance of X; τMY and σMY are the group-level and individual- level covariances between M and Y; τXY and σXY are the group-level and individual-level covariances between X and Y; and σXM is the individual-level covariance between X and M. In the three models, unless the within effects are the same as the between effects for all paths (i.e., , , and ), or the individual-level variances and covariance of X and M are equal to zero, or the group size n is infinite, the between effect estimators are biased (please see Appendix A for the population models, assumptions, parameters of theoretical interest, between and within variance-covariance matrices, and derivations of within and between effect estimators in the manifest aggregation approach for the MLM, 2-1-1 mediation, and 1-1-1 mediation models). If the within effect is enough to answer the research question, the manifest aggregation approach will provide unbiased estimates. When the between effect is of theoretical interest as it is in the current study, the manifest aggregation approach may not be a good choice. The bias due to sampling error not only involves in the estimation of the between effect of the decomposed predictor, but also affects the estimation of other group-level effects (Lüedtke et al., 2011; Mayer et al., 2014). The sampling error in the aggregated group means and the resulting biased between effect estimators in the manifest aggregation approach are criticized by the latent aggregation approach. 2.1.2 Latent aggregation approach To correct for the bias in the between effect estimator due to sampling error, there is a new trend to decompose the between and within effects in a latent aggregation approach (Lüedtke et al., 2011; Lüedtke et al., 2008; Marsh et al., 2012; Marsh et al., 2009; Mayer et al., 2014; Preacher, 2011; Preacher, Zhang, & Zyphur, 2011, 2016; Preacher et al., 2010; Ryu, 7 xwxbmwmbwb 2015b), which models the group-level and individual-level variance-covariance matrices explicitly with multilevel structural equation modeling (MSEM, Muthén, 1990). In this approach, the group-level and individual-level latent (or random) components are directly modeled to examine the between and within effects. Derived from the random effect analysis of variance (ANOVA) model, Schmidt (1969) first discussed the theories to decompose individual scores into their grand means, between and within random components, and estimate the between and within relationships of the random components. This idea was widely adopted in multilevel factor analyses to explore different between and within factor structures (Kim, Kwok, & Yoon, 2012; Longford & Muthén, 1992; Muthén, 1990, 1991). For the MLM (Lüedtke et al., 2011; Lüedtke et al., 2008), 2-1-1 mediation (Preacher et al., 2011; Preacher et al., 2010), and 1-1-1 mediation models (Preacher et al., 2010), the models in the latent aggregation approach can be built as follows. MLM model: 2-1-1 mediation model: 1-1-1 mediation model: 8 (9) (10) (11) (12) (13) (14) (15) ijXXijXjXUU000ijxwXijxbXjjijYUUueijMMijMjMUUjXXjXU000ijmwMijmbMjxbXjjijYUUUue0MjbXjjUUwijMMijMjMUU (16) (17) (18) (19) Note: µX, UXij, and UXj represent the grand mean, individual-level random component, and group- level random component of X for ith individual in jth group respectively; µM, UMij, and UMj represent the grand mean, individual-level random component, and group-level random component of M for ith individual in jth group respectively. When 1) the model is correctly specified, 2) error terms follow a multivariate normal distribution with means of zero and constant variances, 3) error terms are uncorrelated with each other as well as group-level and individual-level latent components of the predictors, 4) group- level latent components are uncorrelated with individual-level latent components, and 5) each group has an infinite population, the latent aggregation approach provides approximately unbiased within and between effects for the MLM (Lüedtke et al., 2011; Lüedtke et al., 2008), 2- 1-1 mediation (Preacher et al., 2011; Preacher et al., 2010), and 1-1-1 mediation models (Preacher et al., 2010). The between and within effect estimators in the latent aggregation approach are discussed in Appendix A. The latent aggregation approach outperforms the manifest aggregation approach in the estimation of between effects with sampling error correction under appropriate assumptions. It has been shown that the between effect estimators in the latent aggregation approach had smaller biases and root mean square errors (RMSE) than those in the manifest aggregation approach, when the assumptions in the latent aggregation approach were satisfied (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011). 9 ijXXijXjXUU000ijmwMijmbMjxwXijxbXjjijYUUUUue0MjbXjjUUwMijwXijijUUd However, the latent aggregation approach did not always perform better than the manifest aggregation approach. Intuitively, when the sampling ratio within each group is 100%, the manifest group mean is the population group mean and free from sampling error. The manifest aggregation approach should provide approximately unbiased between effects under this condition. Lüedtke et al. (2008) found when the within-group sampling ratio approached 100%, the manifest aggregation approach outperformed the latent aggregation approach in the estimation of between effects in the MLM model. Lüedtke et al. (2011) indicated when only limited information on the group-level construct was available (e.g., low ICC, small group number, and small individual number within groups), the manifest aggregation approach could outperform the latent aggregation approach in the RMSE of between effect estimators in the MLM model. In the 2-1-1 mediation model, McNeish (2017) found the manifest aggregation approach outperformed the latent aggregation approach with a small number of groups. The contradiction comes from the different assumptions made by the manifest and latent aggregation approaches: in the manifest aggregation approach, the entire group is assumed to be sampled or the within-group sampling ratio is assumed to be 100%, while in the latent aggregation approach, the population in each group is assumed to be infinite4 or the within-group sampling ratio is assumed to be close to 0. When designing a study, sampling of the entire 4 Lüedtke et al. (2008) discussed the assumptions of finite and infinite population in each group and distinguished the formative and reflective group-level constructs. When the group is the referent in aggregation, group-level construct is a reflective measure and the assumption of infinite population in each group is reasonable. One example is school climate, in which students within a school rate the climate for the target school. Following the domain sampling theory, students are exchangeable and can be assumed infinite in that school. However, it does not fit the contextual studies in which the group construct is a composition of individual characteristics. When the referent is the individual and individual characteristics are used for aggregation, the group construct is a formative measure and the population in each group cannot be assumed infinite. One example is school gender ratio, in which students are not exchangeable in terms of their own gender. When the sampling ratio in a school is 100%, manifest gender ratio in each school is free from sampling error. In the current study, group constructs are group compositions which are aggregated from the individual characteristics. Individual is the referent and within-group population cannot be assumed infinite. 10 groups is not generally applied. Sampling with replacement is hardly conducted either, and the number of individuals per group is hardly infinite. When the groups are naturally of small or moderate sizes, like classrooms and schools, not to mention “infinite”, the number of individuals per group is even further away from “large enough”. This problem was mentioned in some previous studies. Lüedtke et al. (2008) and Preacher et al. (2010) limited their discussion of the latent aggregation approach to the situations where the within-group sampling ratio was low (e.g., lower than 5%). In the cases where the within-group sampling ratio approached 100%, Lüedtke et al. (2008), Marsh et al. (2009), Preacher et al. (2010), Lüedtke et al. (2011) and Marsh et al. (2012) suggested the manifest aggregation approach might be a natural choice. When the within-group sampling ratio was moderate, Lüedtke et al. (2008) and Marsh et al. (2009) suggested the “best” between effect estimate was between the estimates in the manifest and latent aggregation approaches. In summary, when the within-group sampling ratio is 100%, the manifest aggregation approach is a natural choice to decompose the between and within effects. When the within- group sampling ratio is low and close to 0, the latent aggregation approach is a good choice. Except in the two special sampling cases, the latent aggregation approach overcorrects sampling error in the aggregation by assuming an infinite group size, while the manifest aggregation approach does not correct sampling error in the aggregation at all as if the entire population in each group is sampled. Although the within-group sampling ratio determines the appropriate way to correct sampling error in the aggregation and to decompose the between and within effects, neither the manifest nor latent aggregation approach takes it into consideration. The role of within-group sampling ratio in the decomposition of between and within effects is unclear in either manifest or latent aggregation approach. 11 2.1.3 Within-group finite population selection With a probability sample, it is possible to quantify the sampling error with a consideration of the within-group sampling ratio, and correct it in the decomposition of between and within effects. When the sampling ratio exceeds 5% of the population, the selection cannot be treated as if it comes from an infinite population (Cochran, 1977), and a correction is needed. The correction made for finite population selection is called finite population correction (fpc). It is calculated as (1-n/N), where n is the sample size, and N is the population size. Intuitively, with a larger sampling ratio, there is more information and less uncertainty about the population mean. The variance of the mean estimator should be smaller than it is with a smaller sampling ratio (Lohr, 2009). When the sampling ratio is 100%, the population mean is known and the variance of the mean estimator is 0. With a simple random sampling (SRS) of n individuals from a population of N, if the population variance of Y is S2, the variance of in the sample is , which is corrected with fpc. In the decomposition of between and within effects, a within-group fpc is needed to correct for the sampling errors in the estimation of variances and covariances of the aggregated group constructs. However, there is no available modeling or analysis approach for adopting within-group fpc in the decomposition of between and within effects. 2.2 Estimation Methods 2.2.1 ML estimation In general, a MSEM can be used for the MLM, and 2-1-1 mediation, and 1-1-1 mediation models in both manifest and latent aggregation approaches. A maximum likelihood (ML) estimation is usually used to estimate the fixed and random effects simultaneously. For the 12 Y2(1)nSNn MSEM, the general fitting function to be minimized (Muthén, 1997) for both approaches can be expressed as , (20) where ΣW is the within and ΣB is the between variance-covariance matrices based on the model; nj is the number of individuals in jth group; g is the number of groups in the sample; Zj is (njk)×1 vector with k variables used in the model observed from nj individuals in group j; µj is (njk)×1 vector of grand means for each of the k variables used in the model for nj individuals in group j. The manifest aggregation approach includes group means and individual deviations from group means of the decomposed predictors as well as other variables in Zj, while the latent aggregation approach includes only the original decomposed predictors and other variables in Zj. There are more challenges in the ML estimation for the latent aggregation approach as more random effects are directly modeled. The ML estimation for MSEM was first developed for the balanced data by Schmidt (1969). For the unbalanced data, the ML estimation with the expectation–maximization (EM) algorithm was developed by Bianchi (1987), and the Fisher scoring algorithm was discussed by Longford and Muthén (1992). A large number of groups is generally necessary for the ML estimation, and inadmissible problems are frequently observed for the singular between variance- covariance matrices and negative variance estimates on the group level (Muthén, 1994). 2.2.2 Within-group finite population selection and MUML estimation In general, neither the manifest nor latent aggregation approach takes the within-group finite population selection into sampling error correction. There is no available approach to incorporate within-group fpc in the MSEM. Muthén’s ML-based estimator (MUML) might provide some ideas. As the ML estimation is computationally intensive for MSEM, Muthén 13 1{log'()'(')()}gWBWBjFnjnjnjjjnjnjnjjjIΣ11ΣZμIΣ11ΣZμ (1989, 1990) suggested an ad hoc estimator which treated the within and between data in a multiple-group fashion with a fitting function of The common cluster size n0 is . (21) , (22) and the pooled within variance-covariance matrix and variance-covariance matrix of group means are , (23) , (24) where Zij is a k×1 vector of k variables used in the model observed from individual i in group j; is a k×1 vector of the jth group means of k variables used in the model; is a k×1 vector of the grand means of k variables used in the model; and µ is a k×1 parameter vector of k variables’ means. This simpler estimator is called MUML (Muthén, 2004), limited information maximum likelihood estimator, or pseudo-balanced maximum likelihood estimator (Hox & Maas, 2001), which gives estimates close to ML estimators (Longford & Muthén, 1992; Muthén, 2004). 14 11000{log([][()()'])}(){log()}gWBWBBjWWPWjFgntrnSnngtrSΣΣΣΣZμZμΣΣ220()()(1)ggjjjjgjjnnnng()()'(1)jngjiPWgjjSnjjijijZZZZ()()'1jngjiBSgjjZZZZjZZ The statistical inference of MUML estimators has been derived, and its performance under different sample sizes and ICCs have been examined. Yuan and Hayashi (2005) showed that the MUML leaded to correct model inference asymptotically when the level 2 sample size went to infinity, and the coefficient of variation of the level 1 sample sizes went to zero. Hox and Maas (2001) examined the robustness of MUML with unequal groups, small sample sizes at two levels, and low and high ICCs. The within-group results posed no problem. Inadmissible estimates were found in the between-group part when the number of groups was small and the ICC was low. They suggested the MUML remained a useful tool when the number of groups was at least 100. It was also shown that the MUML reached unbiasedness when the number of groups reached 200 (Hox, Maas, & Brinkhuis, 2010). Ryu (2015a) compared the ML and MUML estimators through simulation for a multiple-group two-level path model where group memberships were at the individual level. The MUML estimation yielded unbiased estimates in all sample size conditions. However, the MUML estimation consistently resulted in underestimated standard errors for the group-level variance estimates, and larger coverage rates than the nominal value of individual-level mean estimates. Since the MUML estimation showed minor problems in the estimation and could be easily implemented using standard SEM software packages (Ryu, 2015a), it was used for MSEM in empirical studies. Cheung and Au (2005) applied MUML to a cross-cultural study using the data from the International Social Survey Program. They demonstrated that the within-group results from MUML were quite stable even when the number of groups was small. Stapleton (2006) examined the effect of hours spent on TV on student achievement using MSEM via MUML and ML estimations. There was no large difference in the between and within estimates between the two estimation approaches. For the two-level confirmatory factor analysis (CFA), 15 Wu, Lee, and Lin (2018) found the differences in the estimates between MUML and ML estimations were trivial in their empirical studies. Using the data from Programme for International Student Assessment (PISA) 2003, Ryu (2014) tested the multilevel factorial invariance using MUML and ML estimations. The MUML estimation was shown to be a viable alternative to ML estimation. Ryu (2015a) illustrated how to apply MUML estimation for a multiple-group two-level path model and a multiple-group two-level CFA with group memberships at the individual level using the data from Trends in International Mathematics and Science (TIMSS) 2003 and PISA 2003. In the MUML, the between and within effects are estimated by treating SPW and SB in a multiple-group fashion. This approach ends up with the same fitting function as ML estimation under a balanced design (Muthén, 1989, 1990). Under an unbalanced design, the common group size is used to “psudo-balance” the data. Following this logic, it is possible to use the within- group fpc in the estimation of SPW and SB, and estimate the between and within effects via MUML based on the adjusted SPW and SB with within-group fpc. A similar idea was adopted by Stapleton (2002) to incorporate sampling weights into the estimation of MSEM via MUML, in which the weighted SPW and SB were treated in a multiple-group fashion. The MUML provides a possibility to decompose the between and within effects with within-group fpc. However, it is unclear what the pros and cons of MUML with within-group fpc and psudo-balancing data are, compared to the manifest aggregation approach which does not correct for the sampling error in aggregation, and the latent aggregation approach which overcorrects for the sampling error in aggregation. Under different within-group sampling ratios, it is not easy to tell which is a “better choice” in the decomposition of between and within effects. 16 Chapter 3 The Current Study The literature review showed that, in the decomposition of between and within effects in contextual models, the manifest and latent aggregation approaches made different assumptions about the within-group sampling in the sampling error correction for the aggregated group constructs. When the entire population was sampled within each sampled group, the aggregated group mean was free from sampling error and the manifest aggregation approach was suitable. When the within-group sampling ratio was extremely small (e.g., smaller than 5%), the within- group finite population selection was not a major problem in sampling error correction and the latent aggregation approach was appropriate. However, when the within-group sampling ratio was moderate, which is commonly seen in education research, the within-group finite population selection was of concern in the sampling error correction in the decomposition of between and within effects. The between effect estimators from the manifest aggregation approach may be biased as the sampling error in aggregation is not corrected at all. The between effect estimators from the latent aggregation approach may also be biased as the sampling error is overcorrected by assuming an infinite group size. The resulting between effect estimators may be conflations of between and within effects in both approaches. As there was no available approach dealing with finite population selection in sampling error correction in aggregation, the current study first discussed the role of within-group sampling ratio in the decomposition of between and within effects in the manifest and latent aggregation approaches. An approach with within-group fpc was proposed using MUML estimation based on the adjusted SPW and SB. Then, the new approach was compared to the 17 manifest and latent aggregation approaches with ML estimation in a Monte Carlo simulation study. The research questions were: 1) When the within-group finite population selection was of concern, how did the within- group sampling ratio involve in the decomposition of between and within effects in the manifest and latent aggregation approaches? 2) How did the new approach with within-group fpc via MUML estimation perform compared to the manifest and latent aggregation approaches via ML estimation under different within-group sampling ratios, group sizes in the population, ICCs of decomposed predictors or mediators, ratios of between to within effects, and numbers of groups in the sample for the MLM, 2-1-1 mediation, and 1-1-1 mediation models? In the current study, to resemble the data structure typically found in education research, an extremely large number of groups with small to moderate group sizes was assumed in the population. A two-stage cluster sampling design fit the framework of contextual studies, and was often used for data collection in education research. Thus, a two-stage cluster sampling design was assumed to be used for data collection in the current study for mathematical derivation and simulation. The current study focused on the between and within effects of the decomposed predictors and/or mediators in the MLM, 2-1-1 mediation, and 1-1-1 mediation models. An empirical example using the dataset from PISA 2012 was also used to illustrate and compare the three analysis approaches. 18 Chapter 4 The Role of Within-group Sampling Ratio in the Decomposition of the Between and Within Effects In previous studies, the within-group fpc was not used in the sampling error correction in the decomposition of between and within effects in either manifest or latent aggregation approach. In this chapter, a derivation was used to show how within-group sampling ratio involve in the decomposition of between and within effects in the manifest and latent aggregation approaches, and how to adjust SPW and SB with within-group fpc. A two-stage cluster sampling design with equal probability of selection was assumed to be used for data collection. First, a simple random sample of g groups was selected from G groups. Each group was of the same size N. Then, a simple random sample of size n was selected within each sampled group. The inclusion probability was the same across individuals and across groups. The within-group fpc was 1-n/N. Assuming G was infinite and N was finite. In general, the ML should be used in the manifest and latent aggregation approaches to estimate the between and within effects, but an iterative procedure such as EM or Fisher scoring algorithm is often required. In order to directly show how the within-group sampling ratio involve in the between and within effects estimation, the ordinary least squares (OLS) was used for the manifest aggregation approach, following the studies conducted by Lüedtke et al. (2008) and Preacher et al. (2010). A method of moment (MOM) was firstly used to decompose the between and within variance-covariance matrices in the latent aggregation approach (i.e., W   PWS and    B ( S B  S PW ) / n ), and the OLS was used to estimate the between and within effects based on the estimated W and  B 5.                                                              5 The OLS estimates in the manifest and latent aggregation approaches were not the same as the ML estimates, and would not be used for evaluation in the simulation study. The purpose of the current derivation was to directly show how within-group sampling ratio affected the decomposition of between and within effects in each approach, and the necessity to consider within-group fpc in the decomposition of between and within effects.   19 In the MLM model, the variances and covariances of the used variables, and expectations of SPW and SB of Xij and Yij are: Cov X      Y ij  X j ij X j              2 Y fpc 2    Y 1 )  XY n (1  fpc   XY  XY 1 n (1  fpc 1 n ) 2  X 0 fpc 1 n 2   X  2 X        (25) ( Y ij  Y j 2 )  n i g j 1 1) ( g n  g n  1) j i 1 ( g n  ( Y ij  Y X j )( ij  X ) j 1 ( g n  1)       ( X ij  X 2 ) j  n  n fpc 1  2   Y  2    X XY    (26)  n i g j ( E S PW )  E        n i ( E S B )  E       1  g g 1 1 g  g  n j j i 1 ( Y j  Y 2 ) ( Y j  Y X )( j  X ) 1  1 g  n i g j           ( X j  X 2 ) 2 Y fpc fpc 2   Y   XY   n n XY    (27) fpc 2   X  n 2 X By the OLS estimation, the expectations of between and within effect estimators of X in the manifest aggregation approach are: E  (  xw )  E [ 1 gn  1 1 1 gn  g n  j i g n  i j (X - X ij (Y -Y )(X - X ij ij  X - X ) j ij j  j X - X ) X - X ( ij ij j  X - X ) j ij j ]  (1  fpc (1  fpc 1 )  XY n 1 n ) 2  X   xw (28) E  (  xb )  E [ 1 -1 ng n ng  n  g j i ( g  j 1 ( Y Y X ij )( - - X ) j ]  X j - X X )( - X ) j  xb ICC X   xw ICC X  fpc 1 fpc n 1 (1  n (1  ICC ) X ICC X ) (29) In the latent aggregation approach, treating N as finite, the expectations of between and within effect estimators can be expressed as:   20 E  (  xw )  [ E 1 ( g n  1 ( g n  g n j  1)  n g i j i 1) (Y -Y )(X - X ) j ij ij j ]  (X - X )(X - X ) j ij ij j n fpc  1 n  n fpc  1 n   XY 2  X   xw (30)   ( xb E )  E [ g { [  g n -1  g j 1 j g n  ( Y Y X )( - j ( X j - X X )( - X )  - X )  j j 1 ( g n  1 ( g n  1) 1) g n  i n i  j g j (Y - Y )(X - X ) ] / n ij j ij j (X - X )(X - X ) ] / j ij ij j   xw 2 xb X  2  X  1 N 1 N (  (  n  n n 1 }  n ) 2  X 1 n  ) 2  X Using the same analyses, the expectations of estimators in the three models are shown in Table 1 (31) (please see Appendix B for details). As shown in Table 1, although within-group fpc was needed to construct SPW, it did not actually affect the within effect estimates in either approach. It did bring problems in the between effects estimation. The sampling error was not corrected appropriately in either approach. In the manifest aggregation approach, the sampling error in the aggregation was not corrected at all. Only when the entire groups were selected and within-group sampling ratio was 1, the manifest aggregation approach provided unbiased between effects. In contrast, the sampling error in the aggregation was overcorrected in the latent aggregation approach. Only when the groups were of extremely large sizes in the population, and the within-group sampling ratio was close to 0, the latent aggregation approach provided approximately unbiased between effects. When the within- group sampling ratio was moderate, as the within-group sampling ratio occurred in both numerators and denominators of the between effects in both approaches, the directions and magnitudes of the biases in the between effects depended on the specific estimators to be evaluated.   21 Table 1. Expectations of between and within effect estimators in the MLM, 2-1-1 mediation, and 1-1-1 mediation models Manifest aggregation approach  xw  E  ( )xw Latent aggregation approach  xw  ( E  )xw MLM   xb E ( )   xb ICC X   xw ICC X  fpc  ( E  )mw  mw fpc 1 n (1 1 n  ICC X )  ( E   )mw mw (1  ICC ) X   xb E ( )   xb ICC X   xw ICC X  1 N 1 N (  (  n  n n 1 n 1  )(1 )(1  ICC ) X  ICC X ) 2-1-1 mediation   mb E ( )  2   M mb (  2  XM 2  X 2  XM 2  X )   mw fpc 1 n 2  M )  fpc 1 n 2  M 2  XM 2  X 2  XM 2  X )   mw 1 N )  1 N (  n  1 ) 2  M n 1 (  n  n ) 2  M 2   M mb (    mb E ( )  ( 2  M   ( E   )xw mw ( 2  M   xw )  [  XY  fpc 1 n  XY    xb E ( )  {  XY  1 N (  n  1 n )  XY  )xw (  E    xb E ( )mw  mw  ( E    mb E ( 1-1-1 mediation (  MY  fpc   XM MY )(  fpc 1 n 2  M  fpc 1 n 2  M [ 2  X  fpc 1 n 2  X  (  XM  fpc 2  M  fpc 1 n  XM ) ] / [  MY  1 N (  n  n [ 2  M 1  )   XM ][ MY  1 N (  n  1 n 1 N ) 2  M n  1 n )  XM ] } / (  ] 1 n 1 n  XM 2  M 2 ) ] { 2  X  1 N (  n  1 n ) 2  X   ( E   )mw mw [  XM  [ 2  M  1 N 1 N n  n  (  n (  n )  XM 2 ] } 1 ) 2  M ] 1 )  [  MY  fpc 1 n  MY    mb E ( )  {  MY  1 N (  n  1 n )  MY  (  XY  fpc   XM )( XY  fpc 1 n 2  X  fpc 1 n 2  X 1 n  XM ) ] / [  XY  1 N (  n  n [ 2  X 1  )   XM ][ XY  1 N (  n  1 n 1 N (  n  1 n )  XM ] } / ) 2  X ] [ 2  M  fpc 1 n 2  M  (  XM  fpc 2  X  fpc  E   )w ( w 1 n 1 n  XM 2  X 2 ) ] { 2  M  1 N (  n  1 n ) 2  M   ( E   )w w [  XM  [ 2  X  1 N 1 N (  (  n n n  n  )  XM 2 ] } 1 ) 2  X ] 1  (  b E )      w 2 b X  fpc 2  X  fpc 1 n 1 n 2  X  (  b E )  1 N (    w 2 b X  2  X  1 N ) 2  X 1 n  ) 2  X n n  1 (  n 2  X 22 In fact, when an appropriate correction of the sampling error with the within-group fpc was used to estimate the variance-covariance matrices, SPW, and SB, the OLS estimators of the between effects were the same in the manifest and latent aggregation approaches. By using the within-group fpc to the original predictors (rather than the manifest group means and individual deviations), the adjusted between and within variance-covariance matrices from the MOM are:  _  W fpc  n  1  fpc n S PW   B fpc _  S B     n    n 1  fpc      fpc S PW / n    (32) (33) The adjusted SPW and SB are in equations (34) and (35), which can be used in the MUML estimation for the new approach with within-group fpc.     S PW fpc _  n  1  fpc n S PW S B fpc _  S B  1 n     n fpc   1    fpc S PW   (34) (35) 23 Chapter 5 The Performances of the Manifest, Latent, and New Aggregation Approaches under Different Within-group Sampling Ratios 5.1 Simulation Design To assess the performances of manifest aggregation approach (i.e., did not correct sampling error), latent aggregation approach (i.e., overcorrected sampling error), and new approach with within-group fpc in the decomposition of between and within effects under a two- stage cluster sampling design, a Monte Carlo simulation study was conducted to explore the properties of their between and within effect estimators in the MLM, 2-1-1 mediation, and 1-1-1 mediation models. The parameters of research interests in the current study were the between and within effects of the decomposed predictors and/or mediators. In the current study, to resemble the data structure typically found in education research, an extremely large number of groups with small to moderate group sizes was assumed in the population. A two-stage cluster sampling design with equal probability of selection across groups and individuals was assumed to be used for data collection. The conditions manipulated were balanced or unbalanced design (BAL), average group size in the population (N, 20 and 100), ICC of the predictor X or mediator M (ICCX/ICCM, .05 and .25), the ratio of between to within effects of the predictor X and/or mediator M (RX/RM, .10 and 10), number of groups in the sample (g, 50 and 200), and within-group sampling ratio (r, .1, .3, .5, .7, and .9). 5.1.1 Population The average group size (N) in the population varied at 20 and 100. In the balanced case, all groups were of N individuals; in the unbalanced case, half of the groups were of individuals, and the other half of the groups were of individuals. 24 32N12N 5.1.2 Population model The MLM, 2-1-1 mediation, and 1-1-1 mediation models were considered in this simulation. The ICC of predictor X or mediator M, and the ratio of between to within effects in the population models were manipulated in the simulation. ICC. In the MSEM, the ICCs of the predictors with both between and within effects were of importance (Hsu, Lin, Kwok, Acosta, & Willson, 2016; Kim et al., 2012; Lachowicz, Sterba, & Preacher, 2015; Muthén & Satorra, 1995). A low proportion of between-level variability (i.e., ICC < .05) could lead to the lack of model convergence (Lachowicz et al., 2015). An ICC of .10 was also considered as low, which might result in a large RMSE (Lüedtke et al., 2011) and an extremely negative bias (Lüedtke et al., 2008; Preacher et al., 2011) with a small group size in the latent aggregation approach. The ICC of the decomposed predictor or mediator was simulated in previous studies. In a two-level random intercept model with only one predictor, Muthén (1994) set the ICC of the predictor as .40, and the ICC of the residual as .05, .10, and .20, which resulted in the ICC of the outcome varying from .31 to .73. Lüedtke et al. (2008) and Lüedtke et al. (2011) thought the ICCs rarely showed values greater than .30 in education and organizational research, and set the ICCs of the decomposed predictors as .05, .10, .20, and .30 in their studies. To compare the performances of multilevel modeling and MSEM using the manifest aggregation approach, Pham (2018) set the ICCs of the mediator and outcome in a 2-1-1 mediation model as .05, .20, and .40. In the multilevel CFA, ICCs of the indicators or factors were also manipulated in previous studies. Hox and Maas (2001) argued that most ICCs in education research were below .20, but the ICCs of group characteristics in family research might be a little higher. In their simulation of a two-level CFA, ICCs of the factors were set as .20 and .33. In a two-level CFA, Hox et al. 25 (2010) set the ICCs of indicators as .05 and .15, and Kim et al. (2012) set the ICCs of their indicators as .09, .20, and .33 in their simulations. Hsu et al. (2016) directly specified the variances of the within and between factors in a two-level CFA, which resulted in the observed ICCs of indicators between .36 and .50. In these simulation studies, the ICC generally ranged from .05 to .50. The ICCs found in the empirical studies might also provide some ideas for the simulation design in the current study. An example of large ICCs of the predictors and mediators were found in the study conducted by Lachowicz et al. (2015). Lachowicz et al. (2015) demonstrated the application of MSEM to the partially nested data using the dataset from Nohe, Michaelis, Menges, Zhang, and Sonntag (2013). In their study, ICCs of the predictor (team leader-rated change-promising behavior), mediator (team member-rated perceived charisma), and outcome (team member-rated commitment to change) were .51, .59, and .20. Lüdtke, Trautwein, Kunter, and Baumert (2006) provided examples of low ICCs of the decomposed predictors in education research. Lüdtke et al. (2006) examined students’ ratings of mathematics instruction using the German national extension dataset of TIMSS 1995. They found the ICCs of rule clarity and pacing were .10 and .06. For the empirical part in this dissertation, SES, OTL, and mathematics performance were used with the data from PISA 2012. In the US dataset, the ICCs of these three variables were .28, .13, and .24, which were moderate. High and low ICCs could be found in the US dataset in PISA 2012 as well. For example, the ICC of race (1= “White”, 0= “Minority”) was .41, and the ICC of gender (1= “girl”, 0= “boy”) was .01. Based on these studies, the ICC was set as .05 and .25 for X in the MLM model and 1-1-1 mediation model in the current study, and the ICC was set as .05 and .25 for M in the 2-1-1 26 mediation model in the simulation. The ICC of Y was equal to .25 across all conditions in the three models, and the ICC of M in the 1-1-1 mediation model ranged from .20 to .25. Ratio of between to within effects. As discussed in the literature review, when the between and within effects were exactly the same, the between effect estimators in the manifest and latent aggregation approaches were unbiased. The research interest in contextual models focused on the between effects which were different from the within effects. In the current study, the within effects were fixed at the certain values in the simulation, with the ratio of between to within effects of X (RX) and M (RM) being varied at .10 and 10. Population model. In the MLM model, µY=µX=0, β00=0, βxw=.2, βxb=RXβxw, and Var(Xij)=1. In the 2-1-1 mediation model, µY=µX=µM=0, β00=0, βmw=.1, βmb=RMβmw, βxb=.2, αb=.2, and Var(Xj)=1. In the 1-1-1 mediation model, µY=µX=µM=0, β00=0, βmw=.05, βmb=RMβmw, βxw=.1, βxb=RXβxw, αw=.2, αb=RXαw, and Var(Xij)=1. The individual-level and group-level error terms were assumed to follow multivariate normal distributions in the three models. Considering the ICCs found in the previous simulation and empirical studies, the variances of individual-level and group-level error terms were set at different values for different RX/RM and ICCX/ICCM in the three models. It guaranteed that the ICCY was equal to .25 in the three models, and the ICCM was between .20 and .25 in the 1-1-1 mediation model across different conditions. Please see Table 2 for the distributions of the error terms in the three models. 27 Table 2. The population distributions of error terms in the MLM, 2-1-1 mediation, and 1-1-1 mediation models RX/RM ICCX/ICCM 1-1-1 MLM 2-1-1 0.05 0.25 0.05 0.25 0.1 10 28 001.35~,004jijMVN0001~0,01.360004jjijMVN0001001.34~,000400004jjijijMVN001.34~,004jijMVN0001~0,01.300004jjijMVN0001001.34~,000400004jjijijMVN001.15~,004jijMVN0001~0,00.240004jjijMVN0001000.89~,000400004jjijijMVN000.34~,004jijMVN0001~0,00.180004jjijMVN0001000.09~,000400004jjijijMVN 5.1.3 Data generation In the MLM model, group components were generated from , and Nj were generated from for each j. The mean of Nj UXij was reset to 0 by using a scale parameter to each UXij in group j, which guaranteed the mean of UXij in group j equal to 0. In the 2-1-1 mediation model, group components were generated from , and Nj were generated from for each j. The mean of Nj UMij was reset to 0 by using a scale parameter to each UXij in group j, which guaranteed the mean of UMij in group j equal to 0. In the 1-1-1 mediation model, group components were generated from , and Nj were generated from for each j. The mean of Nj UXij was reset to 0 by 29 0XjjU00,0()0XjICCMVNVarXijijU10,0()0XijICCMVNVar00XjjjU00010,0()000()jjMVNVarVarMijijU11.040,00()MMijICCICCMVNVar00XjjjU0000,0()000()XjjICCMVNVarVarXijijijU010,0()000()XijijICCMVNVarVar using a scale parameter to each UXij in group j, which guaranteed the mean of UXij in group j equal to 0. 5.1.4 Sample Number of groups. In the multilevel analysis, a sufficient number of groups was needed for the admissible solutions and asymptotic properties of the group-level estimators (Kim et al., 2012). In a multilevel factor analysis with MUML, 50 groups were considered as a “small number of groups” under the conditions with a low ICC, and at least 100 groups were suggested as sufficient for the model test and confidence interval (CI) estimates (Hox & Maas, 2001). Hox et al. (2010) indicated a moderate size of 50 was sufficient if only the factor loadings were of interest. Based on these results, the number of sampled groups (g) was set as 50 and 200 in this study. Groups were randomly drawn with equal probability of selection from an infinite population of groups. Within-group sampling ratio. The latent aggregation approach showed an unacceptable bias when the group size was small (e.g., 5), and its efficiency increased with the increase of group size. For a small bias, a group size of 20 was recommended (Preacher et al., 2011). To compare the new approach with within-group fpc to the manifest and latent aggregation approaches, the within-group sampling ratio was manipulated in this study. The within-group sampling ratio (r) was moderate and ranged from .1, .3, .5, .7, to .9. For jth group, nj individuals were randomly drawn from the group of Nj, with nj equal to the product of Nj and within-group sampling ratio r. 5.1.5 Estimation method The simulation was conducted under 2×2×2×2×2×5=160 conditions for each model. Under each condition, the manifest and latent aggregation approaches, as well as the new 30 approach with within-group fpc were applied. The ML was used for the manifest and latent aggregation approaches, and the MUML was used for the new approach. Under each condition, 100 replications were conducted. 5.1.6 Evaluation criteria The estimators to be evaluated were the between and within effect estimators of the decomposed predictor and/or mediator in each model. The performances of the between and within effect estimators from the three analysis approaches were evaluated in terms of model convergence, accuracy in parameter estimate, variability of estimator, and accuracy of standard error. The model convergence rate across replications was used to evaluate the model convergence for each analysis approach under each simulation condition. The accuracy of estimator was evaluated by relative bias and absolute relative bias. Relative bias is the average difference between the estimate and population parameter in relative to the population parameter over replications. Absolute relative bias is the average absolute difference between the estimate and population parameter in relative to the population parameter over replications. RMSE was used to evaluate the variability of estimator, which is the square root of the mean square difference between the estimate and parameter over replications. The observed coverage rate reflects the accuracy of standard error in each analysis approach. It is the proportion of times in which the true parameter is in the estimated 95% CI. 31 5.2 Results To evaluate the performances of the three analysis approaches in the decomposition of between and within effects, the model convergence rate, and the relative bias, absolute relative bias, RMSE, and observed coverage rate for the between and within effects were firstly obtained across 100 replications under each simulation condition for each analysis approach. The current study focused on the between and within effects of the decomposed predictors and/or mediators in the three models. The results were summarized and presented separately for the between and within effects in each model. Table 34-38 in the Appendix C provided detailed results for these between and within effect estimates in each model under different conditions. The ANOVA was conducted to examine the contributions of the seven design factors: 1) analysis approach, 2) between-to-within-effect ratio of predictor X and/or mediator M (RX/RM), 3) ICC of predictor X or mediator M (ICCX/ICCM), 4) number of groups in the sample (g), 5) balanced or unbalanced design (BAL), 6) average group size in the population (N), and 7) within- group sampling ratio (r), in explaining the variances of model convergence rate, and the variances of relative bias, absolute relative bias, RMSE, and coverage rate for the between and within effects. With only one observation in each cell in the ANOVA, the seven-way interaction term could not be distinguished from the error term. All main effects and other interaction effects were estimated in the ANOVA, and their F statistics and effect sizes (η2) were calculated. 5.2.1 Convergence rate The three analysis approaches generally showed good model convergence rates, which were close to 100%, for the MLM, 2-1-1 mediation, and 1-1-1 mediation models across simulation conditions. For the manifest aggregation approach, the convergence rates were 100% across all conditions for the three models. As expected, the model convergence rates dropped 32 down from the MLM model, 2-1-1 mediation model, to the 1-1-1 mediation model for the latent aggregation approach and new approach, given the complexities of the three models. For the latent aggregation approach, the convergence rate was 100% across all conditions for the MLM model. Its model convergence rate ranged from 89% to 100% for the 2-1-1 mediation model (M=99.19%, SD=2.01%), and from 83% to 100% for the 1-1-1 mediation model (M=98.67%, SD=2.77%). For the new approach with within-group fpc, the convergence rate ranged from 97% to 100% (M=99.88%, SD=0.45%) for the MLM model, from 87% to 100% (M=99.45%, SD=1.86%) for the 2-1-1 mediation model, and from 88% to 100% (M=99.42%, SD=1.90%) for the 1-1-1 mediation model. The non-convergence problems with the latent aggregation approach and new approach were caused by the non-positive definite estimated between variance-covariance matrices, in which the sampling errors in the aggregation were moved out either without or with fpc. The between variance-covariance matrices in the manifest aggregation approach were estimated using the raw group means, which provided positive definite variance-covariance estimates across all conditions. The contributions of these design factors on model convergence rates were examined using ANOVA. The results are illustrated in Table 3. In general, the three analysis approaches showed similar convergence rates for the MLM, 2-1-1 mediation, and 1-1-1 mediation models. The main effect of analysis approach only explained a small proportion of variance in the model convergence rate for each model. For the MLM model, the factors which were of medium effect sizes (η2s > .059) in explaining the variance in the convergence rate were: 1) the interaction between analysis approach and sampling ratio, 2) the interaction between analysis approach, sampling ratio, and ICCX, and 3) the interaction between analysis approach, group size, and 33 sampling ratio. For the 2-1-1 mediation model, 1) the main effect of ICCM, 2) the interaction effect of analysis approach and sampling ratio, and 3) the interaction effect of analysis approach, sampling ratio, and ICCM, explained medium proportions (η2s > .059) of variance in the model convergence rates. For the 1-1-1 mediation model, 1) the main effect of analysis approach, 2) the main effect of group size, and 3) the interaction effect of analysis approach and group size, were of medium effect sizes (η2s > .059) in explaining the variance of model convergence rate. Table 3. Proportion of variance in model convergence rate explained by the design factors Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r Other interactions Approach × ICCX/ICCM × r Approach × N× r MLM 2-1-1 1-1-1 mediation mediation 0.044 0.000 0.014 0.005 0.000 0.014 0.059 0.000 0.027 0.010 0.000 0.027 0.117 0.066 0.066 0.044 0.000 0.062 0.012 0.000 0.045 0.028 0.015 0.036 0.007 0.000 0.033 0.083 0.062 0.031 0.074 0.018 0.028 0.018 0.001 0.078 0.055 0.020 0.026 0.009 0.006 0.061 0.049 0.025 0.036 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The convergence rates are summarized by analysis approach, ICCX/ICCM, and within- group sampling ratio for the three models in Table 4. 34 Table 4. Model convergence rate by analysis approach, ICCX/ICCM, and within-group sampling ratio r M MLM L 2-1-1 mediation 1-1-1 mediation N M L N M L N ICCX/ICCM =0.05 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.989 0.988 0.986 0.986 0.974 0.1 0.3 0.5 0.7 0.9 0.983 0.1 1.000 0.3 1.000 0.5 1.000 0.7 0.9 1.000 Note. M= manifest aggregation approach; L=latent aggregation approach; N=the new approach with within-group fpc; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; r = within-group sampling ratio. 0.999 0.999 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.958 0.991 1.000 1.000 1.000 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.979 0.987 0.976 0.976 0.977 0.979 0.999 0.996 0.999 0.998 0.991 0.998 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 0.964 0.994 1.000 1.000 1.000 ICCX/ICCM =0.25 The manifest aggregation approach converged across all conditions for the three models. For the latent aggregation approach and new approach, the convergence rates in the three models improved slightly with an increasing ICCX/ICCM and within-group sampling ratio. For the three analysis approaches, the convergence rates were 100% in most conditions and across the three models. 5.2.2 Bias As different values were used for different between and within effect parameters in the three models, relative bias and absolute relative bias were used to evaluate the accuracy in the between and within effect estimates from the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc. Within effects. As predicted from the mathematical derivations in Table 1, the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc provided similar within effect estimates. The means and standard deviations of the relative bias 35 and absolute relative bias in the within effect estimates across the 160 simulation conditions from the three analysis approaches are shown in Table 5. For the three analysis approaches, the relative bias and absolute relative bias in the within effect estimates increased with model complexity. The relative biases in the within effects from the new approach with within-group fpc were closer to those from the manifest aggregation approach, and both of them were generally of smaller degrees than those from the latent aggregation approach. The new approach produced the largest average absolute relative bias for these within effects among the three analysis approaches. Table 5. Relative bias and absolute relative bias in within effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models Manifest Latent New MLM βxw 2-1-1 mediation βmw M SD M SD βxw M Relative bias SD 1-1-1 mediation βmw αw M SD M SD 0.004 0.014 0.004 0.048 0.059 0.048 0.005 0.027 0.005 0.040 0.077 0.039 -0.010 0.013 -0.010 0.093 0.103 0.100 -0.007 0.006 -0.007 0.089 0.099 0.095 -0.004 0.000 -0.002 0.057 0.053 0.057 Absolute relative bias 0.314 0.305 0.335 0.268 0.248 0.271 0.227 0.223 0.240 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. 0.633 0.614 0.672 0.576 0.566 0.617 0.227 0.215 0.232 0.552 0.501 0.556 0.499 0.475 0.513 0.315 0.307 0.336 0.268 0.245 0.273 The sources of relative bias and absolute relative bias in within effect estimates in the three models were explored through seven-way ANOVA. The effect sizes for the main and interaction effects are in Table 6 and Table 7. The factors which explained medium proportions (η2s > .059) of variance in the relative bias in within effect in the MLM model were: 1) the two- way interaction between group size and within-group sampling ratio, 2) the three-way interaction of between-to-within-effect ratio, number of groups in the sample, and balanced or unbalanced design, and 3) the three-way interaction of between-to-within-effect ratio, number of groups in the sample, and group size. The two-way interaction of between-to-within-effect ratio and group 36 size accounted for a medium proportion of variance (η2 = .066) in relative bias in the within effect of mediator M in the 2-1-1 mediation model. In the 1-1-1 mediation model, there was no main or interaction effect of medium or large effect sizes for the relative bias in the within effect of predictor X or mediator M on outcome Y. For the relative bias in the within effect of X on M in the 1-1-1 mediation model, the factors of medium effects (η2s > .059) were: 1) the interaction between ICCX, between-to-within-effect ratio, and number of groups in the sample, and 2) the interaction between ICCX, between-to-within-effect ratio, number of groups in the sample, and within-group sampling ratio. In summary, the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc only showed small or negligible differences in relative biases in within effect estimates in the three models. There was no significant or consistent pattern which largely explained the variance in the relative bias in any within effect estimate. 37 Table 6. Proportion of variance of relative bias in within effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r RX/RM × N N × r Other interactions ICCX/ICCM × RX/RM × g RX/RM × g × BAL RX/RM × g × N ICCX/ICCM × RX/RM × g × r βxw 0.007 0.010 0.008 0.006 0.018 0.020 0.034 0.013 0.002 0.001 0.000 0.005 0.008 0.010 0.065 0.004 0.085 0.072 0.005 βmw βxw 0.012 0.008 0.002 0.001 0.005 0.017 0.003 0.012 0.002 0.001 0.000 0.008 0.010 0.029 0.025 0.010 0.014 0.012 0.004 βmw 0.004 0.018 0.001 0.005 0.007 0.002 0.009 0.004 0.001 0.003 0.000 0.002 0.010 0.006 0.001 0.004 0.005 0.003 0.007 αw 0.001 0.000 0.006 0.002 0.035 0.001 0.026 0.002 0.000 0.001 0.000 0.001 0.002 0.001 0.016 0.073 0.007 0.005 0.094 0.035 0.047 0.018 0.014 0.010 0.017 0.012 0.039 0.002 0.002 0.000 0.019 0.022 0.066 0.011 0.002 0.001 0.020 0.007 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. 38 2-1-1 mediation Table 7. Proportion of variance of absolute relative bias in within effect estimates explained by the design factors Main effects Approach RX/RM ICCX/ICCM g BAL N r 1-1-1 mediation βxw βxw βmw βmw αw 0.002 0.000 0.009 0.157 0.008 0.181 0.433 0.000 0.000 0.000 0.002 0.000 0.001 0.003 0.091 0.002 0.000 0.000 0.160 0.009 0.181 0.441 0.000 0.000 0.000 0.001 0.000 0.001 0.000 0.092 MLM 0.002 0.000 0.006 0.162 0.006 0.193 0.433 0.000 0.000 0.000 0.001 0.000 0.001 0.004 0.083 0.001 0.000 0.189 0.107 0.004 0.129 0.321 0.000 0.000 0.000 0.001 0.000 0.000 0.061 0.068 0.002 0.000 0.004 0.162 0.006 0.177 0.444 0.000 0.000 0.000 0.001 0.000 0.001 0.001 0.092 Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r ICCX/ICCM × r N × r Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. Similar results were found for the absolute relative biases in these within effect estimates. The analysis approach accounted for less than 1% of the variances in absolute relative biases in these within effect estimates. The differences in the absolute relative biases among the three analysis approaches by other design factors (i.e., interaction terms) were also ignorable for these within effect estimates. Different from the results for relative biases, some patterns were apparent in explaining the variances in absolute relative biases in these within effect estimates. First, the main effects of number of groups in the sample, group size, and within-group sampling ratio were of medium or 39 large effect sizes (η2s > .059) in explaining the variances in absolute relative biases for all within effects. Among the three design factors, the within-group sampling ratio contributed to the largest proportions of variances in absolute relative biases in these within effect estimates, which ranged from 32% to 44%. The interaction between group size and within-group sampling ratio also explained medium proportions of variances (η2s > .059) in absolute relative biases for all within effects. Table 8. Absolute relative bias in within effect estimates by number of groups, group size, and within-group sampling ratio r 0.1 0.3 0.5 0.7 0.9 MLM βxw 2-1-1 mediation 1-1-1 mediation βmw βxw βmw αw N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 g=50 1.211 0.586 0.433 0.356 0.306 0.512 0.283 0.217 0.176 0.154 0.857 0.429 0.288 0.250 0.221 0.369 0.209 0.157 0.137 0.116 2.508 1.150 0.832 0.700 0.610 1.038 0.585 0.439 0.367 0.328 2.316 1.042 0.777 0.628 0.565 0.958 0.540 0.387 0.338 0.294 1.225 0.547 0.414 0.348 0.308 0.519 0.292 0.228 0.183 0.166 g=200 0.608 0.281 0.216 0.177 0.156 0.261 0.141 0.110 0.094 0.083 0.474 0.205 0.150 0.122 0.109 0.1 0.3 0.5 0.7 0.9 Note. g = number of groups in the sample; N = average group size; r = within-group sampling ratio. 1.228 0.559 0.413 0.345 0.305 0.189 0.106 0.082 0.072 0.060 0.528 0.288 0.221 0.190 0.161 1.125 0.528 0.396 0.316 0.277 0.468 0.261 0.197 0.166 0.150 0.607 0.298 0.217 0.183 0.160 0.250 0.146 0.114 0.096 0.086 As shown in Table 8 and Figure 1, the absolute relative biases in these within effect estimates decreased with a larger number of groups, a larger group size, and a larger within- group sampling ratio. In addition, the ICCM, and the interaction between ICCM and within-group sampling ratio explained medium or large amounts of variance (η2s > .059) in the absolute relative bias for the within effect in the 2-1-1 mediation model. 40 Figure 1. Absolute relative bias in within effect estimates by number of groups, group size, and within-group sampling ratio Note. g = number of groups in the sample; N = average group size; r = within-group sampling ratio. Between effects. Consistent with the mathematical derivations in Table 1, large differences in the between effect estimates were found among the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc. The means and standard deviations of relative bias and absolute relative bias in between effect estimates are calculated and presented in Table 9 for each analysis approach. 41 Table 9. Relative bias and absolute relative bias in between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models Manifest Latent New MLM βxb 2-1-1 mediation βmb M SD M SD βxb M Relative bias SD 1-1-1 mediation βmb αb M SD M SD 0.940 2.636 -6.148 20.523 -0.825 6.328 1.011 2.279 -9.375 32.569 -0.084 2.735 1.565 -5.694 1.993 3.810 32.916 25.231 1.250 -0.046 0.359 2.659 6.807 4.818 0.979 2.285 -4.646 16.170 -0.699 6.505 Absolute relative bias 8.879 7.531 5.248 14.958 36.412 70.614 27.857 58.108 82.063 16.222 30.054 8.954 13.431 34.988 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. 5.010 17.680 10.105 11.179 7.883 162.208 20.742 29.729 28.633 52.891 68.164 15.879 24.427 14.166 26.890 6.850 For most between effect estimates in the three models, the new approach showed the smallest degrees of relative biases among the three analysis approaches. Different from previous studies on latent aggregation approach, which assumed the within-group population was infinite (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011), the current study simulated moderate to large within-group sampling ratios with small to moderate group sizes in the population. The simulation condition in the current study was more in favor of the manifest aggregation approach compared to the latent aggregation approach. For example, when the group size was 20 and within-group sampling ratio was .90, the manifest aggregation approach was assumed to perform better than the latent aggregation approach according to the previous studies (Lüedtke et al., 2011; Lüedtke et al., 2008; Marsh et al., 2012; Marsh et al., 2009; Preacher et al., 2010). This was reflected in the current results, in which the degrees of relative biases in between effect estimates from the manifest aggregation approach were generally smaller than those from the latent aggregation approach. On average, manifest aggregation approach overestimated the between effects, and latent aggregation approach underestimated the between effects. As the new approach considered the within-group finite population selection for the sampling error 42 correction, its relative biases in between effect estimates were between the statistics from the manifest and latent aggregation approaches. The differences among the three analysis approaches were more obvious in the absolute relative biases in between effect estimates. With smaller absolute relative biases for between effect estimates, the manifest aggregation approach outperformed the latent aggregation approach and the new approach with within-group fpc. The latent aggregation approach produced the largest absolute relative biases in between effect estimates. The absolute relative biases in between effect estimates from the new approach were in the middle. The sources of relative biases and absolute relative biases in between effect estimates in the three models were explored using ANOVA. The effect sizes of the design factors are in Table 10 and Table 12. For the relative biases in between effect estimates, only the interaction between analysis approach and between-to-within-effect ratio were of medium effect sizes (η2s > .059). The cell means of relative biases in between effect estimates by analysis approach and between-to-within-effect ratio are shown in Table 11. 43 Table 10. Proportion of variance of relative bias in between effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r βxb 0.055 0.028 0.024 0.000 0.001 0.010 0.003 0.071 0.039 0.001 0.003 0.029 0.005 βmb βxb 0.021 0.002 0.004 0.002 0.001 0.002 0.004 0.031 0.017 0.002 0.001 0.005 0.004 βmb 0.011 0.018 0.020 0.000 0.014 0.000 0.007 0.011 0.002 0.003 0.006 0.005 0.018 αb 0.052 0.022 0.018 0.000 0.004 0.014 0.009 0.068 0.031 0.001 0.003 0.022 0.023 0.058 0.020 0.019 0.000 0.003 0.015 0.014 0.061 0.047 0.002 0.006 0.036 0.017 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The new approach with within-group fpc generally produced the smallest degrees of relative biases among the three analysis approaches under different between-to-within-effect ratios for most between effect estimates. The relative biases in between effect estimates by between-to-within-effect ratio and analysis approach are also plotted in Figure 2. The new approach with within-group fpc lined around zero and fell between the lines of manifest aggregation approach and latent aggregation approach in terms of the relative biases in between effect estimates. The manifest aggregation approach tended to overestimate the between effects, while the latent aggregation approach tended to underestimated the between effects. The only exception was found for the between effect of mediator M on Y in the 1-1-1 mediation model, 44 when the between-to-within-effect ratio was .10. Under this condition, the latent aggregation approach outperformed the manifest aggregation approach and the new approach in terms of relative bias. Table 11. Relative bias in between effect estimates by analysis approach and between-to-within- effect ratio MLM βxb 2-1-1 mediation 1-1-1 mediation βmb βxb βmb αb RX =0.1 2.135 -13.007 -1.617 RX =10 -0.254 0.710 -0.033 RM =0.1 2.276 -18.908 -0.157 RM =10 -0.254 0.157 -0.012 RX =0.1 3.302 -12.934 3.867 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio. RX =10 -0.171 1.546 0.119 RX =0.1 2.665 0.152 0.813 RX =10 -0.165 -0.244 -0.094 RX =0.1 2.211 -9.879 -1.357 RX =10 -0.253 0.586 -0.041 For the five between effects, the degrees of relative biases dropped down when the between-to-within-effect ratio went up from .10 to 10, no matter which analysis approach was used. When the between-to-within-effect ratio was 10, the three analysis approaches provided similar relative biases in these between effect estimates. As seen in the equations in Table 1, the biases in between effect estimates from the manifest and latent aggregation approaches came from the additional parts containing within effects. When the between-to-within-effect ratio was large, i.e., RX/RM =10 in the current study, the additional parts containing within effects were relatively small compared to the between effects. The additional parts slightly affected the relative biases in between effect estimates under this condition. 45 Figure 2. Relative bias in between effect estimates by analysis approach and between-to-within- effect ratio Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio. 46 Table 12. Proportion of variance of absolute relative bias in between effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r ICCX/ICCM × RX/RM RX/RM × g Other interactions Approach × ICCX/ICCM × RX/RM βxb 0.069 0.180 0.081 0.017 0.000 0.029 0.004 0.066 0.057 0.003 0.001 0.037 0.010 0.076 0.016 0.054 βmb βxb 0.071 0.165 0.075 0.016 0.000 0.026 0.005 0.067 0.058 0.004 0.001 0.038 0.011 0.071 0.016 0.055 βxb 0.034 0.422 0.006 0.065 0.001 0.018 0.051 0.033 0.004 0.006 0.000 0.015 0.026 0.005 0.064 0.004 αb 0.065 0.204 0.084 0.013 0.000 0.029 0.005 0.062 0.052 0.001 0.000 0.032 0.013 0.080 0.012 0.050 0.077 0.146 0.039 0.006 0.001 0.033 0.001 0.077 0.060 0.001 0.003 0.050 0.007 0.038 0.006 0.060 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. For the absolute relative biases in between effects (in Table 12) , the main effects of analysis approach, between-to-within-effect ratio, and ICCX/ICCM, and their two-way interaction effects were of medium to large effect sizes (η2s > .059) for some between effects. The largest variances in absolute relative biases in between effect estimates were explained by the between- to-within-effect ratio (η2s > .138). As shown in Table 13, when the between-to-within-effect ratio was .10, all three analysis approaches showed large absolute relative biases in between effect estimates across different levels of the ICC of predictor X or mediator M. 47 Table 13. Absolute relative bias in between effect estimates by analysis approach, between-to- within-effect ratio, and ICCX/ICCM RX =10 RM =0.1 RM =10 2-1-1 mediation 1-1-1 mediation RX =0.1 RX =0.1 RX =0.1 RX =0.1 RX =10 MLM βxb RX =10 RX =10 αb βmb βxb βmb Manifest Latent New 19.007 127.403 49.839 0.428 9.237 1.676 94.292 0.529 21.551 0.373 37.370 0.319 287.777 0.255 106.638 0.500 20.300 4.157 48.294 1.090 34.333 0.255 0.763 0.326 17.036 98.620 43.453 0.422 1.291 0.461 ICCX/ICCM = 0.05 ICCX/ICCM = 0.25 10.523 16.422 14.395 0.165 10.270 0.148 16.680 0.127 13.897 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator. 0.186 19.697 0.369 33.694 0.352 28.635 21.778 35.949 31.874 9.756 14.434 12.609 0.159 0.136 0.112 0.169 0.216 0.222 0.184 0.187 0.141 The absolute relative biases in between effect estimates from the three analysis approaches at different levels of between-to-within-effect ratio and ICCX/ICCM are presented in Table 13 and Figure 3. When the between-to-within-effect ratio was .10, manifest aggregation approach provided the smallest absolute relative biases in these between effect estimates across different levels of ICCX/ICCM. The absolute relative bias from the new approach was between the ones from the manifest and latent aggregation approaches. When the between-to-within- effect ratio was 10, the differences in absolute relative biases in between effect estimates among the three analysis approaches were smaller. For some between effect estimates, the new approach with within-group fpc outperformed the other two approaches in terms of absolute relative bias when the between-to-within-effect ratio was 10. Across all levels of between-to-within-effect ratio and ICC, the latent aggregation approach performed the worst in absolute relative biases in between effect estimates. The new approach with within-group fpc provided similar or smaller absolute relative biases in these between effect estimates compared to the manifest aggregation approach. The interaction effects of ICCX/ICCM and between-to-within-effect ratio on the absolute relative biases in the between effect estimates are plotted in Figure 4. The degrees of absolute 48 relative biases in between effect estimates were inversely related to the ICCX/ICCM and between- to-within-effect ratio for the three analysis approaches. With a larger ICCX/ICCM and between- to-within-effect ratio, all three analysis approaches showed smaller absolute relative biases in these between effect estimates. The differences in the absolute relative biases by ICCX/ICCM was also inversely related to the between-to-within-effect ratio. Figure 3. Absolute relative bias in between effect estimates by analysis approach and between- to-within-effect ratio Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio. 49 Figure 4. Absolute relative bias in between effect estimates by between-to-within-effect ratio and ICCX/ICCM Note. RX/RM = between-to-within-effect ratio; ICC = intraclass correlation coefficient of the decomposed predictor or mediator. The relative biases and absolute relative biases in between effect estimates were scaled by the population values of between effects. The main effects of between-to-within-effect ratio, and interaction effects of analysis approach and between-to-within-effect ratio were of medium to large effect sizes (η2s > .059) in explaining the variances in the relative biases and absolute relative biases for the between effects. To understand how the between-to-within-effect ratio 50 influenced the accuracies of between effect estimation in the three analysis approaches, a further look was taken at the biases and absolute biases in between effect estimates. As shown in Table 14, the new approach showed the smallest degrees of biases in between effect estimates among the three analysis approaches, which was consistent with the results on relative biases. The sources of biases in between effect estimates were examined using ANOVA, and the effect sizes of the design factors are presented in Table 15. The factors which showed medium to large effects (η2s > .059) on the biases in these between effect estimates were: 1) the analysis approach, 2) the interaction between analysis approach and between-to- within-effect ratio, 3) the interaction between analysis approach, between-to-within-effect ratio, and ICCX/ICCM, and 4) the interaction between analysis approach, between-to-within-effect ratio, and group size. The biases in between effect estimates from the three analysis approaches are presented and plotted by different levels of between-to-within-effect ratio and ICCX/ICCM (or group size) in Table 16, Figure 5, and Figure 6. Table 14. Bias and absolute bias in between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models Manifest Latent New MLM βxb 2-1-1 mediation 1-1-1 mediation βmb βxb βmb αb M SD M SD M SD M SD M SD -0.233 0.580 -0.050 0.445 1.911 0.242 -0.116 -0.016 -0.007 0.222 0.374 0.102 -0.069 0.708 0.079 0.214 2.283 0.286 -0.034 -0.061 -0.022 0.067 0.172 0.045 -0.231 0.487 -0.055 0.448 1.427 0.300 Absolute bias Bias 0.047 0.202 0.127 0.360 1.514 0.665 0.444 1.631 0.649 0.364 2.102 0.771 0.182 0.391 0.180 0.187 0.544 0.192 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. 0.319 1.941 0.707 0.103 0.225 0.147 0.179 2.699 0.865 0.437 1.304 0.581 51 Table 15. Proportion of variance of bias in between effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r RX/RM × N RX/RM × r Other interactions Approach × ICCX/ICCM × RX/RM Approach × RX/RM × N βxb 0.085 0.023 0.010 0.000 0.000 0.005 0.024 0.160 0.052 0.001 0.000 0.029 0.009 0.014 0.029 0.102 0.062 βmb βxb 0.060 0.036 0.018 0.001 0.000 0.018 0.014 0.091 0.048 0.004 0.002 0.035 0.020 0.024 0.020 0.072 0.047 βxb 0.021 0.165 0.024 0.004 0.001 0.060 0.007 0.018 0.024 0.007 0.001 0.020 0.048 0.060 0.002 0.022 0.016 αb 0.108 0.019 0.005 0.000 0.000 0.004 0.026 0.192 0.057 0.001 0.001 0.031 0.003 0.013 0.042 0.104 0.060 0.036 0.001 0.021 0.000 0.001 0.019 0.011 0.238 0.006 0.000 0.006 0.006 0.027 0.002 0.078 0.087 0.047 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. 52 Table 16. Bias in between effect estimates by analysis approach, between-to-within-effect ratio, and ICCX/ICCM (or group size) Manifest Latent New Manifest Latent New Manifest Latent New MLM βxb 2-1-1 mediation 1-1-1 mediation βmb βxb βmb αb RX =0.1 RX =10 RM =0.1 RM =10 0.056 -0.493 -0.059 -0.732 2.652 -0.124 0.033 -0.363 0.001 -0.368 0.225 -0.031 0.029 -0.027 -0.005 -0.285 0.187 -0.010 0.013 -0.015 -0.004 -0.140 0.090 0.008 RX =10 0.047 -0.261 0.053 RX =0.1 ICCX/ICCM = 0.05 -0.306 2.889 0.113 ICCX/ICCM = 0.25 -0.036 0.203 0.126 0.019 0.003 0.025 RX =0.1 RX =10 RX =0.1 RX =10 0.009 -0.008 -0.004 -0.093 -0.196 -0.045 0.062 -0.364 -0.052 -0.739 2.099 -0.196 0.017 0.010 0.012 -0.072 -0.048 -0.050 0.027 -0.031 -0.003 -0.274 0.245 0.031 0.065 -0.443 -0.032 -0.672 2.353 -0.149 0.030 -0.340 0.002 -0.330 0.186 -0.038 0.041 -0.213 0.033 -0.213 2.723 0.179 0.018 -0.002 0.002 -0.121 -0.217 -0.076 0.057 -0.339 -0.050 -0.660 1.887 -0.154 N = 20 N = 100 0.021 -0.077 -0.033 -0.346 0.486 0.015 0.015 -0.038 -0.005 -0.353 0.457 -0.011 -0.178 0.129 0.015 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 0.025 -0.045 0.044 -0.130 0.369 0.060 -0.044 -0.026 -0.019 0.032 -0.056 -0.004 0.008 0.004 0.006 Across different levels of ICCX/ICCM and group size, the new approach with within- group fpc generally outperformed the manifest and latent aggregation approaches in terms of biases in between effect estimates in the three models. In opposite to the results on relative biases, the differences in biases in between effect estimates among the three analysis approaches were more pronounced when the between-to-within-effect ratio was 10. No matter which analysis approach was used, the degrees of biases in between effect estimates increased when the between-to-within-effect ratio went up from .10 to 10. The differences among the three analysis approaches in biases in between effect estimates became smaller with a larger ICCX/ICCM and a larger group size. 53 Figure 5. Bias in between effect estimates by between-to-within-effect ratio and ICCX/ICCM Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICC = intraclass correlation coefficient of the decomposed predictor or mediator. 54 Figure 6. Bias in between effect estimates by between-to-within-effect ratio and group size Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; N = average group size. 55 In Table 14, the manifest aggregation approach showed the smallest absolute biases for most between effect estimates, and the latent aggregation approach produced the largest absolute biases. These results were consistent with the ones on absolute relative biases in between effect estimates. A different story was found when exploring the resources of variance in absolute biases in between effect estimates. The main and interaction effects of between-to-within-effect ratio were no longer of medium or large effect sizes in explaining the variance in absolute biases in these between effect estimates. For the absolute biases in between effect estimates (in Table 17), the factors which accounted for medium to larger proportions of variance (η2s > .059) were: 1) analysis approach, 2) ICCX/ICCM, 3) group size, 4) the two-way interaction between analysis approach and ICCX/ICCM, 5) the two-way interaction between analysis approach and group size, and 6) the three-way interaction between analysis approach, ICCX/ICCM, and group size. The largest proportions of variances in these absolute biases in between effect estimates were explained by the analysis approach, ICCX/ICCM, and their interaction. 56 Table 17. Proportion of variance of absolute bias in between effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r Other interactions Approach × ICCX/ICCM × RX/RM Approach × ICCX/ICCM × N βxb 0.136 0.007 0.197 0.016 0.000 0.077 0.010 0.003 0.126 0.003 0.003 0.081 0.028 0.003 0.074 βmb βxb 0.152 0.005 0.151 0.011 0.000 0.066 0.005 0.006 0.129 0.002 0.002 0.085 0.029 0.006 0.076 βxb 0.115 0.001 0.070 0.081 0.001 0.108 0.104 0.007 0.051 0.008 0.001 0.059 0.031 0.018 0.033 αb 0.132 0.013 0.212 0.015 0.000 0.078 0.015 0.005 0.115 0.002 0.001 0.069 0.034 0.002 0.060 0.075 0.005 0.101 0.014 0.000 0.072 0.030 0.081 0.059 0.003 0.003 0.048 0.029 0.060 0.037 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The absolute biases in between effect estimates from the three analysis approaches by different levels of ICCX/ICCM and group size are presented in Table 18 and Figure 7. When ICCX/ICCM was .05, manifest aggregation approach provided the smallest absolute biases for most between effect estimates. The absolute biases from the new approach were between the ones from the manifest and latent aggregation approaches. When ICCX/ICCM was .25, the new approach with within-group fpc provided smaller or similar absolute biases in between effect estimates compared to those from the manifest aggregation approach. Across all levels of ICCX/ICCM and group size, the latent aggregation approach performed the worst in terms of 57 absolute biases in between effect estimates. The absolute biases in between effect estimates were inversely related to ICCX/ICCM and group size for all analysis approaches. In addition, with a larger ICCX/ICCM and a larger group size, the differences in absolute biases in between effect estimates among the three analysis approaches became smaller. When ICCX/ICCM was .25 and group size was 100, the differences among the three analysis approaches in absolute biases in between effect estimates became trivial. Table 18. Absolute bias in between effect estimates by analysis approach, ICCX/ICCM, and group size MLM βxb 2-1-1 mediation 1-1-1 mediation βmb βxb βmb αb N=20 N=100 N=20 N=100 N=20 N=100 N=20 N=100 N=20 N=100 0.688 4.751 1.262 0.548 1.148 0.792 0.279 0.997 0.287 0.186 0.265 0.184 0.129 0.470 0.206 0.100 0.153 0.129 0.659 3.537 1.111 0.526 1.018 0.679 Manifest Latent New 0.448 5.762 1.294 ICCX/ICCM = 0.05 0.426 1.272 0.862 ICCX/ICCM = 0.25 0.177 0.242 0.220 0.226 0.486 0.450 0.324 0.416 0.347 0.216 0.209 0.195 0.161 0.206 0.162 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 0.102 0.176 0.160 0.101 0.097 0.089 0.081 0.101 0.095 0.337 0.428 0.320 0.226 0.235 0.215 58 Figure 7. Absolute bias in between effect estimates by ICCX/ICCM and group size Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; ICC = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 59 In summary, the new approach with within-group fpc outperformed the manifest and latent aggregation approaches in terms of bias and relative bias in between effect estimates of the decomposed predictors and/or mediators. The differences in biases in between effect estimates among the three analysis approaches, dropped down with an increasing ICCX/ICCM and a decreasing between-to-within-effect ratio. The differences in relative biases in between effect estimates among the three analysis approaches decreased with a larger between-to-within-effect ratio. The three analysis approaches performed better in terms of estimator accuracy for the between effects when the ICCX/ICCM and group size were larger. Compared to the latent aggregation approach and new approach, the manifest aggregation approach provided smaller absolute biases and absolute relative biases for most between effect estimates. When the ICCX/ICCM was .25, the new approach with within-group fpc outperformed the manifest aggregation approach with smaller absolute biases for some between effects. When the between-to-within-effect ratio was 10, the new approach provided smaller absolute relative biases for some between effect estimates compared to the manifest aggregation approach. The differences in absolute biases among the three analysis approaches were less pronounced when the group size was larger and ICCX/ICCM was larger. The differences in absolute relative biases among the three analysis approaches were less pronounced with a larger between-to-within- effect ratio. The performances of the three analysis approaches got better with a larger ICCX/ICCM and a larger group size, in terms of absolute biases in between effect estimates. For the absolute relative biases in between effect estimates, the three analysis approaches performed better with a larger between-to-within-effect ratio and a larger ICCX/ICCM. 60 5.2.3 RMSE Within effects. As expected from the derivations in Table 1, the manifest aggregation approach, latent aggregation approach, and new approach with within-group fpc provided similar RMSE for the within effect estimates in the three models. The means and standard deviations of RMSE from the three analysis approaches are presented in Table 19. On average, the latent aggregation approach provided the smallest RMSE for the within effect estimates among the three analysis approaches, while the differences among the three analysis approaches in RMSE of within effect estimates were trivial. Table 19. RMSE of within effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models MLM βxw 2-1-1 mediation 1-1-1 mediation βmw βxw βmw αw M 0.078 0.076 0.084 SD 0.068 0.062 0.068 M 0.029 0.028 0.030 SD 0.029 0.028 0.030 M 0.079 0.077 0.084 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. SD 0.070 0.063 0.070 M 0.036 0.036 0.039 SD 0.031 0.030 0.032 M 0.079 0.077 0.084 SD 0.067 0.062 0.069 The sources of variances in RMSE for within effect estimates in the three models were explored using seven-way ANOVA. As shown in Table 20, the differences in RMSE of within effect estimates among the three analysis approaches were trivial (η2s < .01). Similar to the results for absolute relative biases in within effect estimates, the within-group sampling ratio explained large proportions of variances in RMSE for the within effect estimates in the three models, which ranged from 31% to 44%. The number of groups in the sample, group size, and the interaction between group size and within-group sampling ratio also contributed medium or large proportions to the variances in the RMSE of within effect estimates (η2s > .059). In other words, the sample size largely explained the variances in RMSE of within effect estimates. In addition, ICCM and the 61 interaction between ICCM and within-group sampling ratio, also showed medium effects (η2s > .059) on the RMSE of the within effect estimates in the 2-1-1 mediation model. Table 20. Proportion of variance in RMSE of within effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r ICCX/ICCM × r N × r βxw 0.002 0.000 0.006 0.162 0.007 0.191 0.428 0.000 0.000 0.000 0.001 0.000 0.001 0.004 0.084 βmw βxw 0.002 0.000 0.005 0.162 0.008 0.174 0.443 0.000 0.000 0.000 0.001 0.000 0.001 0.002 0.093 βmw 0.002 0.000 0.000 0.160 0.009 0.181 0.443 0.000 0.000 0.000 0.001 0.000 0.001 0.000 0.091 αw 0.002 0.000 0.008 0.158 0.007 0.180 0.433 0.000 0.000 0.000 0.002 0.000 0.001 0.002 0.093 0.001 0.000 0.184 0.107 0.004 0.128 0.312 0.000 0.000 0.000 0.001 0.000 0.000 0.063 0.068 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The mean RMSE by different numbers of groups in the sample, group sizes, and within- group sampling ratios are summarized in Table 21 and plotted in Figure 8. The larger within- group sampling ratio, the smaller RMSE of within effect estimates. The RMSE of within effect estimates was also inversely related to the group size and number of groups in the sample. In other words, with more groups in the sample (i.e., g), and a larger number of cases drawn from each group (i.e., N × r), a smaller RMSE of within effect estimates of the decomposed variables 62 would be observed. Relatively high RMSE for within effect estimates were presented when the group size was 20 and the within-group sampling ratio was .10. Table 21. RMSE of within effect estimates by number of groups, group size, and within-group sampling ratio r 0.1 0.3 0.5 0.7 0.9 MLM βxw 2-1-1 mediation 1-1-1 mediation βmw βxw βmw αw N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 g=50 0.307 0.146 0.109 0.088 0.076 0.127 0.070 0.055 0.044 0.039 0.111 0.054 0.036 0.032 0.028 0.046 0.026 0.020 0.017 0.015 0.317 0.145 0.104 0.087 0.075 0.131 0.073 0.055 0.046 0.041 0.146 0.065 0.049 0.039 0.035 0.060 0.034 0.024 0.021 0.018 0.310 0.137 0.104 0.087 0.076 0.130 0.073 0.057 0.046 0.041 g=200 0.150 0.070 0.054 0.044 0.039 0.065 0.035 0.027 0.023 0.021 0.058 0.026 0.019 0.015 0.014 0.1 0.3 0.5 0.7 0.9 Note. g = number of groups in the sample; N = average group size; r = within-group sampling ratio. 0.154 0.071 0.051 0.043 0.038 0.024 0.013 0.010 0.009 0.008 0.066 0.036 0.028 0.023 0.020 0.070 0.034 0.025 0.020 0.017 0.029 0.016 0.012 0.010 0.009 0.152 0.074 0.054 0.045 0.040 0.062 0.036 0.028 0.024 0.022 63 Figure 8. RMSE of within effect estimates by number of groups, group size, and within-group sampling ratio Note. g = number of groups in the sample; N = average group size; r = within-group sampling ratio Between effects. For RMSE in between effect estimates, the three analysis approaches showed large differences. The means and standard deviations of RMSE of between effect estimates across the 160 simulation conditions from the three analysis approaches are presented in Table 22. For all between effect estimates in the three models, the manifest aggregation approach had the smallest RMSE among the three analysis approaches, while the latent aggregation approach gave the largest ones. The RMSE of between effect estimates from the 64 new approach with within-group fpc was between the statistics from the manifest and latent aggregation approaches. Table 22. RMSE of between effect estimates in the MLM, 2-1-1 mediation, and 1-1-1 mediation models MLM βxb 2-1-1 mediation 1-1-1 mediation βmb βxb βxb βmb M 0.514 2.096 1.119 SD 0.374 2.565 1.757 M 0.200 0.556 0.280 SD 0.183 0.846 0.375 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. M 0.390 2.737 1.258 SD 0.209 3.654 2.082 M 0.125 0.324 0.218 SD 0.053 0.303 0.255 M 0.505 1.667 0.905 SD 0.365 1.845 1.406 The sources of variance in RMSE of between effect estimates were explored using ANOVA. The results are in Table 23. For RMSE of between effect estimates, analysis approach accounted for medium to large proportions of variances (η2s > .059), which was consistent with the results on biases, relative biases, absolute biases, and absolute relative biases in between effect estimates. Similar to the results on absolute biases in between effect estimates, the ICC of predictor X or mediator M and group size showed medium to large (η2s > .059) effects on RMSE of between effect estimates. The factors which explained medium to large proportions of variances (η2s > .059) in RMSE of between effect estimates were: 1) the two-way interaction between analysis approach and ICCX/ICCM, 2) the two-way interaction between analysis approach and group size, and 3) the three-way interaction between analysis approach, ICCX/ICCM, and group size also. In addition, 1) the two-way interaction between analysis approach and between-to-within-effect ratio, and 2) the three-way interaction between analysis approach, ICCX/ICCM, and between-to-within-effect ratio, were also of medium effect sizes (η2s > .059) in explaining the variance in RMSE of between effect estimates in the 2-1-1 mediation model. 65 Table 23. Proportion of variance in RMSE of between effect estimates explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r Other interactions Approach × ICCX/ICCM × RX/RM Approach × ICCX/ICCM × N βxb 0.116 0.003 0.202 0.023 0.000 0.070 0.030 0.000 0.104 0.006 0.002 0.063 0.056 0.001 0.058 βmb βxb 0.138 0.003 0.157 0.014 0.000 0.063 0.021 0.004 0.118 0.004 0.002 0.070 0.056 0.005 0.066 βxb 0.111 0.000 0.067 0.064 0.003 0.104 0.128 0.004 0.048 0.008 0.001 0.055 0.055 0.014 0.030 βmb 0.113 0.005 0.211 0.018 0.001 0.072 0.033 0.002 0.095 0.004 0.001 0.054 0.059 0.001 0.045 0.073 0.013 0.094 0.013 0.000 0.061 0.030 0.077 0.058 0.005 0.002 0.050 0.034 0.061 0.041 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The means of RMSE of between effect estimates by analysis approach, ICCX/ICCM, and group size are presented in Table 24 and Figure 9. For most between effect estimates across different levels of ICCX/ICCM and group size, the manifest aggregation approach provided the smallest RMSE, and the latent aggregation approach generated the largest RMSE. The RMSE from the new approach with within-group fpc generally fell between the statistics from the manifest and latent aggregation approaches. The difference in RMSE among the three analysis approaches dropped down with a larger group size and a larger ICCX/ICCM. The RMSE from the latent aggregation approach was more sensitive to the influences of group size and ICCX/ICCM, 66 compared to the RMSE from the other two analysis approaches. The RMSE from the manifest aggregation approach was least affected by the changes in group size and ICCX/ICCM. When the ICCX/ICCM was .05 and the group size was 20, the differences in RMSE of between effect estimates among the three analysis approaches were obvious. The differences in RMSE of between effect estimates among the three analysis approaches were not that striking, with a larger ICCX/ICCM or group size. When the ICCX/ICCM was .25 and the group size was 100, the new approach with within-group fpc provided a smaller or similar RMSE for the between effects compared to the manifest aggregation approach. No matter which analysis approach was used, the RMSE of between effect estimates was inversely related to the group size and ICCX/ICCM. Table 24. RMSE of between effect estimates by analysis approach, ICCX/ICCM, and group size MLM βxw 2-1-1 mediation 1-1-1 mediation βmw βxw βmw αw N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 N =20 N =100 0.778 5.998 2.223 0.646 1.571 1.411 0.294 1.482 0.453 0.203 0.355 0.324 0.155 0.710 0.334 0.122 0.213 0.177 0.740 4.467 1.888 0.620 1.356 1.034 0.262 0.270 0.250 0.184 0.265 0.231 0.370 0.544 0.592 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 0.122 0.244 0.239 0.100 0.128 0.120 0.386 0.551 0.429 0.275 0.295 0.269 Manifest Latent New 0.539 8.064 2.346 ICCX/ICCM =0.05 0.525 1.931 1.560 ICCX/ICCM =0.25 0.222 0.310 0.284 0.275 0.645 0.840 0.120 0.122 0.112 67 Figure 9. RMSE of between effect estimates by analysis approach, ICCX/ICCM, and group size Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; ICC = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 68 5.2.4 Coverage Within effects. The accuracy of the standard errors from the three analysis approaches were evaluated in terms of the observed coverage rates. They were calculated as the proportions of times in which the population values of the parameters were in the estimated 95% CIs. As shown in Table 25, the average coverage rates for within effects from the manifest and latent aggregation approaches were close to the nominal level, i.e., .95. In contrast, the new approach with within-group fpc provided lower coverage rates for within effects than the nominal level. Its average coverage rates for the within effects were about 90%. The differences in coverage rates for within effects among the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc were also reflected in the ANOVA results. As shown in Table 26, the largest proportions of variances (i.e., from 16% to 22%, and from 20% to 21%) in coverage rates for within effects were explained by the analysis approach, and the interaction of analysis approach and balanced or unbalanced design. The factors which were of medium to large effect sizes (η2s > .059) in explaining the variances in coverage rates for the within effects in the three models were: 1) the balanced or unbalanced design, 2) the two-way interaction of analysis approach and group size, 3) the two- way interaction of balanced or unbalanced design and group size, and 4) the three-way interaction of analysis approach, balanced or unbalanced design, and group size. 69 Table 25. Observed coverage rate for within effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models MLM βxw 2-1-1 mediation 1-1-1 mediation βmw βxw βmw αw M 0.951 0.952 0.897 SD 0.021 0.020 0.079 M 0.947 0.945 0.895 SD 0.024 0.026 0.078 M 0.948 0.949 0.895 SD 0.021 0.021 0.084 M 0.947 0.947 0.891 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. Table 26. Proportion of variance in observed coverage rate for within effects explained by the design factors SD 0.022 0.023 0.083 M 0.926 0.947 0.888 SD 0.042 0.024 0.084 MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r BAL × N Other interactions Approach × BAL × N βxw 0.219 0.000 0.000 0.000 0.113 0.021 0.004 0.000 0.000 0.000 0.214 0.098 0.001 0.063 βmw βxw 0.198 0.000 0.000 0.001 0.118 0.054 0.007 0.000 0.002 0.000 0.196 0.098 0.002 0.057 βmw 0.213 0.003 0.001 0.000 0.105 0.037 0.007 0.000 0.000 0.000 0.207 0.088 0.002 0.036 αw 0.160 0.001 0.005 0.000 0.066 0.043 0.013 0.000 0.000 0.000 0.198 0.135 0.049 0.023 0.192 0.000 0.004 0.000 0.091 0.051 0.003 0.002 0.000 0.001 0.209 0.087 0.003 0.061 0.096 0.077 0.107 0.094 0.092 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. To understand the main and interaction effects of analysis approach, balanced or unbalanced design, and group size, the cell means of coverage rates for within effects are presented in Table 27, and plotted in Figure 10 by the three design factors. The latent 70 aggregation approach generally showed the most preferable coverage rates for these within effects across conditions. The group size and balanced or unbalanced design did not affect the coverage rates for within effects using either manifest or latent aggregation approach. Under the balanced design, the coverage rates for within effects in the new approach were similar as those from the manifest and latent aggregation approaches. Under the unbalanced design, where the large groups were triple the size of the small groups, the standard errors of within effects from the new approach were underestimated. The coverage rates for within effects from the new approach were even lower when the group size in the population was larger under the unbalanced design. Table 27. Observed coverage rate for within effects by analysis approach, balanced or unbalanced design, and group size 2-1-1 mediation 1-1-1 mediation N=100 N=20 N=100 N=20 N=100 N=20 N=100 N=20 MLM N=20 βmw βmw βxw βxw αw N=100 Manifest 0.945 0.947 Latent New 0.946 0.958 0.958 0.957 0.946 0.940 0.947 0.947 0.947 0.947 Balanced 0.950 0.952 0.948 0.950 0.950 0.951 Unbalanced 0.947 0.945 0.760 0.947 0.949 0.946 0.947 0.946 0.947 0.918 0.951 0.950 0.930 0.935 0.934 0.953 0.954 0.772 0.943 Manifest 0.948 0.951 0.949 Latent 0.756 New 0.913 Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; N = average group size. 0.946 0.950 0.918 0.951 0.951 0.913 0.943 0.944 0.773 0.946 0.943 0.906 0.915 0.952 0.913 0.950 0.949 0.764 71 Figure 10. Coverage rate for within effects by analysis approach, balanced or unbalanced design, and group size Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; BAL = balanced or unbalanced design; N = average group size. 72 Between effects. The coverage rates for the between effects are presented in Table 28. Different from the results on within effects, the manifest and latent aggregation approaches generally produced lower observed coverage rates than .95 for the between effects. In other words, the standard errors for the between effects in the three models were underestimated from the manifest and latent aggregation approaches. The manifest aggregation approach performed the worst in terms of coverage rates for between effects among the three analysis approaches, with an average observed coverage rate lower than .90. The observed coverage rates in the new approach with within-group fpc were closer to the nominal level, i.e., .95, compared to those from the manifest and latent aggregation approaches. The new approach slightly overestimated the standard errors of between effects with the conservatively estimated 95% CIs. Table 28. Observed coverage rate for between effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models 1-1-1 mediation MLM 2-1-1 mediation βxb βmb βxb βmb αb M 0.762 0.864 0.966 SD 0.313 0.171 0.028 M 0.674 0.881 0.963 SD 0.372 0.213 0.055 Manifest Latent New Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc. M 0.866 0.922 0.972 SD 0.178 0.083 0.028 M 0.837 0.949 0.954 SD 0.188 0.053 0.033 M 0.774 0.864 0.965 SD 0.302 0.170 0.024 The sources of variances in coverage rates for between effects were investigated using ANOVA (in Table 29). The main effect of analysis approach was of medium to large effect sizes (η2s > .059) in explaining the variances in coverage rates for these between effects. In addition, the factors which explained medium to large proportions (η2s > .059) of variances in the coverage rates for most between effects were: 1) the between-to-within-effect ratio, 2) the two- way interaction between analysis approach and between-to-within-effect ratio, 3) the two-way interaction between analysis approach and within-group sampling ratio, and 4) the three-way interaction between analysis approach, between-to-within-effect ratio, and within-group 73 sampling ratio. The interaction between analysis approach and group size was also of medium effect size (η2 = .063) in explaining the variance in the coverage rate for between effect in the 2- 1-1 mediation model. Table 29. Proportion of variance in observed coverage rate for between effects explained by the design factors MLM 2-1-1 mediation 1-1-1 mediation Main effects Approach RX/RM ICCX/ICCM g BAL N r Two-way interactions Approach × RX/RM Approach × ICCX/ICCM Approach × g Approach × BAL Approach × N Approach × r Other interactions Approach × RX/RM × r βxb 0.140 0.145 0.003 0.021 0.001 0.016 0.026 0.101 0.004 0.009 0.001 0.014 0.156 0.138 βmb βxb 0.126 0.053 0.000 0.023 0.005 0.002 0.021 0.065 0.028 0.010 0.001 0.016 0.149 0.130 βmb 0.182 0.069 0.005 0.034 0.005 0.017 0.006 0.109 0.003 0.026 0.003 0.063 0.076 0.049 αb 0.131 0.138 0.014 0.013 0.002 0.018 0.026 0.086 0.015 0.007 0.003 0.015 0.152 0.130 0.193 0.157 0.019 0.023 0.001 0.003 0.015 0.122 0.030 0.010 0.002 0.011 0.119 0.079 Note. All main effects and two-way interactions between analysis approach and other design factors are listed in the table. For other interactions, only the ones that explained medium to large proportions of variances (i.e., η2 > .059) for at least one of these outcomes are presented in the table. Approach = manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; ICCX/ICCM = intraclass correlation coefficient of the decomposed predictor or mediator; g = number of groups in the sample; BAL = balanced or unbalanced design; N = average group size; r = within-group sampling ratio. The main and interaction effects between analysis approach, between-to-within-effect ratio, and within-group sampling ratio were further explored. The cell means of coverage rates for between effects are presented in Table 30 and plotted in Figure 11 by analysis approach, between-to-within-effect ratio, and within-group sampling ratio. The coverage rates for between effects from the new approach with within-group fpc were above .95 in most conditions, and 74 they were much closer to the nominal coverage rate, i.e., .95, than those from the manifest and latent aggregation approaches. The coverage rates for between effects from the manifest and latent aggregation approaches were affected by the between-to-within-effect ratio and within-group sampling ratio. The coverage rates for between effects from the manifest aggregation approach got better with an increasing within-group sampling ratio, which was consistent with the established findings (Lüedtke et al., 2011; Lüedtke et al., 2008; Marsh et al., 2012; Marsh et al., 2009; Preacher et al., 2010) and theoretical derivations. As expected, the coverage rates for between effects from the latent aggregation approach dropped down with an increasing within-group sampling ratio. The differences in coverage rates for the between effects among the three analysis approaches decreased with a decreasing between-to-within-effect ratio. When the between-to- within-effect ratio was .10, the differences in coverage rates among the three analysis approaches (or by different levels of within-group sampling ratio) were trivial. When the between-to-within- effect ratio was 10, the differences in coverage rates among the three analysis approaches (or by different levels of within-group sampling ratio) were clearer. For instance, when the between-to- within-effect ratio was 10 and the within-group sampling ratio was .10 or .30, the coverage rates for between effects from the manifest aggregation approach were lower than 80%, which were unacceptable. Under the same conditions, the coverage rates from the new approach were over 95%. When the between-to-within-effect ratio was 10 and the within-group sampling ratio was .70 and .90, the coverage rates for between effects from the latent aggregation approach were unfavorable. The coverage rates for between effects from the new approach were not affected largely by the between-to-within-effect ratio or within-group sampling ratio, and were better than those from the manifest and latent aggregation approaches. 75 Table 30. Observed coverage rate for between effects by analysis approach, RX/RM, and within-group sampling ratio 1-1-1 mediation βmb αb Latent MLM βxb 2-1-1 mediation βmb 0.979 0.979 0.963 0.968 0.965 0.970 0.954 0.968 0.949 0.970 0.952 0.977 0.939 0.963 0.944 0.964 0.941 0.963 0.942 0.965 r Manifest Latent New Manifest Latent New Manifest Latent New Manifest Latent New Manifest 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; r = within-group sampling ratio. 0.981 0.986 0.936 0.982 0.893 0.975 0.859 0.981 0.815 0.982 0.954 0.973 0.949 0.964 0.944 0.962 0.944 0.959 0.945 0.961 0.934 0.935 0.868 0.966 0.770 0.973 0.709 0.974 0.643 0.976 0.958 0.867 0.949 0.951 0.819 0.983 0.701 0.986 0.570 0.990 0.986 0.982 0.964 0.961 0.937 0.943 0.931 0.939 0.899 0.939 0.977 0.976 0.955 0.956 0.949 0.948 0.950 0.949 0.944 0.949 0.947 0.945 0.942 0.937 0.939 0.929 0.856 0.772 0.704 0.665 0.916 0.936 0.942 0.939 0.944 0.109 0.398 0.665 0.843 0.930 0.909 0.933 0.936 0.940 0.941 0.142 0.467 0.691 0.849 0.935 0.831 0.911 0.935 0.944 0.944 0.069 0.238 0.411 0.602 0.854 0.927 0.937 0.936 0.934 0.936 0.493 0.788 0.887 0.918 0.909 0.921 0.935 0.932 0.931 0.929 0.527 0.750 0.814 0.825 0.811 βxb RX/RM =0.1 RX/RM =10 New 0.970 0.969 0.966 0.966 0.970 0.944 0.956 0.966 0.966 0.973 76 Figure 11. Coverage rate for between effects by analysis approach, RX/RM, and within-group sampling ratio Note. Manifest = manifest aggregation approach; Latent = latent aggregation approach; New = the new approach with within-group fpc; RX/RM = between-to-within-effect ratio; r = within- group sampling ratio. 77 5.3 Summary The primary purpose of the current simulation study was to compare the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc in the decomposition of between and within effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models. The simulation factors included ICC of the predictor or mediator, between-to- within-effect ratio, number of groups in the sample, balanced or unbalanced design, group size, and within-group sampling ratio. The three analysis approaches showed acceptable model convergence rates (i.e., above 98%) for the three models. For the manifest aggregation approach, the estimation converged across the 16,000 replications for the three models. The latent aggregation approach showed similar convergence rates across different levels of the design factors. In the new approach with within-group fpc, the model convergence rate increased with an increasing within-group sampling ratio and a larger ICC of the predictor or mediator. Overall, the simulation results confirmed the mathematical derivations in the previous chapter. There were small differences in the relative biases, absolute relative biases, and RMSE for the within effects among the three analysis approaches. Under a balanced design, the new approach provided similar coverage rates for the within effects to those from the manifest and latent aggregation approaches. Under an unbalanced design, the coverage rates for the within effects from the new approach were unfavorable, especially when the group size in the population was large. To directly compare the three analysis approaches under different conditions, the within effect estimates from the three analysis approaches were further summarized in this section. As a result from the simulation study, four factors explained medium or large proportions of variances 78 in at least one of the following evaluation criteria for the within effects: 1) model convergence rate, 2) relative bias, 3) absolute relative bias, 4) RMSE, and 5) coverage rate. The four design factors were: 1) the number of groups in the sample, 2) balanced or unbalanced design, 3) group size, and 4) within-group sampling ratio. The mean within effect estimate, 95% empirical CI over replicates, model convergence rate, and observed coverage rate by the four factors are presented in Figure 12 for each analysis approach in each model. The manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc showed similar point estimates and 95% empirical CIs across all conditions for the within effects in the three models. The model convergence rates were also similar among the three analysis approaches under different conditions. The number of groups in the sample, balanced or unbalanced design, group size, and within-group sampling ratio showed little effects on the point estimates of within effects, and model convergence rates. With a larger sample size (i.e., a larger number of groups, a larger group size, and a larger within-group sampling ratio), the 95% empirical CIs for the within effects were narrower, no matter which analysis approach was used. The obvious differences among the three analysis approaches were found in the observed coverage rates for the within effects under the unbalanced design. The new approach showed lower coverage rates for the within effects compared to the manifest and latent aggregation approaches, and its coverage rates became worse when the group size was larger. In fact, the balanced or unbalanced design only affected the standard error estimation, but not the within effect point estimation in the new approach. The influence of balanced or unbalanced design on the standard error estimation in the new approach might be related with the MUML estimation it used. Further simulations are needed to understand how the balanced and unbalanced designs affect the estimation of standard errors for the within effects in MUML. 79 Cluster robust standard errors and bootstrapped standard errors provide some choices for the standard error estimation in the new approach. For the between effects, the differences among the three analysis approaches was more obvious. In general, the new approach with within-group fpc outperformed the other two analysis approaches in terms of biases, relative biases, and coverage rates for the between effects. The manifest aggregation approach outperformed the other two in the absolute biases, absolute relative biases, and RMSE for the between effects. As discussed before, the current study applied moderate to large within-group sampling ratios with small to moderate groups in the simulation, which better represented the designs in education studies. This population assumption and sampling design did not fit the assumptions made by the latent aggregation approach (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011). It is no surprise that the latent aggregation approach did not show any advantage over the manifest aggregation approach or the new approach with within-group fpc in the decomposition of between and within effects in the current study. The results did not mean that the latent aggregation approach was unfavorable in general, but suggested that under the same or a similar population and sampling scenario, the latent aggregation approach might not be a good choice to separate the individual effects and group compositional effects. In other words, it is necessary to consider the population and sampling procedures to choose the analysis approach in the decomposition of between and within effects. The between-to-within-effect ratio, ICC of the predictor or mediator, group size, and within-group sampling ratio explained medium or large proportions of variances in at least one of the following evaluation criteria for the between effects: 1) model convergence rate, 2) bias, 3) relative bias, 4) absolute bias, 5) absolute relative bias, 6) RMSE, and 7) coverage rate. For each between effect, the mean between effect estimate, 95% empirical CI, model convergence 80 rate, and observed coverage rate are presented by these four design factors in Figure 13 for each analysis approach, in order to compare the three analysis approaches directly. When the between-to-within effect ratio was .10, the three analysis approaches showed less clear differences in the point estimates and coverage rates for the between effects in the three models. The non-negligible differences among the three analysis approaches were found in the 95% empirical CIs. When the between-to-within-effect ratio was .10, the 95% empirical CIs from the manifest aggregation approach were generally the narrowest among the three analysis approaches. The trade-offs were its lower coverage rates, especially when the sample size within each group was small (i.e., a small group size and a small within-group sampling ratio). For example, when the between-to-within-effect-ratio was .10 and the group size was 20, the coverage rates for the between effects from the manifest aggregation approach were around 50% with a within-group sampling ratio of .10, and all below 80% with a within-group sampling ratio of .30. The 95% empirical CIs from the latent aggregation approach were the widest. With a larger ICC of the predictor or mediator, and a larger group size, the 95% empirical CIs for the between effects in the three models from the three analysis approaches became narrower. When the within-group sampling ratio got larger, the 95% empirical CIs from the new approach with within-group fpc became narrower, while the 95% empirical CIs from the manifest and latent aggregation approaches slightly changed. When the between-to-within-effect ratio was 10, the new approach with within-group fpc showed better point estimates and coverage rates for the between effects in the three models compared to the other two analysis approaches. For the point estimates of between effects, manifest aggregation approach generally underestimated the between effects, while latent aggregation approach overestimated the between effects. Although the coverage rates for 81 between effects from the new approach were slightly higher than the nominal rate, i.e., .95, they were much closer to .95 compared to the coverage rates from the manifest and latent aggregation approaches. In contrast, the manifest aggregation approach provided the narrowest 95% empirical CIs for the between effects across all conditions with the lowest coverage rates. Similar to the results when the between-to-within-effect ratio was 0.1, when the group size and within-group sampling ratio (i.e., within-group sample size) were small, the coverage rates from the manifest aggregation approach were unacceptably low. For example, when the group size was 20, and the within-group sampling ratio was .10 or .30, the coverage rate for the between effect in the 2-1-1 mediation model from the manifest aggregation approach was 0. The latent aggregation approach generally showed large biases and large 95% empirical CIs for the between effects. Its coverage rates for the between effects were acceptable when the within- group sampling ratio was low, but became undesirable when the within-group sampling ratio was medium or large. In addition, with a larger ICC of the predictor or mediator, and a larger group size, the biases in between effect estimates became smaller, the 95% empirical CIs became narrower, and the coverage rates became better for each analysis approach. Additionally, for the between effects, with a larger within-group sampling ratio, the biases became smaller, the 95% empirical CIs became narrower, and the coverage rates became higher for the manifest aggregation approach and the new approach with within-group fpc. For the latent aggregation approach, the biases in between effect estimates increased, the coverage rates dropped down, and the empirical 95% CIs were virtually unchanged with an increase in the within-group sampling ratio. 82 Figure 12. Simulation results for within effects by analysis approach, within-group sampling ratio, balanced or unbalanced design, number of groups in the sample, and group size 83 Figure 12. (cont’d) Note. The left labels in the plots indicate the simulation conditions and parameters. The solid vertical lines in the plots represent the true values of the parameters, with their values labeled out on the axes. The short and solid horizontal lines represent the 95% empirical CIs (i.e., 0.025 to 0.975 empirical quantiles over 400 replications) from each analysis approach. The solid dots in the horizontal lines represent the point estimates for the within effects from each analysis approach. The right labels in the plots indicate the analysis approaches, with their model convergence rates and coverage rates in the parentheses. The first number in the parenthesis is the model convergence rate, and the second number in the parenthesis is the coverage rate. M = manifest aggregation approach; L = latent aggregation approach; N = the new approach with within-group fpc; r = within-group sampling ratio; g = number of groups in the sample; N = average group size. 84 Figure 13. Simulation results for between effects by analysis approach, within-group sampling ratio, between-to-within-effect ratio, ICCX/ICCM, and group size 85 Figure 13. (cont’d) Note. The left labels in the plots indicate the simulation conditions and parameters. The solid vertical lines in the plots represent the true values of the parameters, with their values labeled out on the axes. The short and solid horizontal lines represent the 95% empirical CIs (i.e., 0.025 to 0.975 empirical quantiles over 400 replications) from each analysis approach. The solid dots in the horizontal lines represent the point estimates for the within effects from each analysis approach. The right labels in the plots indicate the analysis approaches, with their model convergence rates and coverage rates in the parentheses. The first number in the parenthesis is the model convergence rate, and the second number in the parenthesis is the coverage rate. M = manifest aggregation approach; L = latent aggregation approach; N = the new approach with within-group fpc; r = within-group sampling ratio; RX/RM = between-to- within-effect ratio; ICC = intraclass correlation coefficient of the decomposed predictor or mediator; N = average group size. 86 Chapter 6 The Manifest, Latent, and New Aggregation Approaches: An Application with PISA Dataset In this application of manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc, the between and within direct and indirect effects of SES on student mathematics performance through OTL were examined in a 1-1-1 mediation model using the data from PISA 2012. 6.1 Background Since the publication of the Coleman Report (Coleman et al., 1966), continuous efforts have been given to explore the role of schooling in alleviating student SES gap in mathematics performance. Some researchers look at this problem through OTL, which describes students’ content exposures in mathematics, and is a key factor to understanding schooling. Previous studies showed OTL had a significant impact on student mathematics achievement, regardless of students’ parental education and income (Cogan, Schmidt, & Wiley, 2001; Lleras, 2008; Schmidt et al., 2015). However, the between-school and within-school SES gaps in OTL were found which exacerbated rather than alleviated SES gaps in student mathematics performance (Schmidt et al., 2015). In other words, high SES schools showed more capabilities to provide advanced mathematics courses to their students, which brought benefits to their student performance on average (Schmidt et al., 2015); within schools, high SES students had more opportunities to attend demanding courses (Burger, 2016; Kalogrides & Loeb, 2013; Milner, 2012; Reeves, 2012; Roscigno, 1998), which were further translated into their advantages in mathematics (Schmidt et al., 2015). The direct and indirect effects of SES were highly likely to occur between and within schools, which reflected the institutional-level and individual-level mechanisms. It was necessary 87 to decompose the between and within direct and indirect effects of SES on student mathematics performance via OTL. The main purpose of the current empirical study was to explore the between and within direct and indirect effects of SES on student mathematics performance through OTL in a 1-1-1 mediation model for different countries using the data from PISA 2012 with the three analysis approaches. 6.2 Data and Measures PISA is an international standardized assessment that measures how well-prepared 15- year-old students are for their future lives. In each round of assessment, student mathematics, reading, and science literacies are measured with a major subject measured with greater precision. From the 2012 iteration, PISA started to survey students about their exposures to certain topics in the major subject. In PISA 2012, the major subject was mathematics and students’ content coverage in mathematics were measured (OECD, 2014a, 2014b). There were 34 OECD countries and 31 partner countries participating in PISA 2012. There were about 150 schools drawn from each country, with around 30 sampled students within each sampled school. In each country, a stratified two-stage sampling design was used in PISA, where schools were sampled using probability proportional to size sampling (PPS), and students were sampled with equal probability within sampled schools. Student weights and school weights were created for the sampled students and schools, which reflected how many other students (or schools) they could represent in the population (OECD, 2014a, 2014b). Based on the student weights and school weights, the school size (of 15-year-olds) and within school sampling ratio can be calculated as , (36) 88 __jiNijjjWsizFSTUWTWeFSCHWT and , (37) where is the school size (of 15-year-olds) of jth school; is total number of sampled students in jth school; is the student weight (i.e., product of the inverse of the school’s probability of selection and the inverse of the student’s probability of selection within that school) for ith student in jth school; is the school weight (i.e., inverse of the school’s probability of selection) for jth school; is the within-school sampling ratio for school j; and is actual number of students in jth school used for analyses. In PISA, the mean mathematics performance across OECD countries is set at 500 and the standard deviation is 100. OTL is constructed based on students’ exposures to 13 mathematics topics. The response categories vary from “never heard of it” to “knew it well”. Student socioeconomic background is represented by the economic, social and cultural status (ESCS) in PISA. ESCS is computed as a weighted score of students’ home possessions, parents’ occupations, and parents’ education levels. This variable has an average score of 0 and a standard deviation of 1 across OECD countries (OECD, 2014a, 2014b). In the current study, only students with no missing data on ESCS, OTL and mathematics performance were included in the current study, and only the schools with at least two students were used in the analyses. After excluding two countries (i.e., Albania and Norway) which did not collect ESCS or OTL information, about two thirds of students in each of the 63 countries were used, as the PISA student survey used a random rotated block design and one third of students had missing data on OTL by design. The number of schools ranged from 12 to 1421 in 89 jjjnratioSizejsizejN_ijWFSTUWT_jWFSCHWTjratiojn these 63 countries, the average within-school sample size ranged from 12 to 81, and the within- school sampling ratio ranged from .09 to .65. The manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc were used to examine the between and within direct and indirect effects of SES on student mathematics performance through OTL in the 1-1-1 mediation model. 6.3 Results The descriptive statistics of ESCS, OTL, and mathematics performance in the 63 countries are presented in Table 31. Different from the settings in the simulation study (i.e., ICCX/ICCM = .05 or .25), there were 21 countries with an ICC of ESCS (i.e., the predictor) higher than .30. There were 11 countries with an ICC of OTL (i.e., the mediator) higher than .30, while the ICC of the mediator was set between .20 and .25 in the previous simulation study. Moreover, the ICC of mathematics performance (i.e., the outcome) was over .30 in 48 countries, while it was set as .25 in the simulation study across all conditions. With larger ICCs in the empirical datasets, the larger differences in the between effects among the three analysis approaches were expected holding other conditions constant. 90 Table 31. Descriptive statistics of ESCS, OTL, and mathematics performance in each country Country Finland Liechtenstein Switzerland Iceland Sweden Canada Netherlands Perm United Kingdom Estonia Denmark Montenegro Jordan Ireland Korea Slovenia Serbia Singapore Croatia Spain Russian Federation Kazakhstan Italy New Zealand Japan Australia Belgium Chinese Taipei Czech Republic Lithuania Latvia Qatar Germany Israel France Malaysia United States Austria Portugal Luxembourg Greece United Arab Emirates Poland Turkey N school 296 12 405 130 196 864 174 63 N student 5729 195 7367 2215 3028 14005 2796 1166 ESCS M (SD) 0.349(0.826) 0.291(0.906) 0.119(0.870) 0.790(0.809) 0.285(0.813) 0.414(0.842) 0.237(0.751) -0.078(0.729) OTL M (SD) ICC 0.095 1.618(0.619) 0.130 1.582(0.757) 0.138 1.485(0.695) 0.142 1.142(0.622) 0.146 0.771(0.555) 0.169 1.917(0.615) 0.176 1.482(0.673) 0.182 2.008(0.429) ICC 0.191 0.276 0.356 0.044 0.128 0.136 0.333 0.039 Performance M (SD) 510.038(86.987) 539.189(92.491) 523.189(92.635) 497.925(89.622) 482.015(86.728) 511.478(86.339) 524.421(90.773) 486.952(85.483) ICC 0.091 0.536 0.317 0.118 0.111 0.201 0.642 0.323 507 200 334 50 233 183 156 291 153 172 162 891 224 8201 0.230(0.814) 0.188 1.492(0.668) 0.196 491.312(90.896) 0.283 3134 4776 3073 4540 3315 3336 3788 3062 3671 3321 16602 0.156(0.786) 0.294(0.910) -0.257(0.879) -0.451(1.009) 0.129(0.855) 0.026(0.737) -0.003(0.854) -0.297(0.904) -0.281(0.921) -0.346(0.849) -0.099(1.003) 0.194 2.006(0.451) 0.199 1.601(0.606) 0.199 1.906(0.625) 0.204 2.132(0.749) 0.213 1.467(0.601) 0.223 2.070(0.522) 0.223 1.882(0.562) 0.224 2.039(0.535) 0.225 2.207(0.647) 0.228 2.060(0.560) 0.228 1.911(0.643) 0.078 0.112 0.073 0.161 0.106 0.268 0.209 0.112 0.161 0.130 0.112 522.985(79.526) 489.645(83.660) 408.528(80.605) 385.327(74.317) 502.338(84.184) 553.503(99.359) 488.343(88.404) 449.189(89.516) 568.658(106.427) 470.164(86.539) 497.236(87.862) 0.166 0.177 0.338 0.343 0.196 0.408 0.586 0.444 0.354 0.430 0.170 3431 -0.079(0.736) 0.232 2.104(0.402) 0.058 483.962(86.504) 0.266 214 1167 176 189 768 270 3844 20365 2773 4104 9342 5469 -0.265(0.746) -0.030(0.948) 0.058(0.812) -0.066(0.716) 0.198(0.803) 0.209(0.894) 0.233 1.977(0.559) 0.235 1.836(0.623) 0.236 1.519(0.712) 0.236 2.054(0.473) 0.241 1.625(0.715) 0.245 1.815(0.715) 0.097 0.333 0.181 0.301 0.209 0.300 433.073(70.608) 492.377(90.150) 505.946(99.513) 539.280(92.583) 497.425(97.431) 527.380(98.161) 0.370 0.526 0.234 0.545 0.289 0.514 163 4017 -0.388(0.848) 0.245 1.974(0.571) 0.189 558.129(115.433) 0.445 279 215 201 149 215 169 224 164 161 174 193 42 185 445 179 165 3487 0.087(0.741) 0.249 1.872(0.542) 0.318 524.104(93.093) 0.490 3025 2796 6630 2650 3163 2934 3415 3240 3063 3677 3408 3376 -0.124(0.913) -0.177(0.876) 0.452(0.885) 0.220(0.926) 0.187(0.845) -0.010(0.801) -0.712(0.993) 0.197(0.971) 0.116(0.819) -0.476(1.173) 0.074(1.110) -0.053(0.997) 0.255 1.647(0.518) 0.256 2.058(0.461) 0.260 1.715(0.807) 0.260 1.666(0.653) 0.264 1.821(0.620) 0.278 1.889(0.554) 0.278 1.595(0.596) 0.279 2.008(0.629) 0.279 1.555(0.679) 0.285 1.716(0.606) 0.289 1.448(0.706) 0.292 1.913(0.575) 0.089 0.112 0.259 0.345 0.198 0.344 0.120 0.114 0.433 0.109 0.176 0.084 479.429(88.787) 497.406(80.816) 382.242(99.366) 525.811(93.971) 472.031(103.060) 502.837(95.825) 424.396(81.336) 481.165(89.326) 509.823(90.600) 487.096(93.127) 492.539(94.672) 453.111(87.606) 0.331 0.246 0.477 0.491 0.443 0.562 0.323 0.254 0.506 0.331 0.315 0.364 7402 0.303(0.850) 0.299 2.092(0.696) 0.227 433.542(90.768) 0.474 3031 3194 -0.157(0.915) -1.450(1.102) 0.303 1.848(0.546) 0.308 1.920(0.547) 0.085 0.188 520.680(90.657) 449.345(94.167) 0.257 0.628 91 Table 31. (cont’d) N Country Slovak Republic Tunisia Macao-China Hong Kong- China China- Shanghai Brazil Costa Rica Argentina Hungary Indonesia Romania Uruguay Vietnam Colombia Bulgaria Mexico Peru Thailand Chile N ESCS OTL Performance school student M (SD) ICC M (SD) ICC M (SD) ICC 219 153 45 148 3062 -0.137(0.907) 0.320 1.721(0.558) 0.308 487.454(102.922) 0.499 2804 3534 -1.216(1.268) -0.880(0.873) 0.327 1.228(0.592) 0.328 2.200(0.567) 0.059 0.165 389.257(78.275) 538.136(92.141) 0.513 0.355 3037 -0.805(0.964) 0.328 1.832(0.633) 0.076 563.036(95.217) 0.433 155 3440 -0.366(0.975) 0.340 2.295(0.455) 0.174 610.992(101.208) 0.480 809 192 224 188 208 178 178 162 346 184 1421 237 237 212 12192 2886 3777 3135 3691 3337 3390 3302 5659 3356 22222 3774 4372 4479 -1.236(1.188) -0.992(1.219) -0.611(1.132) -0.195(0.935) -1.754(1.102) -0.465(0.935) -0.867(1.127) -1.829(1.089) -1.057(1.128) -0.211(1.037) -1.016(1.245) -1.210(1.215) -1.096(1.277) -0.225(1.256) 0.354 1.407(0.715) 0.357 1.528(0.709) 0.376 1.401(0.694) 0.380 2.002(0.518) 0.382 1.608(0.567) 0.388 2.026(0.623) 0.398 1.639(0.673) 0.408 1.949(0.468) 0.408 1.772(0.696) 0.442 1.978(0.649) 0.446 1.795(0.640) 0.469 1.787(0.706) 0.502 1.768(0.554) 0.685 1.817(0.597) 0.280 0.179 0.272 0.308 0.174 0.227 0.246 0.188 0.196 0.211 0.195 0.206 0.245 0.333 384.230(78.129) 408.194(66.683) 400.182(77.548) 487.565(90.453) 376.816(70.150) 446.455(80.149) 414.173(86.448) 511.450(83.873) 390.504(75.044) 444.749(92.096) 419.312(74.071) 373.077(83.231) 441.263(92.917) 444.665(87.112) 0.463 0.389 0.477 0.626 0.486 0.463 0.425 0.536 0.377 0.545 0.368 0.451 0.527 0.535 The 63 countries were sorted by the ICC of ESCS, and their results were presented in this order, in order to compare the between and within effects from the three analysis approaches. The within effect estimates and their standard errors from the three analysis approaches are shown in Table 32. In most countries, ESCS showed significant direct and indirect within effects via OTL on student mathematics achievement, which was consistent with previous studies (Schmidt et al., 2015). Within each school, students with higher ESCS demonstrated their advantages in mathematics directly, as well as indirectly through a better mathematics content coverage. As expected from the derivations and simulations, the within effect estimates and their standard errors from the three analysis approaches did not differ much from each other in the 63 countries, regardless of the degrees of ICCs of ESCS, OTL, and mathematics performance. On the school level, significant direct and indirect between effects of ESCS on mathematics performance were also found in most countries, where the magnitudes of the 92 between effects were larger than the corresponding within effects in these countries. In other words, the school-level SES inequality also directly and indirectly contributed to the mathematics performance gap via OTL. In contrast to the within effects, obvious differences in the between effect estimates and their standard errors were found among the three analysis approaches (in Table 33). As expected, the between effect estimates and their standard errors from the new approaches were generally between the statistics from the manifest and latent aggregation approaches. Moreover, in this empirical study, the between effect estimates from the new approach with fpc were generally closer to the ones from latent aggregation approach than those from the manifest aggregation approach. While the between effect estimates from the new approach, on average, were closer to the ones from manifest aggregation approach in the simulation. The difference was related to the setting of within-group sampling ratio. The school population size of 15-year-olds ranged from 26 to 449 in the 63 countries with an average of 123. The within- school sampling ratio ranged from .09 to .61, with a mean of .31 across these countries. The fpc was calculated as one minus within-group sampling ratio, and was generally over .50 in these countries. Based on Equation (36), after the correction, the between variance-covariance matrices from the new approach with these large fpcs, were closer to the ones used in the latent aggregation approach than the ones used in the manifest aggregation approach. In contrast, the within-group sampling ratio was set as .10, .30, .50, .70, and .90 in the simulation. On average, the between variance-covariance matrices from the new approach adjusted with fpc were closer to the ones used in the manifest aggregation approach in the simulation. To compare the empirical study with the simulation study, the results from the similar conditions should be used. The simulation conditions with large between-to-within-effect ratio 93 (i.e., RX/RM =10), large ICC of the predictor (i.e., ICCX/ICCM =.25), large group size in the population (i.e., N=100), and small within-group sampling ratio (i.e., r=.10 or .30) were similar to the conditions in the PISA dataset for most countries. Under these conditions, the between effect estimates from the new approach were closer to the estimates from the latent aggregation approach in the previous simulations. In summary, the results from the empirical study was consistent with the ones from the simulation. Both of them reflected the necessaries to consider the assumptions about the within- group sampling in the decomposition of between and within effects. 94 Table 32. Within effects from the manifest aggregation approach, latent aggregation approach, and the new approach with within- group fpc in the 1-1-1 mediation model Country Manifest βxw Latent Est 23.369 6.077 18.638 20.642 26.397 14.845 3.830 17.349 SE 1.267 4.813 1.054 2.418 1.998 0.805 1.401 3.047 10.290 1.040 14.974 22.649 10.545 8.786 14.579 8.453 1.047 7.883 7.366 8.785 13.989 1.785 1.287 1.433 0.966 1.615 1.915 1.282 1.517 1.504 1.468 0.646 22.719 1.946 10.860 3.967 18.958 2.622 12.071 8.653 1.395 0.538 2.015 1.523 1.085 1.123 14.459 1.770 8.385 1.710 14.426 16.468 3.559 6.953 15.314 19.650 10.332 13.495 9.468 17.731 13.244 17.168 1.651 1.686 1.112 1.621 1.808 1.726 1.280 1.481 1.671 1.191 1.386 1.433 Est 23.430 5.934 18.555 20.333 26.380 14.849 3.821 17.057 10.244 14.715 22.705 10.491 8.704 14.557 8.424 0.981 7.875 7.380 8.752 13.993 22.379 10.751 4.020 18.752 2.615 12.129 8.655 14.471 8.272 14.249 16.406 3.561 7.122 15.280 19.620 10.338 13.551 9.521 17.751 13.217 17.018 Finland Liechtenstein Switzerland Iceland Sweden Canada Netherlands Perm United Kingdom Estonia Denmark Montenegro Jordan Ireland Korea Slovenia Serbia Singapore Croatia Spain Russian Federation Kazakhstan Italy New Zealand Japan Australia Belgium Chinese Taipei Czech Republic Lithuania Latvia Qatar Germany Israel France Malaysia United States Austria Portugal Luxembourg Greece SE 1.266 4.829 1.052 2.409 1.999 0.805 1.401 3.040 New Manifest Est 24.140 9.251 18.300 21.415 27.152 14.633 3.662 17.067 SE 1.272 4.750 1.053 2.289 1.964 0.801 1.397 2.908 Est 61.498 24.188 47.885 19.112 4.781 53.287 43.752 42.971 SE 1.789 6.950 1.539 2.981 2.890 1.084 1.737 4.838 βmw Latent Est 61.429 24.321 47.911 18.582 5.567 53.335 43.758 42.875 SE 1.788 6.955 1.539 2.971 2.880 1.084 1.737 4.826 New Manifest Est 61.515 22.426 44.092 13.665 4.038 53.074 43.394 42.518 SE 1.798 7.036 1.544 2.823 2.825 1.076 1.743 4.508 Est 0.178 0.111 0.149 0.112 0.088 0.156 0.096 0.086 SE 0.009 0.049 0.008 0.017 0.012 0.006 0.015 0.018 αw Latent Est 0.179 0.109 0.149 0.114 0.089 0.157 0.096 0.087 SE 0.009 0.050 0.008 0.017 0.013 0.006 0.016 0.019 New Est 0.185 0.075 0.125 0.134 0.076 0.152 0.096 0.088 SE 0.009 0.050 0.008 0.018 0.013 0.006 0.016 0.019 1.041 10.029 1.042 65.387 1.273 65.387 1.273 65.229 1.277 0.191 0.009 0.191 0.009 0.190 0.009 1.780 1.288 1.431 0.966 1.616 1.914 1.279 1.516 1.504 1.468 0.646 15.234 22.305 6.624 8.088 14.374 9.437 0.652 6.979 7.568 8.861 13.981 1.799 1.293 1.446 0.961 1.619 1.928 1.285 1.507 1.502 1.471 0.648 52.612 42.604 29.987 26.360 56.316 70.506 15.492 34.634 80.063 41.190 59.679 2.923 1.844 1.886 1.270 2.156 2.771 1.927 2.402 2.058 2.097 0.940 52.789 42.646 30.033 26.387 56.486 70.443 15.372 34.720 80.045 41.237 59.771 2.916 1.844 1.883 1.271 2.156 2.769 1.922 2.400 2.058 2.097 0.940 49.391 41.555 30.389 26.154 56.226 72.375 12.617 35.094 80.051 41.651 59.107 2.984 1.865 1.842 1.260 2.166 2.783 1.908 2.361 2.058 2.101 0.938 0.084 0.158 0.083 0.108 0.183 0.125 0.050 0.082 0.173 0.079 0.184 0.011 0.010 0.014 0.011 0.013 0.012 0.011 0.011 0.012 0.012 0.005 0.085 0.158 0.083 0.108 0.183 0.124 0.050 0.081 0.173 0.078 0.185 0.011 0.010 0.014 0.011 0.013 0.012 0.011 0.012 0.012 0.012 0.005 0.079 0.163 0.076 0.108 0.186 0.127 0.067 0.080 0.171 0.079 0.185 0.011 0.010 0.014 0.012 0.013 0.012 0.011 0.012 0.012 0.012 0.005 1.944 22.391 1.872 40.996 3.241 40.650 3.236 40.597 3.137 0.095 0.010 0.094 0.010 0.095 0.010 1.394 0.537 2.012 1.523 1.085 1.123 10.374 3.658 18.549 2.876 11.954 8.183 1.380 0.549 2.036 1.531 1.078 1.108 19.213 29.495 70.883 51.905 68.033 53.545 1.719 0.878 2.243 2.397 1.194 1.455 19.197 29.502 71.016 51.891 68.006 53.593 1.718 0.878 2.241 2.397 1.194 1.455 19.376 27.337 68.278 52.035 67.034 52.715 1.737 0.870 2.257 2.378 1.198 1.446 0.117 0.061 0.209 0.060 0.213 0.176 0.013 0.004 0.017 0.010 0.009 0.010 0.117 0.060 0.207 0.060 0.214 0.176 0.013 0.004 0.017 0.010 0.010 0.010 0.112 0.063 0.207 0.058 0.203 0.175 0.013 0.005 0.017 0.010 0.009 0.010 1.770 14.563 1.774 60.869 2.536 60.903 2.537 60.358 2.535 0.158 0.011 0.158 0.011 0.156 0.011 1.703 1.651 1.680 1.111 1.619 1.807 1.726 1.280 1.481 1.670 1.191 1.386 1.430 7.415 1.737 48.110 2.479 48.143 2.472 46.256 2.470 0.100 0.012 0.100 0.012 0.094 0.012 14.661 14.636 2.526 7.510 15.346 18.626 10.589 13.191 8.790 18.171 11.818 15.778 1.653 1.713 1.115 1.649 1.812 1.703 1.279 1.474 1.663 1.204 1.398 1.419 41.982 51.391 29.865 42.139 54.928 38.699 45.144 61.420 33.947 48.762 30.521 23.721 2.660 2.967 1.229 2.436 2.369 2.595 1.927 2.073 2.271 2.052 2.009 2.232 41.429 51.249 29.836 41.965 55.122 38.396 45.182 61.404 33.926 48.663 30.524 23.647 2.656 2.953 1.228 2.432 2.367 2.594 1.927 2.072 2.271 2.053 2.009 2.225 39.954 53.001 28.278 37.954 50.762 36.888 44.830 60.511 32.909 46.969 30.682 22.676 2.577 2.934 1.262 2.429 2.339 2.477 1.925 2.071 2.294 2.049 1.962 2.146 0.078 0.094 0.117 0.097 0.159 0.115 0.115 0.178 0.131 0.115 0.153 0.097 0.011 0.011 0.011 0.013 0.013 0.012 0.011 0.012 0.013 0.009 0.012 0.011 0.078 0.093 0.118 0.096 0.158 0.115 0.115 0.178 0.132 0.115 0.153 0.097 0.012 0.011 0.011 0.013 0.014 0.013 0.011 0.012 0.013 0.010 0.012 0.011 0.068 0.093 0.115 0.098 0.158 0.127 0.114 0.182 0.123 0.114 0.149 0.105 0.012 0.011 0.011 0.014 0.014 0.013 0.011 0.012 0.013 0.010 0.012 0.012 95 Table 32. (cont’d) Country Manifest βxw Latent New Manifest βmw Latent New Manifest αw Latent New Est SE Est SE Est SE Est SE Est SE Est SE Est SE Est SE Est SE United Arab Emirates Poland Turkey Slovak Republic Tunisia Macao-China Hong Kong- China China- Shanghai Brazil Costa Rica Argentina Hungary Indonesia Romania Uruguay Vietnam Colombia Bulgaria Mexico Peru Thailand Chile 5.966 1.036 25.153 5.618 1.763 1.091 18.601 1.672 5.232 0.091 1.017 1.569 5.912 24.941 5.554 18.664 5.173 0.084 -1.629 1.586 -1.711 6.655 1.553 5.909 8.733 7.643 3.246 3.126 12.738 11.475 5.601 6.617 8.129 3.698 7.304 4.912 8.145 0.548 0.982 1.044 1.372 0.963 1.367 1.260 1.206 0.869 1.339 0.416 1.104 1.060 1.233 6.656 5.869 8.719 7.581 3.115 3.120 12.583 11.507 5.552 6.546 7.735 3.652 7.186 4.687 8.025 1.035 1.765 1.091 5.457 1.032 43.893 1.200 43.914 1.199 42.580 1.185 0.120 0.010 0.121 0.010 0.115 0.010 24.194 6.470 1.764 1.094 46.391 23.220 2.607 2.038 46.698 23.049 2.608 2.036 46.477 22.749 2.607 2.008 0.116 0.051 0.012 0.009 0.117 0.054 0.012 0.010 0.115 0.048 0.012 0.010 1.670 20.977 1.655 47.271 2.723 46.787 2.719 51.536 2.705 0.091 0.011 0.092 0.011 0.098 0.011 1.017 1.568 1.586 1.553 0.547 0.982 1.044 1.367 0.963 1.363 1.260 1.206 0.869 1.334 0.416 1.103 1.061 1.235 4.979 -0.186 1.011 1.559 6.537 67.258 1.840 2.251 6.617 67.320 1.840 2.251 6.607 67.798 1.834 2.276 0.061 0.097 0.010 0.012 0.060 0.097 0.011 0.012 0.067 0.112 0.011 0.011 -2.095 1.599 44.148 2.064 44.180 2.064 43.203 2.071 0.111 0.014 0.111 0.014 0.115 0.014 6.802 1.556 47.228 2.970 47.212 2.970 47.128 2.979 0.081 0.009 0.081 0.009 0.084 0.009 6.121 8.710 7.303 3.360 3.479 11.355 10.598 4.549 6.862 7.124 3.856 7.558 3.917 7.831 0.537 0.981 1.035 1.381 0.984 1.301 1.245 1.210 0.858 1.272 0.424 1.111 1.045 1.218 23.778 21.248 19.483 34.551 11.693 22.240 35.895 36.377 32.208 28.164 29.152 30.755 37.877 40.295 0.860 1.504 1.583 2.329 1.629 1.874 1.863 2.402 1.226 1.861 0.687 1.579 1.970 1.808 23.792 21.279 19.551 34.658 11.714 22.203 35.892 36.245 32.157 27.959 29.042 30.795 37.492 40.390 0.858 1.504 1.583 2.321 1.628 1.872 1.861 2.400 1.224 1.857 0.686 1.576 1.970 1.809 21.402 20.947 20.865 34.269 12.282 23.674 35.165 33.067 31.209 25.650 28.395 29.521 35.844 38.955 0.848 1.504 1.573 2.293 1.644 1.814 1.830 2.310 1.223 1.801 0.679 1.547 1.907 1.760 0.089 0.086 0.084 0.053 0.054 0.091 0.103 0.038 0.113 0.086 0.050 0.120 0.082 0.080 0.006 0.012 0.011 0.010 0.010 0.013 0.011 0.009 0.009 0.012 0.004 0.011 0.008 0.010 0.089 0.086 0.084 0.053 0.054 0.092 0.103 0.038 0.113 0.087 0.050 0.120 0.083 0.079 0.006 0.012 0.011 0.011 0.010 0.013 0.012 0.009 0.010 0.013 0.004 0.012 0.008 0.010 0.079 0.082 0.090 0.065 0.057 0.082 0.111 0.038 0.114 0.078 0.050 0.123 0.075 0.079 0.006 0.012 0.011 0.011 0.010 0.013 0.012 0.009 0.009 0.012 0.004 0.012 0.008 0.011 96 Table 33. Between effects from the manifest aggregation approach, latent aggregation approach, and the new approach with within- group fpc in the 1-1-1 mediation model Country Manifest Finland Liechtenstein Switzerland Iceland Sweden Canada Netherlands Perm United Kingdom Estonia Denmark Montenegro Jordan Ireland Korea Slovenia Serbia Singapore Croatia Spain Russian Federation Kazakhstan Italy New Zealand Japan Australia Belgium Chinese Taipei Czech Republic Lithuania Latvia Qatar Germany Israel France Malaysia United States Austria Portugal Luxembourg Greece Est 56.297 -4.679 64.155 46.385 60.129 36.932 23.330 70.378 SE 4.826 41.606 6.114 8.380 5.621 3.130 9.077 14.830 βxb Latent Est 68.628 -312.463 99.450 64.741 74.188 44.163 11.497 83.882 SE 7.108 1446.695 9.862 17.175 8.125 4.816 19.082 52.976 New Manifest Est 65.418 -7.124 88.714 54.820 68.527 41.200 16.429 89.118 SE 6.914 97.834 8.581 12.278 8.114 4.417 16.902 26.779 Est 29.314 123.303 33.133 13.734 13.194 71.391 146.101 84.779 SE 5.014 44.390 5.605 14.837 8.042 4.593 8.015 37.463 βmb Latent Est 25.028 344.393 20.344 25.776 5.803 76.243 167.820 179.470 SE 6.182 1010.804 6.883 45.983 14.734 7.655 14.760 206.123 New Manifest Est 26.313 133.734 21.060 25.673 8.671 74.292 162.204 114.576 SE 6.158 71.853 6.200 27.472 12.919 6.757 13.247 88.185 Est 0.085 0.535 0.438 0.209 0.291 0.327 0.728 0.203 SE 0.057 0.212 0.047 0.045 0.060 0.021 0.065 0.044 αb Latent Est 0.003 1.375 0.630 0.266 0.220 0.368 0.987 0.239 SE 0.084 0.338 0.072 0.067 0.068 0.027 0.086 0.056 New Est 0.011 1.136 0.567 0.234 0.179 0.350 0.935 0.219 SE 0.078 0.313 0.069 0.063 0.061 0.027 0.088 0.060 53.188 4.529 73.318 7.755 68.907 7.359 74.193 5.421 62.260 9.268 64.901 8.690 0.526 0.029 0.613 0.037 0.599 0.037 57.024 54.830 64.166 35.431 51.369 17.795 77.926 75.762 30.164 51.006 30.911 5.486 3.652 7.994 4.299 4.551 9.109 6.296 7.219 7.752 6.680 2.314 56.133 7.308 31.208 52.449 60.791 60.999 42.330 76.403 7.589 2.783 6.180 9.919 3.248 7.041 64.283 8.777 81.531 69.779 70.355 43.316 60.371 -9.115 110.702 86.863 9.077 43.514 35.237 71.221 26.775 65.489 87.195 67.778 49.040 99.199 55.682 8.106 6.437 10.397 5.474 7.453 17.592 16.866 13.424 16.353 11.429 3.718 70.658 60.973 72.506 43.855 57.532 -0.130 94.118 84.198 13.456 46.955 32.938 7.444 5.666 10.211 5.401 7.106 16.029 12.123 11.891 15.008 10.578 3.300 2.023 36.412 110.016 66.210 72.202 187.427 92.316 97.194 161.407 144.953 66.409 12.586 6.548 16.374 6.347 8.269 11.856 9.729 15.306 12.629 12.549 4.655 -56.256 18.773 135.965 75.207 69.638 236.253 83.270 111.814 214.019 190.575 63.358 25.165 13.916 24.862 8.454 16.209 23.101 27.036 34.480 28.518 24.663 8.865 -22.850 30.662 121.814 70.221 70.618 223.059 94.811 107.208 203.944 177.027 64.526 20.137 11.208 23.689 8.282 14.849 21.013 19.057 29.514 26.024 22.438 7.496 0.139 0.301 0.246 0.182 0.335 0.574 0.406 0.271 0.460 0.311 0.320 0.029 0.026 0.059 0.043 0.033 0.042 0.030 0.032 0.031 0.034 0.013 0.134 0.329 0.285 0.205 0.358 0.654 0.537 0.312 0.509 0.361 0.331 0.040 0.034 0.063 0.053 0.039 0.046 0.036 0.036 0.036 0.039 0.015 0.120 0.311 0.286 0.198 0.340 0.636 0.487 0.299 0.504 0.351 0.319 0.038 0.034 0.060 0.052 0.041 0.047 0.037 0.036 0.037 0.040 0.016 19.012 69.234 10.936 46.909 19.426 66.341 82.487 61.136 38.217 0.141 0.023 0.196 0.027 0.184 0.028 13.902 4.130 11.082 16.741 5.911 10.593 36.089 58.259 77.215 66.495 47.444 87.936 10.683 3.807 10.281 15.878 5.584 9.680 64.631 88.979 67.312 155.400 91.687 71.283 13.676 3.799 7.811 13.481 3.835 8.065 108.510 88.079 42.971 160.941 96.488 55.595 31.899 5.157 14.475 22.275 7.243 12.015 80.174 88.215 54.920 160.304 95.590 61.990 22.022 4.734 13.065 21.238 6.730 10.715 0.280 0.395 0.443 0.556 0.552 0.610 0.033 0.017 0.048 0.035 0.023 0.039 0.333 0.511 0.576 0.627 0.638 0.677 0.039 0.022 0.058 0.040 0.030 0.044 0.320 0.474 0.569 0.606 0.621 0.641 0.039 0.022 0.059 0.042 0.030 0.046 14.082 57.703 13.662 177.670 14.494 213.541 24.657 207.004 23.789 0.451 0.032 0.485 0.036 0.474 0.037 73.126 6.534 110.529 12.344 90.414 10.201 99.940 8.243 77.361 14.588 91.465 12.089 0.463 0.037 0.636 0.047 0.599 0.045 11.546 10.434 8.111 11.152 10.009 13.800 7.383 6.636 14.331 6.493 14.928 11.584 62.572 60.841 36.470 50.933 120.648 54.643 23.616 39.850 30.974 38.845 73.551 47.068 8.370 7.829 7.342 10.042 9.922 12.093 7.135 6.406 12.761 5.613 11.960 8.048 80.151 54.934 110.649 101.002 50.129 126.503 118.274 87.540 89.534 98.447 33.826 89.583 12.910 12.134 7.958 6.857 11.841 9.269 10.811 10.558 9.633 11.524 21.144 12.943 144.836 32.120 128.032 106.692 29.167 151.510 161.027 104.965 94.945 140.660 19.573 189.308 39.562 31.840 9.393 13.599 16.520 18.113 20.711 17.606 13.445 23.042 30.779 41.271 105.968 49.081 131.178 106.492 26.745 142.478 154.183 100.782 91.243 121.186 27.603 124.920 24.805 21.906 7.830 12.263 15.736 16.036 19.694 16.175 12.117 18.770 25.608 25.798 0.190 0.198 0.499 0.527 0.307 0.551 0.267 0.275 0.708 0.199 0.444 0.221 0.024 0.026 0.064 0.039 0.045 0.035 0.027 0.032 0.049 0.023 0.042 0.024 0.242 0.259 0.507 0.677 0.339 0.649 0.284 0.260 0.831 0.213 0.444 0.246 0.030 0.030 0.069 0.047 0.052 0.040 0.030 0.035 0.060 0.024 0.042 0.027 0.231 0.238 0.406 0.643 0.312 0.611 0.281 0.262 0.782 0.207 0.407 0.230 0.031 0.031 0.074 0.048 0.052 0.042 0.030 0.035 0.062 0.025 0.042 0.028 59.919 51.631 37.188 47.214 104.104 57.725 30.509 40.522 30.971 43.121 65.329 48.731 5.235 4.982 7.221 5.384 7.818 7.030 4.671 4.938 9.446 4.225 10.797 5.067 59.659 69.826 36.247 53.303 126.564 52.163 21.887 40.258 31.432 37.906 75.126 35.497 97 Table 33. (cont’d) Country Manifest βxb Latent New Manifest βmb Latent New Manifest αb Latent New Est SE Est SE Est SE Est SE Est SE Est SE Est SE Est SE Est SE United Arab Emirates Poland Turkey Slovak Republic Tunisia Macao-China Hong Kong- China China- Shanghai Brazil Costa Rica Argentina Hungary Indonesia Romania Uruguay Vietnam Colombia Bulgaria Mexico Peru Thailand Chile 53.275 4.521 49.281 48.158 4.912 7.421 71.581 6.685 39.870 31.795 4.945 7.373 44.979 6.495 62.762 6.756 19.233 24.261 40.330 59.764 16.568 46.808 33.959 26.957 29.301 41.961 19.788 31.589 24.121 23.943 2.000 3.183 4.119 5.716 4.348 5.324 3.840 4.551 2.589 5.713 1.098 3.320 4.649 3.192 61.956 52.826 40.780 80.681 41.970 33.210 42.261 62.049 13.684 21.114 45.845 73.098 11.031 54.227 29.227 18.181 29.375 44.072 17.640 27.475 11.559 17.372 5.615 59.203 5.272 82.157 5.979 90.569 8.149 84.757 7.660 0.367 0.032 0.370 0.037 0.311 0.036 7.068 11.553 52.202 48.990 6.525 10.351 85.457 104.595 12.669 16.659 104.034 182.682 25.571 33.343 92.659 165.921 21.343 27.912 0.189 0.314 0.025 0.026 0.184 0.292 0.030 0.027 0.179 0.280 0.031 0.028 12.528 73.765 10.312 87.618 11.209 91.043 20.777 98.939 16.637 0.425 0.028 0.506 0.032 0.502 0.033 7.051 8.166 44.289 32.627 6.253 8.824 65.255 191.109 18.937 15.733 104.154 201.737 44.163 17.801 81.675 182.600 35.288 18.333 0.094 0.005 0.020 0.072 0.106 0.020 0.022 0.076 0.103 0.064 0.022 0.076 8.578 44.920 8.275 143.936 16.758 217.395 33.520 196.147 30.455 0.148 0.030 0.151 0.032 0.151 0.033 9.087 63.199 8.842 142.056 19.098 169.247 28.791 160.766 27.337 0.229 0.022 0.242 0.024 0.239 0.024 3.102 5.228 5.704 10.575 5.988 7.493 5.465 6.038 3.918 11.114 1.476 6.277 7.801 5.216 16.353 22.668 44.285 69.204 15.301 54.022 30.803 21.403 31.063 49.931 19.227 31.442 24.842 21.947 2.823 4.701 5.208 8.885 5.560 7.131 5.039 5.699 3.600 8.889 1.384 5.153 5.984 4.582 83.610 59.084 49.165 106.984 88.272 58.847 85.454 171.811 58.393 83.226 71.505 70.391 100.609 101.163 3.626 7.165 7.694 10.952 11.652 10.135 7.726 14.184 5.572 12.090 2.929 7.932 13.984 9.291 105.028 77.123 46.268 102.494 123.218 62.770 105.348 236.760 70.436 113.449 95.337 93.420 173.369 125.059 5.913 13.331 11.167 22.070 18.532 14.849 12.352 22.622 9.768 26.734 4.721 17.067 28.330 16.072 96.492 69.146 44.105 103.659 107.261 60.477 98.388 212.913 65.725 99.236 85.793 80.156 138.000 107.677 5.380 11.685 10.013 18.581 16.160 13.616 11.332 21.375 8.503 20.583 4.156 13.495 20.628 13.999 0.380 0.289 0.361 0.397 0.193 0.333 0.374 0.183 0.286 0.332 0.196 0.307 0.240 0.284 0.014 0.024 0.026 0.026 0.022 0.028 0.027 0.021 0.020 0.023 0.009 0.019 0.016 0.013 0.423 0.319 0.391 0.431 0.218 0.374 0.374 0.194 0.315 0.378 0.211 0.330 0.248 0.301 0.016 0.028 0.029 0.027 0.024 0.035 0.028 0.022 0.021 0.025 0.009 0.020 0.014 0.013 0.406 0.310 0.377 0.391 0.220 0.369 0.350 0.179 0.316 0.359 0.205 0.326 0.248 0.299 0.016 0.029 0.030 0.028 0.025 0.036 0.028 0.022 0.022 0.027 0.009 0.020 0.014 0.013 98 Chapter 7 Discussion School contextual effects or compositional effects beyond individual effects are of interest in education research. The effects of school-level SES, percent of students eligible for free or reduced-price lunch (FRPL), percent of girls (or boys), and percent of minorities on student learning or development have previously been studied. These school-level compositions are often aggregated from individual data in the sample. To examine the effects of aggregated school compositions on individual outcomes controlling for interindividual differences, and understand the underlying school-level processes, traditionally, manifest aggregation approach is used. In the manifest aggregation approach, the observed group means are used to represent the group compositions. Recently, there is a new trend to adopt the latent aggregation approach in the decomposition of between and within effects, in which the sampling error in aggregation is corrected. There are statistical assumptions about the constructs to be decomposed, the population of research interest, and the sampling procedures used for data collection in both manifest and latent aggregation approaches. However, little attention was given to these assumptions when choosing the analysis approach in the applications. The current study focused on the decomposition of school-level or group-level compositional effects and individual effects based on the individual data in the sample. To resemble the data structure typically found in education research, an extremely large number of groups with small to moderate group sizes was assumed in the population, and a two-stage cluster sampling design with equal selection probability at each sampling stage was assumed to be used for data collection. A new approach was created to deal with the within-group finite population selection problem in the sampling error correction in aggregation. The performances of the manifest aggregation approach, latent aggregation approach, and new approach with 99 within-group fpc were compared in terms of the decomposition of between and within effects in the MLM, 2-1-1 mediation, and 1-1-1 mediation models. From the mathematical derivations, the within effect estimators from the three analysis approaches were unbiased under the assumptions about the population and sampling procedures made in the current study. The between effect estimators in the manifest aggregation approach were biased, as the sampling error in aggregation was not corrected at all. The between effect estimators in the latent aggregation approach were also biased, as the sampling error was overcorrected by assuming an infinite group size. The results from the simulation study confirmed the mathematical derivations. The within effect estimates from the three analysis approaches showed little difference in relative biases, absolute relative biases, and RMSE. The only non-negligible difference among the three analysis approaches was found in the coverage rates for within effects under the unbalanced design. The coverage rates from the new approach were lower than the nominal level, and worse than the ones from the manifest and latent aggregation approaches. The low coverage rates were related to the underestimation of standard errors for the within effects in the new approach, which were caused by the MUML estimation it used. The MUML treated the unbalanced data as if it was the balanced case, and in principal, would perform worse under the unbalanced design. As the accuracy of the within effect estimates in the new approach was not affected, and comparable to those from the manifest and latent aggregation approaches, adjustments of standard errors for the within effects may help with the performance of the new approach under the unbalanced design. For the between effects, large differences were found among the three analysis approaches. The new approach with within-group fpc outperformed the manifest and latent aggregation approaches in terms of biases, relative biases, and observed coverage rates for the 100 between effects. For the point estimates of between effects, the manifest aggregation approach generally underestimated the between effects, while the latent aggregation approach overestimated the between effects in the simulation. The empirical study also found similar results, i.e., the between effect estimates from the latent aggregation approach were the largest among the three analysis approaches for most countries, those from the manifest aggregation approach were the smallest, while the between effect estimates from the new approach were in the middle. These results were consistent with the mathematical derivations. The manifest aggregation approach had a better performance than the other two approaches in the absolute biases, absolute relative biases, and RMSE for the between effects. However, the manifest aggregation approach showed unacceptably low coverage rates for the between effects. The coverage rates for between effects in the manifest aggregation approach were generally lower than 90%. As discussed before, under conditions in which the sample size within each group was small (i.e., small group size and small within-group sampling ratio), the coverage rates for between effects in the manifest aggregation approach were even close or equal to 0. Based on these results, the new approach with within-group fpc was preferred, as it provided more accurate parameter and standard error estimates for between effects than the other two approaches, although it was less efficient than the manifest aggregation approach. The results in the current study also indicated that, to examine the group compositional effects (or the formative group-level constructs) under the conditions with small to moderate group sizes and moderate to large within-group sampling ratios, the latent aggregation approach might not be a good choice. It performed even worse than the manifest aggregation approach. Different results on between effect estimates were found in previous studies, which generally 101 favored the latent aggregation approach over the manifest aggregation approach (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011). These studies paid attention to the reflective group-level constructs, or formative group-level constructs under the conditions with large group sizes and extremely small within-group sampling ratios. Under these conditions, the latent aggregation approach outperformed the manifest aggregation approach in terms of biases in between effect estimates. In contrast, the current study focused on the formative group-level constructs under the conditions with small to moderate group sizes and moderate to large within- group sampling ratios. These assumptions about population and sampling design were more reasonable for the education research on school compositional effects, than the assumptions of an infinite group size and an extremely small within-group sampling ratio. The simulation settings in the current study tended to favor the manifest aggregation approach more in principal. As expected, the latent aggregation approach performed worse than the new approach with within- group fpc, and even worse than the manifest aggregation approach for the between effect estimation in the current study. Lüedtke et al. (2011) also mentioned that when the within-group sampling ratio was 100%, the manifest aggregation approach would perform better than the latent aggregation approach, and the finite sampling correction was needed when a moderate sampling ratio was used in their study. Lüedtke et al. (2008) found even under the suitable population and sampling design assumptions for the latent aggregation approach, the estimates from the latent aggregation approach were more variable than those from the manifest aggregation approach, especially under the conditions with small ICCs, small numbers of groups, and small group sizes. Similar results were also found in the current study. The between effect estimates from the latent 102 aggregation approach showed a larger variability than those from the new approach and manifest aggregation approach. Thus, it is reasonable that the latent aggregation approach performed the worst among the three analysis approaches for the between effects in the current study, which seemed to be completely opposed to the conclusions made in the previous studies (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011). The different results in the current study and previous studies originated from the assumptions about the nature of group-level constructs, population of interests, and sampling procedures. An additional piece of simulation was conducted for the MLM model in the current study. A moderate group size (i.e., N=100) and an extremely small within-group sampling ratio (i.e., r=.02) were used for data generation, which generally favored the latent aggregation approach. Similar to the results from previous studies (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011), the latent aggregation approach outperformed the manifest aggregation approach in term of biases in between effect estimates under this condition, and the new approach provided similar between effect estimates as those from the latent aggregation approach. All these results suggested that it was necessary to consider the assumptions about the population and sampling design, as well as the natures of group-level constructs in the decomposition of between and within effects in contextual models. When the within-group sampling ratio is extremely small (e.g., smaller than 5%) or the within-group population can be assumed to be infinite, the latent aggregation approach is a good choice. When the within-group sampling ratio is extremely large (e.g., close to 100%), the manifest aggregation approach can be used to separate the between and within effects. For the education research in which the within- group sampling ratio is usually moderate, the finite population selection need to be considered in 103 the sampling error correction. The new approach with within-group fpc provides an additional choice to estimate the group compositional effects under this condition. In addition, the current study also indicated that with a larger sample size (i.e., a larger number of groups in the sample, a larger group size, and a larger within-group sampling ratio), the within effect estimates from the three analysis approaches showed less variability. With a larger group size in the sample and a larger ICC of the predictor or mediator in the model, the performances of the three analysis approaches became better in terms of higher accuracies of between effect estimates and standard errors, and less variable between effect estimates. With a larger within-group sampling ratio, the manifest aggregation approach and the new approach performed better, while the latent aggregation approach performed worse. It should be noted that the new approach with within-group fpc used the MUML estimation, which was based on the pooled within and between variance-covariance matrices. Under an unbalanced design, it treated the data as if it was from a balanced design. In principal, it worked fine for the balanced or approximately balanced datasets. Under the unbalanced design, the estimation of standard errors for the within effects was affected. Clustering standard errors and bootstrapping of standard errors may help with the estimation of standard errors for the within effects in the new approach, if MUML estimation is still applied. To completely solve this problem, the development of full information maximum likelihood estimation which incorporates different sampling features is necessary for the new approach. 7.1 Implications The current study developed a new analysis approach with within-group fpc for the decomposition of between and within effects, which considered the finite population selection issue in the sampling error correction for aggregation. The new approach was compared to the 104 manifest and latent aggregation approaches using mathematical derivation, simulation, and empirical study. The results from the three pieces of studies were consistent. They emphasized the importance of considering the natures of the decomposed constructs, as well as the reasonable assumptions about the population and sampling procedures for the decomposition of between and within effects. For the reflective group-level constructs, or under the conditions where the population size within each group can be assumed to be infinite, or under the conditions where the within- group sampling ratio is extremely low (e.g., smaller than 5%), the latent aggregation approach is a good choice in the decomposition of between and within effects. For the formative group-level constructs which are measured at the group level without sampling error, or for the formative group-level constructs under the conditions with an extremely large within-group sampling ratio (e.g., close to 100%), the manifest aggregation approach works better than the other analysis approaches. For the formative group-level constructs under the conditions with a large number of groups and small to moderate group sizes in the population, and a moderate to large within-group sampling ratio, the new approach with within-group fpc provides another choice to separate the between and within effects. It works better than the other two approaches in the parameter and standard error estimation for the between effects. Under the balanced or approximately balanced design, the parameter and standard error estimates for the within effects from the new approach are similar as those from the manifest and latent aggregation approaches. Under the unbalanced design, the standard errors of the within effects from the new approach with within-group fpc are underestimated. 105 With a larger sample size, i.e., a larger number of groups, a larger group size, and a larger within-group sampling ratio, the manifest aggregation approach and the new approach will perform better in terms of the accuracies of parameter and standard error estimation, and variability of the between and within effect estimates. The latent aggregation approach will perform better with a larger number of groups, a larger group size, and a smaller within-group sampling ratio. 7.2 Limitations and Future Study In the current study, the between and within effects of the decomposed predictors and/or mediators were examined for three particular two-level models, i.e., the MLM model, 2-1-1 mediation model, and 1-1-1 mediation model, under certain assumptions about the constructs of research interest, populations of subjects, sampling procedures, and variables used in these models. The results from the mathematical derivation and simulation could only be generated to the same or similar conditions. There were several limitations that need to mention. First, the current study focused on the group compositional effects. The group compositions are usually formative constructs, like school SES, gender ratio, ethnicity compositions, etc. in the education research. To make the conditions close to the ones in education research, the number of groups was assumed to be infinite in the population, while the population within each group was assumed to be finite and of small to moderate size. A moderate to large within-group sampling ratio was assumed to be used for data collection. Under these conditions, the challenge in estimating the group compositional effects was to correctly deal with the sampling error in the aggregation. Only to the conditions with similar population and sampling procedures, and for the formative group-level constructs, the results from the 106 current study can be generalized. More importantly, it is necessary to have the within-group sampling ratio in the dataset, in order to use the new approach with within-group fpc. The current study did not cover the conditions which theoretically favored either manifest or latent aggregation approach. As discussed before, for the formative group-level constructs, when the entire groups are drawn, the manifest aggregation approach is assumed to perform better than the other two approaches. For the reflective group-level constructs under the conditions in which the within-group population can be assumed to be infinite, or for the formative group-level constructs under the conditions with an extremely small within-group sampling ratio (e.g., smaller than 5%), the latent aggregation approach is assumed to be a good choice based on previous studies (Lüedtke et al., 2011; Lüedtke et al., 2008; Preacher et al., 2011). It deserves further simulations to examine the performance of the new approach under these conditions, and compare it to the manifest and latent aggregation approaches. Second, all models were correctly specified in the three analysis approaches in the current simulation study. There was no model misspecification problem involved. The models in the three analysis approaches correctly described the data structures in the population. It is unclear whether the three analysis approaches are sensitive to model misspecifications under different settings, or how they will perform under different types of model misspecifications. The impact of model misspecification needs further examinations, no matter which analysis approach is adopted to decompose the between and within effects. In the current study, there is no random slope or cross-level interaction included in the MLM, 2-1-1 mediation, and 1-1-1 mediation models. In other words, the within variance- covariance matrix was assumed homogeneous across all groups in the population. The results in the current study only provided some information for the studies with the same or similar 107 assumptions. For the manifest aggregation approach, the random slopes and cross-level interactions can be included in the models following the traditional multilevel modeling strategy (Raudenbush & Bryk, 2002). Marsh et al. (2009), Preacher et al. (2010), and Preacher et al. (2011) showed the possibilities to include random slopes and cross-level interactions for the latent aggregation approach. However, when the within effects of the decomposed variables are of random slopes, or there are cross-level interactions involved, the new approach with within- group fpc in the current study cannot be applied. Further work is needed to incorporate the random slopes and cross-level interactions into the new approach. A comparison of the three analysis approaches in the estimation of the random slopes and cross-level interactions is also necessary. Furthermore, when the between and within effects of the decomposed variables, as well as the random slopes and cross-level interactions of the within components are all included in the models, it is necessary to reconsider the meaning of these estimates in the applications, and whether these analyses and estimation approaches are reasonable for the research questions. There were seven design factors in the current simulation. As the research focus is on the performances of different analysis approaches under different within-group sampling ratios, the other design factors only had two levels in the simulation. To be specific, only large and small between-to-within-effect ratios (RX/RM =.10, 10), small to moderate ICC of predictors or mediators (ICCX/ICCM =.05, .25), small to moderate number of groups in the sample (g =50, 200), balanced or moderate unbalanced design (size ratio of large to small groups = 1:1, 3:1), and small to moderate group size (N=50, 200) were included in the simulation. The simulation results can be used as a reference for the studies conducted under the same or similar conditions. Third, all variables used in the simulation were generated from multivariate normal distributions. In practice, the group compositions in education research also include school-level 108 percentages of different types of students, like percent of students eligible for FRPL, gender ratio, and ethnicity compositions in a school. At the individual level, these characteristics usually follow Bernoulli distributions, while at the school level, the group compositions follow binomial distributions. The outcomes may also follow distributions other than the multivariate normal distribution. With different assumptions about the distributions of these variables used in the model, different considerations may be given and different results may come out. In the future study, it is also necessary to consider how to decompose the between and within effects for the variables following distributions other than multivariate normal distribution, with a correction for the sampling errors in the aggregation. In addition, the standard errors for the within effects from the new approach were underestimated under the unbalanced design in the current study. Further simulation study is needed to explore if clustering standard errors and bootstrapping of standard errors help with the estimation of standard errors for the within effects in the new approach, if MUML estimation is still applied. To completely solve this problem, a full information maximum likelihood estimation with within-group fpc is a choice. Future study can work on the development of the estimation method which fully incorporates different sampling design features into the estimation for the multilevel models. Moreover, the within-group sampling ratio is necessary to use the new approach with within-group fpc. As shown in the empirical study, many large-scale datasets in education provide the possibilities. However, different from the current study which assumed a two-stage cluster sampling design with equal selection probability at each stage, the selection probability is not the same either across groups or across individuals in these large-scale datasets. Sampling weights need to be incorporated into estimation for all three analysis approaches. Previous 109 studies indicated that multilevel psedo-maximum likelihood estimation (MPML) provides one way to incorporate the group-level and individual-level sampling weights into multilevel models (T. Asparouhov, 2004; T. Asparouhov, 2006; Rabe-Hesketh & Skrondal, 2006). This estimation method can be adapted and applied to the manifest and latent aggregation approaches. For the new approach with within-group fpc, further studies can work on the incorporation of two-level sampling weights into estimation with the adjustment of finite population selection in the sampling error correction in aggregation. Furthermore, comparisons of the three analysis approaches with different weighting procedures in the decomposition of between and within effects are also of research interest. 110 APPENDICES 111 APPENDIX A: The Between and Within Effects in the Manifest and Latent Aggregation Approaches under the Assumption of an Infinite Group Size In this section, the population models, assumptions, parameters of theoretical interest, between and within variance-covariance matrices, and derivations of between and within effect estimators in the manifest and latent aggregation approaches were presented for the MLM, 2-1-1 mediation, and 1-1-1 mediation models under the assumption of an infinite group size made in the latent aggregation approach. The ML should be used in the manifest and latent aggregation approaches to estimate the between and within effects, but an iterative procedure such as EM or Fisher scoring algorithm is often required. To directly show the between and within effect estimators in the manifest and latent aggregation approaches, following the procedures used by Lüedtke et al. (2008) and Preacher et al. (2010), an OLS was used in the manifest aggregation approach. In the latent aggregation approach, a MOM was firstly used to decompose the between and within variance- covariance matrices (i.e., W   PWS and    B ( S B  S PW ) / n ), and the OLS was used to estimate the between and within effects based on the estimated W and  B 6. A two-stage cluster sampling design was assumed. First, a simple random sample of g groups was selected from G groups, and each group was of the same size N. Then, a simple random sample of size n was selected within each group. Assuming N was infinite. 1. MLM model (1) Population models X ij  X  U Xij  U Xj (38)                                                              6 The OLS estimates in the manifest and latent aggregation approaches were not the same as the ML estimates, and were not used for evaluation in the simulation study. The purpose of this derivation was to directly show the between and within estimators in the manifest and latent aggregation approaches.   112 Y ij  (2) Assumptions    00 U xw Xij  U  xb Xj     ij 0 j (39)  υ0j and εij are independently and identically distributed across i and j,   ij    0 j    ~ MVN 0     0    , 2   Ye  0  .        2  Ye  υ0j and εij are uncorrelated with UXij and UXj.  The between and within random components of X and Y follow multivariate normal distributions: U U    Yij Xij    ~ MVN 0     0    , 2   Y  2    X XY        , and U U    Yj Xj    ~ MVN 0     0    , 2   Y  2    X XY        .  Cov(UXij, UXj)= Cov(UYij, UYj)=Cov(UXij, UYj)=Cov(UYij, UXj)=0. (3) The interested between and within effects of X are:  xb   xw  ( , Cov U Y ij Xj ) ( Var U Xj ( , Cov U Y ij Xij ) ( Var U Xij ) )   XY 2  X   XY 2  X (4) The between and within variance-covariance matrices, ICC of Y, and proportions of variances explained at two levels are: r   xw  xb / WΣ  Cov U U    Yij Xij    2 (1   xw      xw ( ICC Var X (1 ) )  X ( ICC Var X  Var ) ) ij X ij (  ij ) (1  ICC Var X ( ) X    ) ij BΣ  Cov r 2 2  xw U U    Yj Xj        ( ICC Var X r  xw )  X ( ICC Var X Var ) X ij ij (  0 j ) ICC Var X ( X    ) ij 113 (40) (41) (42) (43) (44)   ICC Y  2    xw (1  r ( ICC Var X ) ICC Var X 2 2  X xw ( ) Var    ij ) ij X (   ) r ij   ) Var  2 2  xw (  0 ICC Var X ( X j (45) )  Var (  0 j ij )   R 2 1  2  xw  (1 ) 2  xw (1 ( ICC Var X  ( ICC Var X )  ) ij Var ) X X ij (  ij ) R 2 2  r 2 2  xw 2 2  xw r X ( ICC Var X ( ICC Var X  ) ij Var ) X ij (  0 j ) (5) The variance-covariance matrices used in estimation are: Cov X      Y ij  X j ij X j              1 n 2   Y 2 Y (1  )  XY  1 n   XY  XY (1  ) 2  X 1 n 0 ( E S PW ) 2   Y   2    X XY    ( E S ) B 2 Y 2    Y   2      X   n n  n XY XY 2 X        2   X  2 X 1 n    (6) Manifest aggregation approach  Model: Y ij    xw  00 ( X ij  X ) j   xb X j  u 0 j  e ij  Expectations of estimators: E  (  xb )  E [ 1 -1 ng n ng  n  g j i ( g  j 1 ( Y Y X X ij )( - - j ) ]  X X X X )( - - ) j j  XY    xb XY  2   X  2 X 1 n 1 n ICC X  ICC X  1  xw n 1 (1  n (1  ICC ) X ICC X ) (46) (47) (48) (49) (50) (51) (52) 114   E  (  xw )  [ E 1 gn  1 1 1 gn  g n  j i g n  i j (X - X ij (Y -Y )(X - X ij ij  X - X ) j ij j  j X - X ) X - X ( ij ij j  X - X ) j ij j ]  (1  (1  1 )  XY n 1 n ) 2  X   xw (53) (7) Latent aggregation approach  Models: X ij  X  U Xij  U Xj Y ij     00 U xw Xij  U  xb Xj  u 0 j  e ij  Expectations of estimators: - X )  - X )  j j 1 ( g n  1 ( g n  1) 1) g n  i n i  j g j (Y -Y )(X - X ) n ] / ij j ij j (X - X )(X - X ) n ] / ij j ij j E  (  xb )  [ g { E [  XY   1  XY n 1 n 2 X 2    X   2 X ( Y Y X j )( - ( X j - X X )(   xb  g n -1  g j j g n 1  1 n 1 n   XY 1 ( g n  1 ( g n  g n j  1)  n g i j i 1) (Y -Y )(X - X ) j ij ij j (X - X )(X - X ) j ij ij j ]   XY 2  X   xw E  (  xw )  [ E 2. 2-1-1 mediation model (1) Population models X j U  X Xj M ij  M  U Mij  U Mj Y ij     00 U mw Mij    ij  0 j   U mb Mj  U  xb Xj 115 (54) (55) } (56) (57) (58) (59) (60)   U Mj  (2) Assumptions U  0  Xj b j (61)  υ0j, εij, and ω0j are independently and identically distributed across i and j,   ij    0 j    0 j      ~ MVN 0   0    0        , 2   Ye  0   0            . 2  Ye 0 2  Me  υ0j, εij, and ω0j are uncorrelated with UXj.  υ0j and εij are uncorrelated with UMj and UMij.  The between and within random components of X, M, and Y follow multivariate normal distributions:      U U U Yij Mij Xj      ~ MVN 0   ,0    0        2   Y  2    M  0  MY 0           0 , and U U U      Yj Mj Xj      ~ MVN 0   ,0    0        2   Y  2    M MY  2     X MX XY           .  Cov(UMij, UMj)= Cov(UYij, UYj)=Cov(UMij, UYj)=Cov(UYij, UMj)=Cov(UMij, UXj)=Cov(UYij, UXj)=0. (3) The between and within effects of M on Y are of interest in this model:    mb       2 Y XjMj   MjY 1 XjY 2  XjMj 2  Y 2  M  MY   2  M   XY XM 2  X 2  XM 2  X   mw   MY 2  M 116 (62) (63)   BΣ  U Cov U U      Yj Mj Xj                mw xb b   b ij 2 r ( Var X     mw     r xb b    mw xb b 2  r ) Var   mw  ) ( Var X r  mw  r ) Var X   ( ij ij j (  0 Var ICC Y   2   mw  1 ICC  ICC M 2  Var X (      r xb b mw M [ Var X 2  b ( ij )  Var (  0 j )]  Var (  ij ) (  0 j ) )  (  0 Var ) j ) 2 ( Var X   b ( Var X  b Var ) ij ij (  0 j ) Var X ( ) ij      (66) 2  Var (  0 j )  Var (  0 j )     r xb b mw 2  Var X ( )  ij  r  mw 2  Var (  0 j )  Var (  0 j )   mw ij  ) r          (64)          0 (65) (67) (68) (69) (4) The between and within variance-covariance matrices, ICC of Y, and proportions of variances explained at two levels are: r   /m b  m w M 2 [ Var X  b ( ij )  Var (  0 j )]  Var (  ij ) WΣ  U   Cov U  0  Yij Mij                1 2  mw ICC  ICC 1  mw ICC M  ICC M 2 [ Var X  b ( ij )  Var (  0 j )] 1 ICC  ICC M 2 [ Var X  b ( ij )  Var (  0 j )] M 0 M 0 1 2  mw 1 ICC  ICC M M ICC  ICC M [ Var X 2 M  b ( R 2 1  2  mw [ Var X 2  b ( )  Var (  0 j )] ij )  Var (  0 j ij )]  Var (  ij )     mw    mw Var X xb r r 2  ( Var X      ( ) xb ij b b 2 2 )  ij r  mw   Var r  mw 2  ( ) Var  0 j ( )  0 j Var  (  0 j ) R 2 2   (5) The variance-covariance matrices used in estimation are: 117   Cov M M        Y ij  ij M X j j           j        1 n 2   Y 2 Y (1  )  MY  1 n   MY  MY  XY (1  ) 2  M 1 n 0 0 1 n 2   M 2 M   XM 2  X ( E S PW ) 2   Y  2     M  0  MY 0      0 ( E S B )  2 Y n n 2    Y  2      M     MY n  XY 2  M n  n XM MY      n 2  X (6) Manifest aggregation approach  Models: Y ij    mw  00 ( M M  ij ) j   mb M j   xb X j  u 0 j  e ij M ij  Expectations of estimators:   b  00 X w 0  j  d ij j E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 Y S 2 Mj ]          (70) (71) (72) (73) (74) (75) r MjY  [ n ng  1 g  j ( 1 -1 ng  n i g j ( Y Y M M ij )( - - j M M M M )( - - j j )][ 1 -1 ng  n i g j ) (76) ] (Y -Y )(Y -Y ) ij ij (77) ] (Y -Y )(Y -Y ) ij ij r XjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j  n i g j ( X j - X X )( j - X )][ 1 -1 ng 118   r XjMj  [ n ng  1 n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j )] (78) (79) (80) S 2 Y  1 -1 ng S 2 Mj  n -1 ng  n i g j (Y -Y )(Y -Y ) ij ij g  j ( M M M M )( - - j j )  XM E  (  mb )    MY 1 n 2 M  MY 1 n  [ 2     Y ][ 2 M  2 Y  XY ][ 2 Y [ 2    Y  2 X ]  ] 1  2  XM (1 [ 2      M ][ 2 M  2 X 2  M ] [ 2    M ][ 2 M  2 X 1 n ] 2 2    Y Y 1 n 2   M 2 M    MY  1 n  MY  2   M 2 M  1 n   XY XM 2  X 2  XM 2  X  2   mb M (   ( 2  M  1 n 2  XM 2  X 2  XM 2  X 2   M 1 1 ) n n 1 mw n 21  Mn )  )   (  mw E )  E [ 1 gn  1  n i g j 1 1 gn  n  g j i ij j (Y - Y )(M - M M - M )  ij j ij j (M - M M - M ) M - M M - X )   ( j ij j ij j ij ij ]  (1  (1  1 n 1 n )  MY ) 2  M (7) Latent aggregation approach  Models: M ij  M  U Mij  U Mj X j U  X Xj Y ij     00 U mw Mij   U mb Mj  U  xb Xj  u 0 j  e ij 119 (81)   mw (82) (83) (84)     U Mj U  b Xj  w 0 j  Expectations of estimators: E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 YB S 2 Mj ] g  j ( Y Y M M j )( - - j ) M M M M )( - - )  j j n -1 g g  ( j r MjY  {[ n  1 ({[ {[ g n  g (Y - Y )(M - M ) n ] / } / ij j j ij ij j j ] / } (M - M )(M - M ) n 1/2 ij g  n i j n 1) g  j i  1 ( g n  1 ( g n  g 1) n  j i 1) g  j 1 ( Y Y Y Y j )( - - j )  1 ( g n  (Y - Y )(Y - Y ) n ij ij j j 1/2 ] / } ) (85) (86)  (87) g  j ( Y Y X j )( - - X ) 0] / }/  n j X - X X )( j j - X ) 0] / } n 1/2   (88) g  j 1 ( - Y Y Y Y j )( - j )  1 ( g n  1)  n i g j (Y -Y )(Y -Y ) n ij j ij j 1/2 ] / } ) n -1 g g  ( j r XjY  {[ n  1 ({[ {[ g n  g n -1 g g  ( r XjMj  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( M M X )( - j - X ) 0] / } /  n j X j - X X )( j - X ) 0] / } n  1/2  (89) ( M M M M )( - - j j )  1 ( g n  1)  n i g j (M - M )(M - M ) n ] / } ) 1/2 j ij ij j   g j g j n i n i (Y - Y )(Y - Y ) j ij ij j ] / n (90) (M - M )(M - M ) j ij ij j ] / n (91) S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j S 2 Mj  [ n -1 g g  j ( M M M M )( - - j j )  1 ( g n  1) )  1 ( g n  1) 120    XY XM  MY 2 2 2       M Y Y 2  1 M   2 2 2 X M Y 2  XM 2 2  X M   MY   2  M   XY XM 2  X 2  XM 2  X    mb (92) E  (  mb )  E  (  mw )  E [ (93) (94) (95) (96) (97) (98) 1 ( g n  1 ( g n  1) g n  i j 1)  n g j i (Y -Y )(M - M ) j ij ij j (M - M )(M - M ) j ij ij j ]   MY 2  M   mw 3. 1-1-1 mediation model (1) Population models X ij  X  U Xij  U Xj M ij  M  U Mij  U Mj Y ij     00 U mw Mij   U mb Mj   U xw Xij  U  xb Xj U Mj  U  0  Xj b j    ij  0 j U Mij  U w Xij   ij (2) Assumptions  υ0j, εij, ω0j, and δij are independently and identically distributed across i and j, j   ij    0   ij     0 j        ~ MVN 0   0  0   0          , 2   Ye  0   0  0  2  Ye 0 0 .               2  Me 0 2  Me  υ0j, εij, ω0j, and δij are uncorrelated with UXj and UXij.  υ0j and εij are uncorrelated with UMj and UMij. 121    The between and within random components of X, M, and Y follow multivariate normal distributions:      U U U Yij Mij Xij      ~ MVN 0   ,0    0        2   Y  2    M  2     X XM MY XY           , and U U U      Yj Mj Xj      ~ MVN 0   ,0    0        2   Y  2    MY M  2     X MX XY           .  Cov(UMij, UMj)= Cov(UYij, UYj)=Cov(UMij, UYj)=Cov(UYij, UMj)=Cov(UXij, UXj)=Cov(UMij, UXj)=Cov(UXij, UMj)=Cov(UYij, UXj)=Cov(UXij, UYj)=0. (3) The between and within effects of X and M are:  xb       2 Y XjMj  XjY 1  MjY 2  XjMj 2  Y 2  X  XY   2  X   MY XM 2  M 2  XM 2  M  (99)  xw     XijMij  XijY 1  MijY 2  XijMij 2   Y 2  Y 2  X   XY  2  X XM   MY 2  M 2  XM 2  M   mb       2 Y XjMj   MjY 1 XjY 2  XjMj 2  Y 2  M  MY   2  M   XY XM 2  X 2  XM 2  X  (100) (101)  mw     XijMij   MijY 1 XijY 2  XijMij 2   Y 2  Y 2  M   MY  2  M XM   XY 2  X 2  XM 2  X  (102)  b  ( Cov U U Mj ) , Xj ( Var U Xj )   XM 2  X 122    w  , Cov U U Mij ) ( Xij ( Var U Xij )   XM 2  X (4) The between and within variance-covariance matrices, ICCs of M and Y, and proportions of variances explained at two levels are: U   Cov U   U      Yij Mij Xij       r        w   mw mb xw xb b / / / (103) (  xw   w  mw 2 ) (1  ICC Var X ( ) X )  ij 2  mw Var (  ij )  Var (  ij )  w (  xw   w  mw ) (1  ICC Var X ( ) X )   mw Var (  ij ) ij 2  w (1  ICC Var X ( ) X )  Var (  ij ) ij (  xw   w  mw ) (1  ICC Var X ( ) X ) ij  w (1  ICC Var X ( ) X ) ij (1  ICC ) Var X ( ) ij X (104)      U   Cov U   U 2 r       Yj Mj Xj           xw mw w r  r 2  w    xw mw 2    X ICC r w ( Var X  r ICC  X mw w )  ij  r  mw Var X ( ) ij 2   Var (  0 j )  Var (  0 j ) r   Var ( mw ) j r 2 2  w ICC X Var X ( )  Var (  0 j ) ij 0 r     xw ICC Var X ( ) ij X r  w ICC X Var X ( ) ij ICC Var X ( ) ij X      WΣ  BΣ    ICC M     w 2 (1  r ( ICC Var X 2  w ) ) X ij ) r ij   2 ICC Var X X ( Var    ij ) (   ) Var  2 2  w (  0 ICC Var X ( X j (105) (106) )  Var (  0 j ij )   ICC Y  (     mw  xw w r ICC Var X ) ( X 2 ij 2 ) (1  xw  r    ) Var   2  mw w 2  (  ij mw X ( ICC Var X ) (   ij Var ij   ) )      mw r   r 2 2 Var (  0 j )  (  0 j ) Var  2     r w mw xw ICC Var X ( X )  ij  r  mw 2  Var (  0 j )  Var (  0 j )  R 2 1  (     w mw ) ) (1 ( 2    w mw ( ) (1 ICC Var X 2   ( ICC Var X )   ) ij 2  mw ( ) Var 2    ij mw ( ) Var Var   ij xw xw ) X X ij (  ij ) R 2 2  2 r  2 r  xw r    mw     mw X ( ICC Var X  ( ICC Var X   r 2   xw ) w w X ij 2 2 )  ij r  mw   Var r  mw 2  ( ) Var  0 j ) (  0 j Var  (  0 j ) (107) (108) (109) (5) The variance-covariance matrices used in estimation are: 123   2   Y 2 Y (1  )  MY  1 n   MY  MY 1 n  Cov         M M X j X j Y ij  ij M  X ij j j                       1 n (1 1 n (1  ) 2  M 1 n 0 )  XM 1 n 0 (110)              ) 2  X 1 n 0 1 n 2   X  2 X 1 n 2   M 2 M  0 (1  1 n    XM XM )  XY (1    XY  XY ( E S B )  ( E S PW ) 2   Y  2     M  2     X XM MY XY      2 Y n 2    Y  n 2      MY M  n          2 M XM MY XY XY n n 2    X  n XM 2 X      (6) Manifest aggregation approach  Models: Y ij    mw  00 ( M M  ij ) j   mb M j   xw ( X ij  X ) j   xb X j  u 0 j  e ij M ij    w  00 ( X ij  X )   b X j j  w 0 j  d ij  Expectations of estimators: E  (  xb )  [ E r r r  XjY MjY XjMj 1 r 2  XjMj S 2 Y S 2 Xj ]   (111) (112) (113) (114) (115) r XjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j  n i g j ] (Y - Y )(Y - Y ) ij ij ( X j - X X )( j - X )][ 1 -1 ng 124 (116) r MjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y M M ij )( - - j ) (117) ( M M M M )( - - j j )][ 1 -1 ng  n i g j ] (Y - Y )(Y - Y ) ij ij r XjMj  [ n ng  1 n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j )] (118) (119) (120) 2  X (121) (123) S 2 Y  1 -1 ng  n i g j (Y - Y )(Y - Y ) ij ij S 2 Xj  n -1 ng g  j ( X j - X X )( j - X )  MY  1 n  MY  XM  1 n  XM [ 2  M  1 n 2     X  ][ 2 M 2 Y 2 Y ] [  1 n 2   M ][ 2 X  1 n 2  M ] (  XM   XM 2 ) 2  X  1 n 1 1 ) n n 2  X  (1  2   M ][ 2 X  1 n 2  M ] 2 2   Y Y  1 n  XY  1 n  XY [ 2  X  1 n   xb E ( )  2    Y ][  2 Y 2 X  ] 1  [ 2  X  (  MY  1 n   XM MY )(  XM ) 2  M  2  M  XY  1 n  XY   2  X  1 n 2  X  (  XM  2  M   XM 2  M 1 n 1 n  2 ) 1 n 1 n 1 n E  (  xw )  E [ r  XijY 1  r r MijY XijMij r 2 XijMij S 2 Y S 2 Xij ] (122) r XijY  [ 1 gn  1    n i g j 1 gn  1  g j n i j 125 (Y - Y )(X - X ij ij  j X - X ) ij j (X - X ij j  X - X )(X - X ij ij  j ][ X - X ) ij j 1 gn  1  n i g j ] (Y - Y )(Y - Y ) ij ij r MijY  [ 1 gn  1  n i g j 1 gn  1  n i g j (Y - Y )(M - M M - M )  ij j ij j ij (M - M M - M )(M - M M - M )   j ij j ij j ij j ij ][ 1 gn  1  n i g j (Y - Y )(Y - Y ) ij ij (124) ] 1 gn  1 g n  j i (M - M M - M )(X - X  ij j ij j ij  j X - X ) ij j 1 gn  1 g n  j i ] (M - M M - M )(M - M M - M )   ij j ij j ij j ij j r XijMij  [ 1 gn  1 g n  (X - X ij j  X - X )(X - X j ij ij  j ][ X - X ) ij j j i S 2 Y  g n 1   1 gn i j (Y - Y )(Y - Y ) ij ij E  (  xw )   XY   2  X   (125) (126) (127) (128) (129) S 2 Xij  g n 1   1 gn i j ( X - X ij j  X - X ij )( X - X ij j j  X - X ij j )  (1 1 n ( )(1 2   Y  2 Y )  XY 1 n   ) 2  X 1  2 Y    (1 1 n ( )(1 2   Y 1 n ) 2  M )  MY 1 n 2 2  XM 1 n (1 1 n (1 (1    ) ) 2  M ) 2  X )  XM  1 n ) 2  M (1 1 n (1  (1  1 n 2 X )    2  Y 1 ) 2  X n 2 Y  (1 XM   MY 2  M 2  XM 2  M    xw E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 Y S 2 Mj ] r MjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y M M ij )( - - j ) (130)  n i g j ] (Y - Y )(Y - Y ) ij ij ( M M M M )( - - j j )][ 1 -1 ng 126 r XjY  [ n ng  1 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j g  j ( X j - X X )( j - X )][ 1 -1 ng  n i g j ] (Y - Y )(Y - Y ) ij ij (131) r XjMj  [ n ng  1 n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j )] (132) (133) (134) 2  M (135) (136)    S 2 Y  1 -1 ng S 2 Mj  n -1 ng  n i g j (Y - Y )(Y - Y ) ij ij g  j ( M M M M )( - - ) j j  XY  1 n  XY  XM  1 n  XM [ 2  X  1 n 2     X  ][ 2 Y 2 Y 2 X [ ]  1 n 2   M ][ 2 X  1 n 2  M ] (  XM   XM 2 ) 2  M  2 2   Y Y  1 n 1 n 1 n 1 n 1 n 2   M ][ 2 X   XM ) 2  M  (1  1 1 ) n n 2  M ]  MY  1 n  MY [ 2  M  1 n   mb E ( )  2    Y ][  2 M 2 Y  ] 1  [ 2  X  (  XY  1 n  MY  1 n  MY    XM )( XY 2  X  2  X 2  M  1 n 2  M  (  XM  2  X   XM 2  X  2 ) 1 n 1 n 1 n E  (  mw )  E [ r MijY 1   r r XijY XijMij r 2 XijMij S 2 Y S 2 Mij ] 1 gn  1  n i g j (Y - Y )(M - M M - M )  ij j ij ij j (M - M M - M )(M - M M - M )   j ij j ij j ij j ij ][ 1 gn  1 127 r MijY  [ 1 gn  1  n i g j (137) ] (Y - Y )(Y - Y ) ij ij  n i g j r XijY  [ 1 gn  1  n i g j (X - X ij j  X - X )(X - X ij ij  j ][ X - X ) ij j 1 gn  1  n i g j (138) ] (Y - Y )(Y - Y ) ij ij (Y - Y )(X - X ij ij  j X - X ) ij j 1 gn  1  g j n i j  n i g j (M - M M - M )(X - X  j ij ij ij j  j X - X ) j ij ] / (X - X ij  j X - X )(X - X ij ij j  ] X - X ) 1/2 j ij  j (139) r XijMij  {[ 1 gn  1[ gn  1 1[ 1 gn  g n  i j 1  n g j i (M - M M - M )(M - M M - M ) j   ij ij ij ij j j j ] } 1/2 S 2 Y  n g 1   1 gn i j (Y - Y )(Y - Y ) ij ij S 2 Mij  n g 1   1 gn j i ( M - M M - M M - M M - M )(   j ij ij ij ij j j (140) (141) ) j  (1 1 n ( )(1 2   Y  2 Y )  MY 1 n   ) 2  M 1  E  (  mw )  2 Y    (1 1 n ( )(1 2   Y 1 n ) 2  M )  XY 1 n 2 2  XM 1 n (1 1 n (1    ) (1 ) 2  X ) 2  X )  XM  1 n ) 2  M (1 1 n (1  (1  1 n 2 X )    2  Y 1 ) 2  M n 2 Y  (1 XM   XY 2  X 2  XM 2  X    mw  (  b ) E  E [  MY   2  M   n -1 ng n ng  g  j ( M M X )( - j g  j ( X j - X X )( 1 - X ) - X ) j j ]  128 (142) (143)  XM  2     X 2 b X  XM 1 n 1 n  2   X  2 X 1 n 2   X  w 1 n 2 X  (  w E )  E [ 1 1 gn  1 gn   n i g j (X - X X - X )(M - M M - M ) j   ij ij ij ij j j  n i g j 1 (X - X X - X )(X - X X - X ) j   ij ij ij ij j ]  (1  (1  1 )  MX n 1 n ) 2  X   w (7) Latent aggregation approach  Models: M ij  M  U Mij  U Mj X j  X  U Xij  U Xj Y ij     00 U mw Mij   U mb Mj   U xw Xij  U  xb Xj  u 0 j  e ij U Mj U  b Xj  w 0 j U Mij  U w Xij  d ij  Expectations of estimators: E  (  xb )  [ E r r r  XjY MjY XjMj 1 r 2  XjMj S 2 YB S 2 Xj ] (150) (144) (145) (146) (147) (148) (149) n -1 g g  ( r XjY  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( Y Y X j )( - j X - X X )( - X j j  n i - g ) ) j n  X 1 1) ( g n  1 g  1) ( g n  1 g  ( g n  1)  ij n i j i j ij j (Y -Y )(X - X ) n ] / } / ij j ij j ] / } (X - X )(X - X ) n 1/2 j ij j  (151) ( - Y Y Y Y j )( - j )  (Y -Y )(Y -Y ) n ] / } ) 1/2 ij j 129   (Y -Y )(M - M ) n ] / } / ij j j ij ij j j ] / } (M - M )(M - M ) n 1/2 ij  (152) (Y -Y )(Y -Y ) n ] / } ) 1/2 ij j ij j (X - X )(M - M ) n ] / } / ij j ij j ] / } (M - M )(M - M ) n 1/2 j ij ij j  (153) n -1 g g  ( r MjY  {[ ({[ {[ n  g n  g j 1  g j 1 n -1 g g  ( r XjMj  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( Y Y M M j )( - - j ) M M M M )( - - )  j j ( - Y Y Y Y j )( - j )  1 ( g n  g  n i j n 1) g  j i  1 ( g n  1 ( 1) g n  g n  1) j i g  j ( X - j X M M )( - j M M M M )( - - ) j j  n i g j )  1 1) ( g n  1 g  1) ( g n  1 g  ( g n  1)  j n n i j i ( X j - X X )( j - X )  (X - X )(X - X ) n ] / } ) 1/2 j ij ij j S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j )  1 ( g n  1) S 2 Xj  [ n -1 g g  j ( X j - X X )( j - X )  1 ( g n  1) E  (  xb )   MY XM  XY 2 2 2     X Y Y 2  1 X   2 2 2 M X Y 2  XM 2 2  X M    g j g j n i n i (Y - Y )(Y - Y ) j ij ij j ] / n (154) (X - X )(X - X ) n ] / (155) j ij ij j  XY   2  X   MY XM 2  M 2  XM 2  M    xb (156) E  (  xw )  E [ r  XijY 1  r r MijY XijMij r 2 XijMij S 2 YX S 2 Xij ] 1 ( g n  1)  n i g j (Y - Y )(X - X ) j ij ij j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j 1 ( g n  1)  n i g j (Y - Y )(M - M ) j ij ij j ][ (M - M )(M - M ) j ij ij j 1 ( g n  1) 130  n i g j ] (Y - Y )(Y - Y ) j ij ij j (157) (158) (159) ] (Y - Y )(Y - Y ) j ij ij j r XijY  [ 1 ( g n  1)  n i g j r MijY  [ 1 ( g n  1)  n i g j   r XijMij  [ 1 ( g n  1) E  (  xw )  1 ( g n  1)  n i g j (M - M )(X - X ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j ] (M - M )(M - M ) j ij ij j S 2 YW  n g 1   ( 1) g n j i (Y - Y )(Y - Y ) j ij ij j S 2 Xij  g n 1   ( 1) g n j i (X - X )(X - X ) j ij ij j (160) (161) (162)   XY 2 2      Y Y 2  X  XM 2 2 M X 2 X  MY 2 2 Y M 2  XM 2 2   M X 1   XY   2  X XM   MY 2  M 2  XM 2  M    xw (163) E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 YB S 2 Mj ] n -1 g g  ( r MjY  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( Y Y M M j )( - - j ) M M M M )( - - )  j j ( - Y Y Y Y j )( - j )  1 ( g n  g  n i j n 1) g  j i  1 ( g n  1 1) ( g n  g n  1) j i (Y -Y )(M - M ) n ] / } / ij ij ij j j ] / } (M - M )(M - M ) n 1/ 2 ij (Y -Y )(Y -Y ) n ] / } ) 1/2 ij j ij n -1 g g  ( r XjY  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( Y Y X j )( - j X - X X )( - X j j  n i - g ) ) j n  X 1 1) ( g n  1 g  1) ( g n  1 g  ( g n  1)  ij n i j i j ij j ( - Y Y Y Y j )( - j )  (Y -Y )(Y -Y ) n ] / } ) 1/2 ij j (Y -Y )(X - X ) n ] / } / ij ij j ] / } (X - X )(X - X ) n 1/2 j ij j  (166) (164)  (165) j j j j 131   n -1 g g  ( r XjMj  {[ ({[ {[ n  g n  g j 1  g j 1 g  j ( X - j X M M )( - j M M M M )( - - ) j j  n i g j )  1 1) ( g n  1 g  1) ( g n  1 g  ( g n  1)  j n n i j i ( X j - X X )( j - X )  (X - X )(X - X ) n ] / } ) 1/2 j ij ij j (X - X )(M - M ) n ] / } / ij j ij j ] / } (M - M )(M - M ) n 1/2 ij j ij j  (167)   n i n i g j g j (Y - Y )(Y - Y ) j ij ij j ] / n (168) (M - M )(M - M ) j ij ij j ] / n (169) S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j S 2 Mj  [ n -1 g g  j ( M M M M )( - - j j )  1 ( g n  1) )  1 ( g n  1) E  (  mb )   XY XM  MY 2 2 2       M Y Y 2  1 M   2 2 2 X M Y 2  XM 2 2  X M   MY   2  M   XY XM 2  X 2  XM 2  X  E  (  mw )  E [ r MijY 1   r r XijY XijMij r 2 XijMij S 2 YW S 2 Mij ] 1 ( g n  1)  n i g j (Y - Y )(M - M ) j ij ij j  n i g j ][ (M - M )(M - M ) j ij ij j 1 ( g n  1)  n i g j r MijY  [ 1 ( g n  1)   mb (170) (171) (172) ] (Y - Y )(Y - Y ) j ij ij j r XijY  [ 1 ( g n  1)  n i g j r XijMij  [ 1 ( g n  1)  n i g j 1 ( g n  1)  n i g j (Y - Y )(X - X ) j ij ij j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j ] (Y - Y )(Y - Y ) j ij ij j 1 ( g n  1)  n i g j (M - M )(X - X ) j ij ij j (173) (174) ] (M - M )(M - M ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1) 132      E      [ g [ g XM 1  n 1 n g  j n -1 n 1  1 n 1 n  2    X   2 X 2 X  XM   XM   b  (  b ) E      S 2 YW  n g 1   1) ( g n i j (Y - Y )(Y - Y ) j ij ij j S 2 Mij  n g 1   1) ( g n i j (M - M )(M - M ) j ij ij j (175) (176) E  (  mw )   MY 2 2  Y M  1  2 X  XM 2 2 M X  XY 2 2     Y Y 2 2   XM M 2 2   M X  MY   2  M   XM XY 2  X 2  XM 2  X    mw (177) ( M M X )( - j g  j ( X j - X X )( - X )  - X )  j j 1 ( g n  1 ( g n  1) 1) g n  i n i  j g j (X - X )(M - M ) n ] / ij ij j j ij ij (X - X )(X - X ) n ] / j j        (178) ]   XM 2  X   w (179)  (  w E )  [ E 1 ( g n  1 ( g n  1) 1) g n  i n i  j g j (X - X )(M - M ) j ij ij j (X - X )(X - X ) j ij ij j 133 APPENDIX B: The Between and Within Effects in the Manifest and Latent Aggregation Approaches Considering the Within-group Finite Population Selection In this section, the between and within effect estimators in the manifest and latent aggregation approaches were derived for the MLM, 2-1-1 mediation, and 1-1-1 mediation models considering the within-group finite population selection. A two-stage cluster sampling design was assumed. First, a simple random sample of g groups was selected from G groups. Each group was of the same size N. Then, a simple random sample of size n was selected within each group. The within-group fpc is 1-n/N. Assuming G was infinite and N was finite. The population models, assumptions, parameters of theoretical interest, models in manifest and latent aggregation approaches were the same as those in the Appendix A. When the within-group finite population selection was considered, the variance-covariance matrices used in the estimation and the expectations of estimators were different. 1. MLM model (1) The variance-covariance matrices used in estimation: Cov X      Y ij  X j ij X j               2 Y fpc 2   Y 1 )  XY n (1  fpc   XY  XY 1 n (1  fpc 1 n ) 2  X 0 ( E S PW )  n  n fpc 1  2   Y  2    X XY    ( E S ) B     2 Y fpc n 2   Y fpc n   XY XY   fpc 2 X 2   X  n        (180) (181) (182) 2   X  2 X fpc 1 n    (2) Expectations of estimators in the manifest aggregation approach: 134   E (   xb )  E [ 1 -1 ng n ng   n i g j g  j 1 ( Y Y X ij )( - - X ) j ]   XY  fpc ( X j - X X )( - X ) j 2  X  fpc 1 n 1 n   xb XY  ICC X   xw 2  X ICC X  fpc 1 n (1  fpc 1 n (1  ICC ) X (183) ICC ) X E  (  xw )  E [ 1 gn  1 1 1 gn  g n  j i  n i g j (Y -Y )(X - X ij ij  X - X ) j ij j (X - X ij  j X - X ) X - X ( ij ij j  X - X ) j ij j ]  (1  fpc (1  fpc 1 )  XY n 1 n ) 2  X   xw (184) (3) Expectations of estimators in the latent aggregation approach: E  (  xb )  [ g { E [ g n    XY  fpc 2  X  fpc 1  XY n 1 n 2  X  g j g n -1  j 1 1   1   ( Y Y X j )( - ( X j - X X )( - X )  - X )  j j 1 ( g n  1 ( g n  1) 1)   xb XY  fpc n n  fpc n n  / 1 / 1 2  X ICC X  IC C X  g n  j g i n i j  1 ( N   xw 1 ( N n (Y -Y )(X - X ) n ] / ij j ij j (X - X )(X - X ) n ] / ij ij j j } (185) n 1 n  )(1  1  n  )(1  ICC ) X ICC X ) E  (  xw )  [ E 1 ( g n  1 ( g n  g n j  1)  n g i j i 1) (Y -Y )(X - X ) j ij ij j ]  (X - X )(X - X ) j ij ij j n fpc  1 n  fpc n  1 n   XY 2  X   xw (186) 2. 2-1-1 mediation model (1) The variance-covariance matrices used in estimation: Cov       M M Y ij  ij M X j j j                 2 Y fpc 2    Y 1 )  MY n   MY  MY  XY (1  fpc 1 n          2  X (187) 2   M 2 M  1 n  XM (1  fpc 1 n ) 2  M fpc 0 0 135   ( E S PW )  n  n fpc 1  2   Y  2    M  0  MY 0      0 ( E S B )       2 Y   fpc n 2   Y fpc n   MY MY n  XY fpc n  2 2   M M n  XM      n 2  X (2) Expectations of estimators in the manifest aggregation approach: E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 Y S 2 Mj ] (188) (189) (190) r MjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y M M ij )( - - j ( M M M M )( - - j j )][ 1 -1 ng  n i g j ) (191) ] (Y - Y )(Y - Y ) ij ij (192) 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j r XjY  [ n ng  1 g  j ( X j - X X )( j - X )][ 1 -1 ng  n i g j ] (Y - Y )(Y - Y ) ij ij r XjMj  [ n ng  1 n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j S 2 Y  1 -1 ng S 2 Mj  n -1 ng  n i g j (Y - Y )(Y - Y ) ij ij g  j ( M M M M )( - - ) j j 136 )] (193) (194) (195)    XY ][ [ 2    Y  2 Y 2 X  ] ] [ 2   M ][ 2 X  XM 2  XM  fpc 1 n 2  M ] [ 2   M ][ 2 X  fpc 1 n 2  M ] 2  M  2 2   Y Y 1 n fpc  2  M  MY  fpc 1 n  MY [ 2  M  fpc 1 n   mb E ( )  2    Y ][  2 M 2 Y 1  2  XM 2  X 2  XM 2  X )   mw fpc 1 n 2  M )  fpc 1 n 2  M 2   M mb (   ( 2  M    mw E ( )  E [ 1 gn  1 1 1 gn  n   g j i   n i g j (Y - Y )(M - M ij ij  j M - M ) ij j (M - M ij  j M - M ) M - M ( ij j ij  j M - M ) ij j ]  (1  fpc (1  fpc 1 n 1 n )  MY ) 2  M   mw (3) Expectations of estimators in the latent aggregation approach: E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 YB S 2 Mj ] (196) (197) r MjY  {[ n -1 g  j ( Y Y M M j )( - - j ) (Y -Y )(M - M ) n ] / }/ ij j j ij ij j j 1/2  n i g j 1) g  j n i  1 ( g n  1 ( g n  g 1) n  j i 1) g  j 1 ( - Y Y Y Y j )( - j )  1 ( g n  (Y -Y )(Y -Y ) n ij j ij j 1/2 ] / } ) ( M M M M )( - - )  j j ] / } (M - M )(M - M ) n ij  (198) g g n   j 1 ({[ {[ g n  g n -1 g g  ( j r XjY  {[ n  1 ({[ {[ g n  g g  j ( Y Y X j )( - - X ) 0] / } /  n j X - X X )( j j - X ) 0] / } n  1/2  (199) (Y - Y )(Y - Y ) n ij j ij j 1/2 ] / } )  n i g j g  j 1 ( Y Y Y Y j )( - - j )  1 ( g n  1) 137   g  j ( M M X )( - j - X ) 0] / } /  n j n -1 g g  ( j r XjMj  {[ n  1 ({[ {[ g n  g X - X X )( j j - X ) 0] / } n  1/2  (200) g  j 1 ( M M M M )( - - j j )  1 ( g n  1)  n i g j (M - M )(M - M ) n ij j ij j 1/2 ] / } ) S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j S 2 Mj  [ n -1 g g  j ( M M M M )( - - j j )  1 ( g n  1) )  1 ( g n  1) g j g j   n i n i (Y - Y )(Y - Y ) j ij ij j ] / n (201) (M - M )(M - M ) j ij ij j ] / n (202) E  (  mb ) {   MY 1 n 2  M  [ 2  M  fpc fpc  1 n  n n 1 n fpc 1   MY  1 n  n n ][ 2    Y 2 M fpc 1  fpc 2  Y  MY 1 n  1 n XM 2  M  1 n  n n fpc 1  2  Y ] }  1 n  n n fpc 1  2  M ] (203) fpc 1  2    M ][ 2 Y 2 X [ ]  fpc 2  XM 1 2  M n  1 n  n n fpc 1  2  M } ]  fpc [ 2   M ][ 2 X / {1  ) 2  M    [ 2   Y ][ 2 X  fpc 2  Y  2  M  fpc fpc 2   mb M (  ( 2  M  2  Y 1 n 1 2  M n 2  XM 2  X 2  XM 2  X ) )  XY 1 n 2  Y    1 n  n n 1 n fpc  1 n n  1 fpc n  1 n n  1 N  (  n  n 1    mw 1 N ( 2  Y 2  M n  1 n ) 2  M n fpc  1 n  fpc n  1 n   MY 2  M   mw (204) E  (  mw )  E [ 1 ( g n  1 ( g n  g n  i j 1)  n g j i 1) (Y -Y )(M - M ) j ij ij j ]  (M - M )(M - M ) j ij ij j 138         3. 1-1-1 mediation model (1) The variance-covariance matrices used in estimation: Cov         X ij M M ij Y ij  M  X X j j j j                       2 Y  2   Y 1 n fpc )  MY (1  fpc 1 n   MY  MY (1  fpc 1 n ) 2  M 0 fpc 1 n 2   M  2 M (1  fpc 1 n )  XY (1  fpc 1 n )  XM 0 (1  fpc 1 n ) 2  X fpc 1 n   XY  XY 0 fpc 1 n   XM  XM ( E S PW )  n  n fpc 1  2   Y  2    M  2     X XM MY XY 0      ( E S B )       2 Y fpc n 2   Y n fpc   MY MY fpc n   XY    XY 2 M n fpc 2   M fpc n     XM XM fpc 2   X  n 2 X (2) Expectations of estimators in the manifest aggregation approach: E  (  xb )  [ E r r r  XjY MjY XjMj 1 r 2  XjMj S 2 Y S 2 Xj ]                (205) (206) (207) (208) 2   X  2 X fpc 1 n      (209) r XjY  [ n ng  1 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j g  j ( X j - X X )( j - X )][ 1 -1 ng  n i g j r MjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y M M ij )( - - j ( M M M M )( - - j j )][ 1 -1 ng 139 ] (Y - Y )(Y - Y ) ij ij ) (210)  n i g j ] (Y - Y )(Y - Y ) ij ij r XjMj  [ n ng  1  XY  fpc 1 n  XY [ 2  X  fpc 1 n   xb E ( )  2    Y ][  2 Y 2 X  ] n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j S 2 Y  1 -1 ng  n i g j (Y - Y )(Y - Y ) ij ij S 2 Xj  n -1 ng g  j ( X j - X X )( j - X ) )] (211) (212) (213)  MY  fpc 1 n  MY  XM  fpc 1 n  XM  fpc 1 n 2   M ][ 2 X 1 n [ 2  M  fpc 2     X  ][ 2 M 2 Y 2 Y ] [ 1  (  XM [ 2  X  fpc 1 n  fpc 1 n 2   M ][ 2 X  XM 2 )  fpc fpc   XM MY )(  fpc 1 n 1 n 2  M ]  XM ) 2 Y  2   Y 1 n fpc   2  X 2  X (  MY   XY  fpc 1 n  XY  1 n 2  M  fpc 2  M 1 n 1 n 1 n  XM 2 ) 2  M 2  X  fpc 1 n 2  X  (  XM  fpc 2  M  fpc E  (  xw )  E [ r  XijY 1  r r MijY XijMij r 2 XijMij S 2 Y S 2 Xij ] r XijY  [ 1 gn  1  n i g j (X - X ij j  X - X )(X - X ij ij  ][ X - X ) j ij j 1 gn  1  n i g j ] (Y - Y )(Y - Y ) ij ij (Y - Y )(X - X ij ij  X - X ) j ij j 1 gn  1  g j n i j  fpc 1 n 2  M ]  (214) (215) (216)   140 r MijY  [ 1 gn  1  n i g j 1 gn  1  n i g j (Y - Y )(M - M M - M ) j  ij ij ij j ][ (M - M M - M )(M - M M - M ) j   ij ij ij ij j j j 1 gn  1  n i g j (217) ] (Y - Y )(Y - Y ) ij ij 1 1 gn  g n  r XijMij  [ {[ 1 gn  1 gn  1 [ i j 1  n g j i  n i g j (M - M M - M )(X - X  j ij ij j ij  j X - X ) j ij ] / (X - X ij  j X - X )(X - X ij ij j  j ] X - X ) 1/ 2 j ij  (218) (M - M M - M )(M - M M - M ) j   ij ij ij ij j j j ] } 1/ 2 S 2 Y  n g 1   1 gn i j (Y - Y )(Y - Y ) ij ij S 2 Xij  n g 1   1 gn j i ( X - X ij j  X - X ij )( X - X ij j j  X - X ij j ) (219) (220)  (  xw E )   2 Y fpc 2   Y 1 ) 2  X n  (1   (1  fpc (1  fpc 1 n ) 2  M )  XM 1 n (1  fpc 1 n ) 2  X  (221) )  MY 1 n 2 2  XM fpc ) ) 2  M 1 n (1 fpc ) 2  M  fpc 1 n ) 2  X (1  fpc 1 n )  XY ( 2   Y  2 Y )(1  fpc  1 n ) 2  X (1  fpc 1 n ( 2   Y  2 Y )(1  1  (1 (1  fpc  1 n  XY   2  X   MY XM 2  M 2  XM 2  M    xw E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 Y S 2 Mj ] (222) r MjY  [ n ng  1 g  j 1 -1 ng  n i g j ( Y Y M M ij )( - - j ( M M M M )( - - j j )][ 1 -1 ng 141  n i g j ) (223) ] (Y - Y )(Y - Y ) ij ij r XjY  [ n ng  1 1 -1 ng  n i g j ( Y Y X ij )( - - X ) j g  j ( X j - X X )( j - X )][ 1 -1 ng  n i g j ] (Y - Y )(Y - Y ) ij ij (224) n -1 ng g  j ( M M X )( - j - X ) j g  j ( X j - X X )( j - X )][ n ng  1 g  j ( M M M M )( - - j j S 2 Y  1 -1 ng S 2 Mj  n -1 ng  n i g j (Y - Y )(Y - Y ) ij ij g  j ( M M M M )( - - ) j j )] (225) (226) (227) r XjMj  [ n ng  1  MY  fpc 1 n  MY [ 2  M  fpc 1 n   mb E ( )  2    Y ][  2 M 2 Y  ] [ 2  X  fpc 2     X  ][ 2 Y 2 Y 2 X ] [ 1 n  XY  fpc 1 n  XY  XM  fpc 1 n  XM  fpc 1 n 2   M ][ 2 X  fpc 1 n 2  M ]  (228) 1  (  XM [ 2  X  fpc 1 n  fpc 1 n 2   M ][ 2 X  XM 2 )  fpc 1 n 2  M ]  XM ) (  XY   MY  fpc 1 n  MY  fpc   XM )( XY 1 n 2  X  fpc  fpc 1 n 2  X 2  M  fpc 1 n 2  M  (  XM  fpc 2  X  fpc  XM 2 ) 2  X 1 n 1 n 1 n E  (  mw )  E [ r MijY 1   r r XijY XijMij r 2 XijMij S 2 Y S 2 Mij ] (229) 142  2 2   Y Y 1 n fpc   2  M 2  M   1 gn  1  n i g j (Y - Y )(M - M M - M ) j  ij ij ij j ][ (M - M M - M )(M - M M - M ) j   ij ij ij ij j j j 1 gn  1  n i g j (230) ] (Y - Y )(Y - Y ) ij ij  n i g j 1 gn  1  g j n i j  n i g j (X - X ij j  X - X )(X - X ij ij (Y - Y )(X - X ij ij  X - X ) j ij j  ][ X - X ) j ij j 1 gn  1  n i g j (231) ] (Y - Y )(Y - Y ) ij ij r MijY  [ 1 gn  1 r XijY  [ 1 gn  1  n i g j (M - M M - M )(X - X  j ij ij ij j  j X - X ) j ij } / (X - X ij  j X - X )(X - X ij ij j  ] X - X ) 1/ 2 j ij  j (232) r XijMij  {[ 1 gn  1 gn  1 [ 1{ 1 gn  g n  i j 1  n g j i     (M - M M - M )(M - M M - M ) j   ij ij ij ij j j j ] } 1/ 2 S 2 Y  g n 1   1 gn i j (Y - Y )(Y - Y ) ij ij S 2 Mij  n g 1   1 gn j i ( M - M M - M M - M M - M )(   j ij ij ij j j ij (1  fpc (1  fpc 1 n ) 2  M )  XY fpc ) 1 n (1 ) 2  X 1 n 2 2  XM  fpc 1 n ) 2  X (1  fpc 1 n ( 2   Y  2 Y )(1  )  MY 1 n fpc  ) 2  M (1  fpc 1 n ( 2   Y  2 Y )(1   (  mw E )   2 Y fpc 2   Y 1 ) 2  M n  (1 1  (1 (1  fpc fpc ) 2  M  1 n  MY   2  M   XM XY 2  X 2  XM 2  X    mw 143 (233) (234) ) j )  XM 1 n (1  fpc 1 n ) 2  X  (235)  (  b ) E  E [ n -1 ng n ng  g  j ( M M X )( - j g  j ( X j - X X )( 1 - X ) - X ) j j ]   XM  fpc 2  X  fpc 1 n 1 n    w 2 b X  XM 2  X  2  X 2  X  fpc 1 n 2  X fpc 1 n (236)  (  w E )  E [ 1 1 gn  1 gn   n i g j (X - X X - X )(M - M M - M ) j   ij ij ij ij j j  n i g j 1 (X - X X - X )(X - X X - X ) j   ij ij ij ij j ]  (1  (1  fpc 1 )  MX n 1 fpc n ) 2  X   w (237) (3) Expectations of estimators in the latent aggregation approach: E  (  xb )  E [ r  XjY 1  r r MjY XjMj r 2 XjMj S 2 YB S 2 Xj ] (238) r XjY  {[ n -1 g  j ( Y Y X j )( - j (Y -Y )(X - X ) n ] / }/ ij j ij j 1/2 ( X j - X X )( - X j ] / } (X - X )(X - X ) n j ij j  (239) g  j 1 ( - Y Y Y Y j )( - j )  (Y -Y )(Y -Y ) n ij j ij 1/2 ] / } ) r MjY  {[ n -1 g  j ( Y Y M M j )( - - j ) ( M M M M )( - - )  j j ] / } (M - M )(M - M ) n ij  (240) (Y -Y )(M - M ) n ] / }/ ij j j ij ij j j 1/2 g  j 1 ( - Y Y Y Y j )( - j )  1 ( g n  (Y -Y )(Y -Y ) n ij j ij j 1/2 ] / } ) r XjMj  {[ n -1 g  j ( X - j X M M )( - j  n i g j (X - X )(M - M ) n ] / } / ij j ij j 1) g g n   j 1 ({[ {[ g n  g ( M M M M )( - - ) j j g  j 1 ( X j - X X )( j - X )  ] / } (M - M )(M - M ) n j ij ij j  (241) 1/2 (X - X )(X - X ) n j ij ij j 1/2 ] / } ) g g n   j 1 ({[ {[ g n  g g g n   j 1 ({[ {[ g n  g    n i g j n i  j 1) g - ) )  X 1 ( g n  1 1) ( g n  1 g ( g n  1)  j n i  ij j  n i g j 1) g  j n i  1 ( g n  1 ( g n  g 1) n  j i 1) )  1 ( g n  1 g ( g n  1 ( g n  1) g 1)  j j i  n  n i 144 2  Y fpc 1  fpc  MY 1 n fpc 1  fpc  1 n 2  M XM  1 n  n n fpc 1  2  Y ] }  1 n  n n fpc 1  2  M ]  MY  fpc  XY  1 n  n n 1 n fpc 1  1 n  n n ][ 2    Y 1 n  n n 2    M ][ 2 M  XM 2 X  2  Y 2  X / [ 2  M  fpc 1 n 2  M [ 2  X  fpc 2  X  1 n 1 n 1 2  X n 2  Y 2  Y  fpc 2  X  fpc   MY  fpc 1  n 1 n fpc  1 n n  1 fpc  XM n 1 fpc n  1 n n  1 fpc n  1 n n  1 fpc n  1 n n  fpc  XM 1 n  n n [  MY 1 n fpc 1     (  {1  [ 2  X  fpc 1 n 2  X  XY  1 N (  n  1 n )  XY      S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j )  1 ( g n  1) S 2 Xj  [ n -1 g g  j ( X j - X X )( j - X )  1 ( g n  1) g j g j   n i n i (Y - Y )(Y - Y ) j ij ij j ] / n (242) (X - X )(X - X ) n ] / (243) j ij ij j E  (  xb ) {   XY 1 n 2  X  [ 2  X  fpc  1 n  n n 2    Y ][ 2 X fpc 1  fpc  XY 1 n 2  Y  1 n  n n fpc 1  2  Y ]  XM  1 n  n n 2    M ][ fpc 1  1 fpc n  XM 2 ) 2  M  2 X 1 N  (  n n  [ 2  M 1 n  n n 1 N ) 2  M  1 )  ] 2 XM ) 2  M ] 1 XM n  (  ][ )   MY 1 1  N 1 ( N 1 N n  n n  n   n  ( 1  2  X  1 N (  n  1 n ) 2  X  [  XM [ 2  M } ] )  XM ] fpc 1  (  n 2  M n  1 ] (244) E  (  xw )  [ E r  XijY 1  r r MijY XijMij r 2 XijMij S 2 YW S 2 Xij ] (245) 145   r XijY  [ 1 ( g n  1) 1 ( g n  1)  n i g j (Y - Y )(X - X ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j ] (Y - Y )(Y - Y ) j ij ij j (246) (247) r MijY  [ 1 ( g n  1) 1 ( g n  1)  n i g j (Y - Y )(M - M ) j ij ij j  n i g j ][ (M - M )(M - M ) j ij ij j 1 ( g n  1)  n i g j ] (Y - Y )(Y - Y ) j ij ij j r XijMij  [ 1 ( g n  1) 1 ( g n  1)  n i g j (M - M )(X - X ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j ] (M - M )(M - M ) j ij ij j S 2 YW  g n 1   ( 1) g n i j (Y - Y )(Y - Y ) j ij ij j S 2 Xij  n g 1   ( 1) g n j i (X - X )(X - X ) j ij ij j (248) (249) (250) fpc 1   XY n  n fpc 1  n  n   ( xw E )  2  Y n  n fpc 1  2  X  n  n 2  Y n  n fpc 1  2  M n  n n  n fpc 1  n  n fpc 1  ( fpc 1   MY fpc 1  2  M ) 2 2  XM n  n fpc 1  2  X 1  n  n  XY   2  X   XM MY 2  M 2  XM 2  M    mw fpc 1   XM n  n fpc 1  2  M n  n fpc 1  2  X n n  n  n fpc 1  fpc 1  2  Y 2  X (251)   E  (  mb )  [ E r  MjY 1  r r XjY XjMj r 2 XjMj S 2 YB S 2 Mj ] (252) 146 g g n   j 1 ({[ {[ g n  g g g n   j 1 ({[ {[ g n  g j j j j  n i g j 1) g  j n i  1 ( g n  1 ( g n  g 1) n  j i 1)  n i g j n i  j 1) g - ) )  X 1 ( g n  1 ( 1) g n  1 g ( g n  1)  j n i  ij j r MjY  {[ n -1 g  j ( Y Y M M j )( - - j ) ( M M M M )( - - )  j j ] / } (M - M )(M - M ) n ij  (253) (Y -Y )(M - M ) n ] / }/ ij ij ij j j 1/2 g  j 1 ( - Y Y Y Y j )( - j )  1 ( g n  (Y -Y )(Y -Y ) n ij j ij 1/2 ] / } ) r XjY  {[ n -1 g  j ( Y Y X j )( - j (Y -Y )(X - X ) n ] / }/ ij ij j 1/2 ( X j - X X )( - X j ] / } (X - X )(X - X ) n j ij j  (254) g  j 1 ( - Y Y Y Y j )( - j )  (Y -Y )(Y -Y ) n ij j ij 1/2 ] / } ) r XjMj  {[ n -1 g  j ( X - j X M M )( - j  n i g j (X - X )(M - M ) n ] / } / ij j ij j 1) g g n   j 1 ({[ {[ g n  g ( M M M M )( - - ) j j g  j 1 ( X j - X X )( j - X )  )  1 ( g n  1 g ( g n  1 ( g n  1) g 1)  j j i  n  n i (X - X )(X - X ) n j ij ij j 1/2 ] / } ) ] / } (M - M )(M - M ) n j ij ij j  (255) 1/2   n i n i g j g j (Y - Y )(Y - Y ) j ij ij j ] / n (256) (M - M )(M - M ) j ij ij j ] / n (257) S 2 YB  [ n -1 g g  j ( Y Y Y Y j )( - - j S 2 Mj  [ n -1 g g  j ( M M M M )( - - j j )  1 ( g n  1) )  1 ( g n  1) 147    1 n  n n fpc 1  2  Y ] }  1 n  n n fpc 1  2  M ] E  (  mb ) {  [ 2  M   MY  1 n  n n ][ 2    Y fpc 1  fpc  MY 1 n 2  Y  1 n  n n fpc 1  2  Y ]  fpc  1 n  n n 1 n fpc 1  1 n  n n 2    Y 1 n  n n 2    M ][ ][  XM 2 X 2 X  2 M fpc 1  fpc 2  Y  XY 1 n fpc 1  fpc  1 n 2  M XM  XY  XY  MY     fpc 1 n fpc  1 n  n n 2  M 1 n fpc 1  1 fpc  XM n 1 fpc n  1 n n  1 fpc n  1 n n  1 fpc n  1 n n  1 fpc  XM n 1 n fpc  1 n n  [  XY  )  MY    (  2  Y 2  M / [ 2  X  fpc 1 n 2  X [ 2  X  fpc 2  Y  fpc 2  M  fpc 2  X 1 n 1 n 1 2  M n 2  Y {1  [ 2  X  fpc 1 n 2  X  MY  1 N (  n  1 n      XM  1 n  n n 2    M ][ fpc 1  1 fpc n  XM 2 ) 2  M  2 X 1 N  (  n n  [ 2  X ( XM n   ][ )   XY 1 1  N 1 ( N 1 N n  n n  n   n  ( 1  ) 2  X ] 1 } ] )  XM ] fpc 1  (  n 2  M n  1 ] ] 2 (258) 1 n  n n 1 N ) 2  X  1 )  XM 2  M  1 N (  n  1 n ) 2  M  [  XM [ 2  X E  (  mw )  E [ r MijY 1   r r XijY XijMij r 2 XijMij S 2 YW S 2 Mij ] (259) r MijY  [ 1 ( g n  1)  n i g j 1 ( g n  1)  n i g j (Y - Y )(M - M ) j ij ij j  n i g j ][ (M - M )(M - M ) j ij ij j 1 ( g n  1) 148 (260) ] (Y - Y )(Y - Y ) j ij ij j   r XijY  [ 1 ( g n  1) 1 ( g n  1)  n i g j (Y - Y )(X - X ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j (261) ] (Y - Y )(Y - Y ) j ij ij j r XijMij  [ 1 ( g n  1) 1 ( g n  1)  n i g j (M - M )(X - X ) j ij ij j  n i g j ][ (X - X )(X - X ) j ij ij j 1 ( g n  1)  n i g j ] (M - M )(M - M ) j ij ij j S 2 YW  g n 1   ( 1) g n j i (Y - Y )(Y - Y ) j ij ij j S 2 Mij  n g 1   ( 1) g n j i (M - M )(M - M ) j ij ij j (262) (263) (264) n  n fpc 1  fpc 1   MY 2  Y n  n fpc 1  2  M n  n   ( mw E )  n  n fpc 1  2  X n  n fpc 1   XY n  n fpc 1  n  n fpc 1  ( 2  Y fpc 1  2  M  n  n 1  n  n ) 2 2  XM n  n fpc 1  2  X fpc 1   XM n  n fpc 1  2  M n  n fpc 1  2  X n n  n  n fpc 1  fpc 1  2  Y 2  M (265) (X - X )(M - M ) n ] / ij ij j j ij ij (X - X )(X - X ) n ] / j j ] (266) ) 2  X n 1  ) 2 X  MY   2  M   XM XY 2  X 2  XM 2  X    mw [ g  (  b ) E  E [   XM  2  X    n -1 n  [ fpc g 1  n 1 fpc n 2  X XM g  j ( M M X )( - j - X )  j g n  j g i n 1) 1 ( g n  1 ( g n  j j i ) -  X 1)  1 (    w N 1 (  N  n n  2 b X 2  X    n XM 2  X 1 g  j ( X - j 1 X X )(   1 n  n n 1 n  n n fpc 1  fpc 1  149  (  w E )  E [       1 ( g n  1 ( g n  1) 1) j g j g n  i n i  (X - X )(M - M ) j ij ij j (X - X )(X - X ) j ij ij j ]  n fpc  1 n  fpc n  1 n   XM 2  X   w (267) 150 APPENDIX C: Detailed Results from the Simulation Study Table 34. Model convergence rate using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions g BAL Balanced 50 Unbalanced Balanced Unbalanced 200 N 20 100 20 100 20 100 20 100 r MLM 2-1-1 mediation 1-1-1 mediation M L N M L N M L N RX/RM=0.1, ICCX/ICCM=0.05 0.1 1.000 1.000 0.990 1.000 1.000 0.940 1.000 0.950 0.930 0.3 1.000 1.000 0.980 1.000 0.970 0.980 1.000 0.970 0.990 0.5 1.000 1.000 1.000 1.000 0.950 1.000 1.000 0.950 1.000 0.7 1.000 1.000 1.000 1.000 0.920 1.000 1.000 0.980 1.000 0.9 1.000 1.000 1.000 1.000 0.940 1.000 1.000 0.960 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.970 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.980 1.000 0.980 0.970 1.000 1.000 0.910 0.3 1.000 1.000 0.990 1.000 0.970 0.970 1.000 0.950 1.000 0.5 1.000 1.000 1.000 1.000 0.950 1.000 1.000 0.950 1.000 0.7 1.000 1.000 1.000 1.000 0.980 1.000 1.000 0.950 1.000 0.9 1.000 1.000 1.000 1.000 0.940 1.000 1.000 0.980 1.000 0.1 1.000 1.000 0.990 1.000 0.970 0.970 1.000 1.000 0.980 0.3 1.000 1.000 1.000 1.000 0.980 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.990 1.000 0.950 0.990 1.000 0.940 0.980 0.3 1.000 1.000 1.000 1.000 0.940 1.000 1.000 0.990 1.000 0.5 1.000 1.000 1.000 1.000 0.910 1.000 1.000 0.990 1.000 0.7 1.000 1.000 1.000 1.000 0.920 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 0.970 1.000 1.000 0.990 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.990 1.000 0.980 0.940 1.000 0.970 0.940 0.3 1.000 1.000 1.000 1.000 0.960 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 0.970 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 0.980 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 151 Table 34. (cont’d) g BAL 50 200 Balanced Unbalanced Balanced Unbalanced N 20 100 20 100 20 100 20 100 r MLM 2-1-1 mediation 1-1-1 mediation M L N M L N M L N RX/RM =0.1, ICCX/ICCM =0.25 0.1 1.000 1.000 0.990 1.000 1.000 0.990 1.000 0.990 0.970 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 0.970 1.000 0.980 0.950 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 152 Table 34. (cont’d) g BAL 50 200 Balanced Unbalanced Balanced Unbalanced N 20 100 20 100 20 100 20 100 r MLM 2-1-1 mediation 1-1-1 mediation M L N M L N M L N RX/RM =10, ICCX/ICCM =0.05 0.1 1.000 1.000 0.970 1.000 1.000 0.880 1.000 0.980 0.910 0.3 1.000 1.000 1.000 1.000 1.000 0.960 1.000 0.990 0.950 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.920 1.000 0.7 1.000 1.000 1.000 1.000 0.990 1.000 1.000 0.830 1.000 0.9 1.000 1.000 1.000 1.000 0.890 1.000 1.000 0.900 1.000 0.1 1.000 1.000 0.990 1.000 1.000 0.940 1.000 0.990 0.990 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.990 1.000 0.980 0.870 1.000 0.970 0.920 0.3 1.000 1.000 1.000 1.000 0.990 0.970 1.000 0.960 0.970 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.940 1.000 0.7 1.000 1.000 1.000 1.000 0.970 1.000 1.000 0.980 1.000 0.9 1.000 1.000 1.000 1.000 0.900 1.000 1.000 0.930 1.000 0.1 1.000 1.000 1.000 1.000 1.000 0.920 1.000 0.980 0.970 0.3 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.980 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.7 1.000 1.000 1.000 1.000 0.990 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 0.980 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 0.940 1.000 0.950 0.970 0.3 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.980 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.940 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.910 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.870 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.970 1.000 0.960 0.980 1.000 0.960 0.950 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.970 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.940 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.970 1.000 0.9 1.000 1.000 1.000 1.000 0.980 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 0.990 1.000 1.000 0.980 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 153 Table 34. (cont’d) g BAL 50 200 Balanced Unbalanced Balanced Unbalanced N 20 100 20 100 20 100 20 100 r MLM 2-1-1 mediation 1-1-1 mediation M L N M L N M L N RX/RM =10, ICCX/ICCM =0.25 0.1 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.870 0.930 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 0.990 1.000 1.000 0.980 1.000 0.910 0.880 0.3 1.000 1.000 1.000 1.000 0.990 1.000 1.000 0.990 1.000 0.5 1.000 1.000 1.000 1.000 0.970 1.000 1.000 0.950 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.980 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 0.990 1.000 1.000 0.950 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 0.990 1.000 1.000 0.980 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 154 Table 35. Relative bias using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions 1-1-1 mediation βmw βmb 2-1-1 mediation βmw βmb MLM βxw N r βxw βxb βxb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.353 0.303 0.354 0.063 0.062 0.063 0.075 0.077 0.075 0.077 0.077 0.077 0.088 0.088 0.088 -0.001 -0.002 -0.001 0.074 0.074 0.073 0.009 0.009 0.009 0.028 0.028 0.028 0.013 0.013 0.013 0.012 -0.123 -0.036 -0.006 -0.019 -0.005 0.027 0.032 0.006 0.012 0.014 0.016 0.011 0.011 0.012 0.049 0.055 0.023 -0.032 -0.031 -0.054 0.006 0.005 0.008 -0.009 -0.009 0.002 -0.019 -0.019 -0.013 8.046 -14.055 8.883 5.014 -47.504 -5.222 7.882 -11.194 8.823 4.552 -11.780 3.499 5.611 13.288 5.358 3.853 -21.084 23.592 -1.982 -15.489 -8.805 -7.554 -17.237 -11.180 -6.156 -12.423 -7.572 -6.466 -11.288 -6.832 1.720 -57.473 15.508 1.930 -39.953 -16.085 5.914 -18.048 0.861 2.771 -76.338 -6.166 -0.308 -61.549 -4.569 10.383 6.460 -44.728 3.580 -25.259 -8.591 7.126 2.969 6.017 2.240 -6.239 0.840 -0.925 -6.824 -0.595 -0.162 -0.150 -0.137 0.006 -0.001 0.004 -0.001 0.002 -0.001 -0.025 -0.024 -0.025 -0.007 -0.011 -0.007 -0.012 -0.012 -0.012 0.006 0.006 0.007 0.003 0.003 0.003 0.001 0.001 0.001 0.001 0.001 0.001 -0.174 -0.178 -0.147 -0.019 -0.025 -0.018 0.003 -0.004 0.016 0.006 0.006 -0.003 -0.007 0.001 -0.016 0.016 0.018 0.014 0.017 0.021 0.018 0.014 0.013 0.014 -0.010 -0.011 -0.004 -0.012 -0.012 -0.012 RX/RM=0.1, ICCX/ICCM =0.05, g=50, Balanced 0.115 0.116 0.030 0.074 0.078 0.080 0.126 0.108 0.126 0.128 0.125 0.128 0.171 0.186 0.172 -0.072 -0.036 -0.072 -0.013 -0.013 -0.013 0.026 0.026 0.026 -0.004 -0.004 -0.004 0.025 0.025 0.024 7.766 18.215 -5.565 5.779 -38.324 5.864 3.285 -86.857 -3.508 2.017 -63.522 -0.940 0.321 -160.122 -0.685 4.934 7.951 11.036 2.100 -6.270 -1.853 0.411 -4.637 -1.445 0.306 -2.870 -0.437 -0.896 -3.666 -1.108 7.275 34.619 71.253 0.837 -163.231 24.238 14.375 -12.313 21.985 8.586 70.741 7.097 13.993 5.307 14.540 4.829 -16.562 -115.774 -1.582 -14.736 -7.820 -4.817 -14.051 -8.257 -4.634 -11.841 -6.209 -7.889 -14.520 -8.375 -0.190 -0.249 -0.124 -0.045 -0.068 -0.035 -0.158 -0.128 -0.157 -0.105 -0.093 -0.106 -0.089 -0.083 -0.089 0.112 0.113 0.111 -0.028 -0.028 -0.028 0.028 0.028 0.028 0.017 0.017 0.017 -0.009 -0.009 -0.009 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Unbalanced 7.593 8.716 17.583 6.027 -25.226 -13.786 4.040 -49.874 -4.859 3.405 -59.813 -0.398 1.213 -69.215 -0.701 6.240 -7.355 -10.606 2.932 -21.734 -3.365 -0.279 -12.172 -3.302 -1.003 -12.997 -2.554 -2.469 -11.280 -2.312 0.084 0.142 -0.085 0.305 0.342 0.441 -0.073 -0.079 -0.108 0.097 0.099 0.042 0.051 0.049 0.019 0.284 0.286 0.210 -0.006 -0.004 -0.043 0.021 0.023 0.032 0.091 0.091 0.067 0.081 0.081 0.086 7.485 38.956 103.747 8.009 12.993 -30.653 2.096 38.748 -16.291 -8.990 -228.776 -16.647 -4.850 -95.886 -10.998 7.456 -110.116 231.666 13.842 -1.588 -0.934 10.957 10.136 9.788 11.155 1.851 7.953 6.645 -0.253 6.134 -0.084 -0.182 -0.093 0.024 0.001 -0.040 0.037 0.044 0.061 -0.049 -0.042 -0.012 -0.007 -0.001 0.016 -0.072 -0.084 -0.173 -0.026 -0.028 -0.008 -0.013 -0.014 -0.006 -0.050 -0.050 -0.076 -0.070 -0.071 -0.086 155 6.620 1.794 0.307 5.878 8.140 -6.154 4.564 10.977 4.525 3.906 14.265 3.429 3.216 -1.577 3.098 1.496 0.019 3.992 -2.036 -3.930 -3.280 -3.134 -4.448 -3.719 -2.980 -3.992 -3.244 -3.794 -4.654 -3.867 1.016 13.880 -44.822 3.820 -27.827 0.753 -1.486 -12.112 -3.094 0.636 -18.802 -1.028 0.816 -28.982 -0.713 3.832 23.836 2.699 2.884 3.430 3.259 -1.516 -3.940 -1.193 -1.472 -4.474 -1.076 -1.054 -2.092 0.177 -0.144 -0.053 -0.135 -0.085 -0.078 -0.083 -0.026 -0.025 -0.025 -0.009 -0.020 -0.009 -0.011 -0.025 -0.011 -0.003 -0.008 -0.003 -0.004 -0.004 -0.004 -0.026 -0.026 -0.025 -0.002 -0.002 -0.002 -0.011 -0.011 -0.011 0.122 0.053 0.094 -0.126 -0.128 -0.099 0.044 0.052 0.001 0.075 0.056 0.049 0.008 0.008 -0.019 -0.116 -0.115 -0.083 0.018 0.019 0.025 -0.019 -0.019 -0.026 0.023 0.024 0.023 -0.003 -0.003 -0.009 9.736 12.917 -71.432 7.382 -9.375 5.009 2.134 -40.397 -5.903 0.613 -64.052 -3.585 0.114 -49.769 -0.828 4.433 -3.223 4.523 0.591 -7.455 -3.745 -1.951 -7.368 -4.058 -1.320 -5.086 -2.190 -1.770 -4.595 -1.996 7.491 -1.489 -2.467 5.528 -22.163 3.276 5.702 -21.803 0.602 -0.294 -36.563 -0.265 0.636 -21.080 3.085 7.947 21.158 19.674 1.529 -22.198 -8.441 4.670 -8.443 -0.883 1.806 -8.680 -1.649 0.223 -10.072 -2.162 Table 35. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.037 0.041 0.034 -0.034 -0.037 -0.034 0.006 0.006 0.006 -0.006 -0.006 -0.006 -0.010 -0.010 -0.010 0.018 0.018 0.018 -0.013 -0.013 -0.013 -0.010 -0.010 -0.010 -0.007 -0.007 -0.006 -0.005 -0.005 -0.005 0.093 0.064 0.090 0.021 0.022 0.023 0.034 0.033 0.035 0.032 0.032 0.034 0.023 0.022 0.023 -0.003 -0.008 0.021 0.019 0.018 0.004 0.004 0.004 -0.001 0.000 0.000 0.000 0.008 0.008 0.007 8.166 -16.815 -36.043 2.993 -98.373 -18.104 3.576 -81.523 -1.387 1.422 -115.174 -1.550 -1.083 -146.830 -2.104 5.792 0.039 1.245 1.407 -4.355 -1.826 0.222 -3.919 -1.430 -1.340 -4.799 -2.172 -1.990 -4.830 -2.220 7.571 -8.745 -22.425 7.833 -1.319 7.054 5.783 -17.370 2.331 4.955 -12.297 3.042 1.361 -23.003 0.964 6.834 1.655 -0.470 4.504 -4.941 0.747 4.309 -3.021 1.359 1.728 -5.284 0.381 1.419 -4.482 0.823 0.027 0.029 0.030 0.001 0.001 0.001 -0.013 -0.011 -0.013 0.000 -0.001 -0.001 -0.004 -0.003 -0.004 -0.008 -0.008 -0.007 -0.003 -0.003 -0.003 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 -0.002 -0.002 -0.002 -0.004 -0.008 -0.004 -0.007 -0.009 -0.003 0.002 0.004 0.002 -0.018 -0.019 -0.018 -0.014 -0.014 -0.012 0.022 0.020 0.017 0.003 0.002 0.006 0.007 0.007 0.004 0.010 0.010 0.010 0.012 0.012 0.012 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Balanced 0.062 0.048 0.065 -0.013 -0.010 -0.013 -0.034 -0.039 -0.035 0.011 0.011 0.011 0.005 -0.002 0.006 -0.004 -0.004 -0.004 0.006 0.006 0.006 -0.017 -0.017 -0.016 -0.020 -0.020 -0.019 -0.017 -0.017 -0.018 2.456 -68.004 -32.630 7.454 29.945 3.109 7.748 -80.895 6.261 2.555 -82.935 0.021 0.560 -189.302 -0.290 6.073 2.254 2.448 2.995 -2.413 0.058 5.440 3.749 4.736 2.671 0.361 2.112 1.834 -0.168 1.664 8.642 5.313 4.206 6.858 -68.467 2.306 4.985 -106.808 1.259 2.738 -197.361 0.197 1.328 -259.365 0.525 6.025 -1.149 1.061 2.717 -2.599 -0.201 1.513 -2.287 0.017 0.897 -2.043 0.204 0.102 -2.404 -0.097 -0.108 -0.089 -0.091 -0.062 -0.069 -0.062 -0.034 -0.034 -0.033 -0.075 -0.076 -0.076 -0.067 -0.068 -0.067 0.007 0.007 0.007 -0.001 -0.001 -0.001 0.012 0.012 0.012 0.001 0.001 0.001 0.010 0.010 0.010 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Unbalanced 8.176 3.322 11.386 6.606 -24.770 1.513 4.711 -42.745 0.815 3.546 -43.269 0.868 1.727 -55.245 0.847 6.970 -28.877 -1.524 4.716 -7.140 -0.220 3.348 -3.799 1.328 1.756 -5.384 0.409 0.685 -5.814 0.308 0.031 0.068 0.114 -0.043 -0.051 -0.008 0.067 0.064 0.087 0.005 0.004 -0.010 0.028 0.026 0.040 -0.003 0.009 -0.037 0.075 0.076 0.090 -0.019 -0.020 0.004 -0.018 -0.018 -0.006 -0.024 -0.024 -0.010 10.443 -63.048 -107.174 4.698 -21.115 -4.321 6.221 -65.010 2.519 5.661 2.930 5.626 4.278 -35.125 4.507 9.382 -4.515 34.785 6.569 -3.317 0.858 7.197 2.699 5.595 6.003 0.267 2.663 4.422 -1.216 2.098 -0.064 -0.062 -0.110 -0.019 -0.013 0.056 -0.044 -0.045 -0.012 -0.055 -0.055 -0.045 -0.036 -0.036 -0.027 0.009 -0.002 0.018 -0.043 -0.043 -0.072 -0.027 -0.025 -0.033 -0.013 -0.014 -0.017 -0.029 -0.028 -0.025 156 7.560 19.996 5.537 6.264 9.929 5.528 7.446 4.574 7.374 6.612 6.972 6.423 6.026 -2.669 5.938 2.766 0.138 0.619 3.286 2.546 2.795 3.626 3.220 3.428 2.956 2.608 2.861 2.916 2.662 2.899 4.116 -32.946 -9.268 3.670 -0.674 -0.205 1.624 -7.426 -0.502 0.670 -3.374 -0.107 0.154 -5.256 -0.418 1.444 -8.326 -5.277 0.044 -3.070 -1.920 -1.596 -4.122 -2.529 -2.536 -4.382 -2.045 -2.430 -3.982 -2.226 -0.027 -0.038 -0.046 -0.026 -0.028 -0.026 -0.028 -0.029 -0.028 -0.041 -0.042 -0.041 -0.027 -0.028 -0.027 -0.008 -0.008 -0.008 -0.010 -0.010 -0.010 -0.011 -0.011 -0.011 -0.010 -0.010 -0.010 -0.006 -0.006 -0.006 -0.080 -0.111 -0.080 -0.035 -0.040 -0.067 -0.034 -0.039 -0.039 -0.019 -0.022 -0.028 -0.012 -0.014 -0.021 -0.018 -0.023 -0.036 0.022 0.020 0.007 0.019 0.018 0.010 0.013 0.013 0.007 0.018 0.018 0.010 7.943 11.297 -4.656 5.531 -32.053 -0.839 3.378 -87.589 -2.186 1.993 -80.695 -0.567 0.680 -120.282 -0.171 6.467 0.574 2.336 3.510 -1.437 0.821 0.637 -3.472 -0.993 1.255 -1.486 0.600 0.717 -1.520 0.537 7.261 -9.534 -27.873 7.386 11.593 7.652 4.277 -8.018 2.718 3.439 -35.148 2.584 1.618 -18.797 2.802 5.645 -14.241 -9.909 3.818 -6.662 -0.957 2.627 -6.262 -0.273 0.213 -7.034 -0.764 -0.514 -6.961 -0.492 Table 35. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.149 0.156 0.136 0.068 0.067 0.068 0.073 0.073 0.073 0.083 0.083 0.083 0.020 0.020 0.020 -0.111 -0.111 -0.111 0.013 0.013 0.013 -0.010 -0.010 -0.010 0.017 0.017 0.017 0.003 0.003 0.003 0.013 -0.020 0.013 -0.024 -0.025 0.033 0.006 0.006 -0.040 -0.017 -0.018 -0.026 -0.033 -0.033 -0.047 -0.105 -0.095 -0.041 -0.044 -0.044 -0.065 -0.022 -0.022 -0.018 -0.042 -0.042 -0.035 -0.028 -0.028 -0.021 8.088 1.680 6.888 3.850 -0.967 1.407 3.286 0.789 2.313 2.245 0.158 1.752 1.381 -0.391 1.243 2.145 0.261 0.541 2.168 1.439 1.681 1.054 0.559 0.820 1.055 0.696 0.952 1.024 0.746 0.996 3.310 -16.723 -18.879 0.557 -6.287 -3.644 1.529 -2.091 -0.491 -1.039 -4.083 -1.848 -0.827 -3.420 -1.747 4.224 -0.048 0.866 -0.046 -2.311 -1.964 0.111 -1.436 -1.395 -0.386 -1.499 -1.554 -0.855 -1.811 -1.820 0.029 0.045 0.016 0.111 0.110 0.111 -0.032 -0.033 -0.032 0.021 0.020 0.021 0.004 0.004 0.004 0.050 0.050 0.050 0.008 0.008 0.008 0.068 0.068 0.067 0.062 0.062 0.062 0.062 0.062 0.063 0.036 -0.024 0.050 0.011 0.013 0.047 -0.042 -0.044 -0.024 0.034 0.032 0.017 0.004 0.004 0.029 0.011 0.017 0.009 -0.008 -0.006 -0.029 0.000 0.000 0.008 -0.017 -0.016 -0.027 0.001 0.001 -0.002 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Balanced -0.309 -0.352 -0.271 -0.337 -0.337 -0.337 -0.148 -0.148 -0.148 -0.212 -0.212 -0.213 -0.187 -0.187 -0.187 0.065 0.065 0.065 -0.048 -0.048 -0.048 0.003 0.003 0.002 -0.021 -0.021 -0.021 -0.022 -0.022 -0.022 6.897 14.108 14.645 6.480 6.074 6.225 0.688 -3.654 -0.799 3.428 1.497 3.013 2.055 0.384 1.930 -2.274 -6.050 -5.454 -4.715 -6.279 -5.754 -4.897 -5.802 -5.326 -5.001 -5.669 -5.188 -5.187 -5.708 -5.236 6.457 1.939 2.122 0.879 -6.211 -2.681 -1.652 -6.484 -3.428 -1.221 -4.333 -1.935 -2.643 -5.312 -2.854 -0.432 -4.020 -3.480 -1.731 -2.962 -2.550 -0.573 -1.248 -0.893 -1.091 -1.615 -1.242 -0.708 -1.094 -0.744 -0.326 -0.392 -0.307 -0.182 -0.182 -0.182 -0.143 -0.143 -0.143 -0.092 -0.092 -0.092 -0.081 -0.081 -0.081 0.044 0.044 0.044 -0.016 -0.016 -0.016 0.019 0.019 0.019 0.015 0.015 0.014 0.026 0.026 0.026 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Unbalanced 6.130 -5.821 28.088 5.788 5.731 5.996 2.777 0.292 3.690 3.785 1.734 4.739 3.814 2.660 5.454 7.930 11.474 9.279 3.157 2.393 4.587 2.656 2.224 4.834 2.404 1.966 4.654 3.170 2.904 5.339 -0.280 -0.145 -0.275 0.090 0.058 0.136 0.082 0.076 0.138 0.113 0.110 0.133 0.089 0.086 0.108 -0.240 -0.244 -0.278 0.006 0.007 -0.045 0.079 0.078 0.100 -0.013 -0.013 -0.019 0.010 0.010 0.008 4.776 -0.056 -5.677 4.194 0.638 2.478 1.226 -2.300 0.476 1.343 -1.283 1.478 0.390 -1.728 1.162 0.398 -8.883 -5.582 2.632 0.784 0.761 1.973 0.683 0.519 1.882 0.837 0.474 1.902 1.139 1.082 0.037 -0.049 0.070 -0.264 -0.274 -0.391 -0.068 -0.075 -0.131 0.150 0.146 0.105 0.003 0.000 -0.051 0.075 0.073 0.017 -0.030 -0.034 0.005 -0.010 -0.010 0.011 0.054 0.054 0.063 0.039 0.039 0.071 157 0.668 -3.622 -19.842 6.626 7.670 7.099 0.692 -2.454 -0.419 0.242 -1.952 -0.244 0.268 -1.460 0.135 10.630 12.100 11.766 4.766 4.276 4.450 5.974 5.782 5.888 5.146 4.980 5.103 5.590 5.476 5.578 12.312 36.420 14.102 1.018 -4.508 -0.723 2.546 -0.286 1.665 3.330 1.678 3.871 1.822 0.048 2.570 4.654 1.726 2.342 6.302 5.118 6.211 0.516 -1.866 -0.270 1.578 0.484 2.001 1.046 0.014 1.619 0.125 0.093 0.102 -0.060 -0.060 -0.060 0.052 0.052 0.052 0.005 0.005 0.005 0.002 0.002 0.002 0.117 0.117 0.117 -0.003 -0.003 -0.003 -0.024 -0.024 -0.024 -0.003 -0.003 -0.003 -0.022 -0.022 -0.022 0.119 0.156 0.089 -0.043 -0.038 -0.020 0.165 0.161 0.209 0.090 0.089 0.092 0.080 0.080 0.086 0.020 0.023 0.092 0.004 0.004 -0.002 0.009 0.010 0.015 0.045 0.045 0.054 0.010 0.010 0.017 5.446 0.266 3.820 1.208 -2.474 -0.839 0.060 -3.665 -1.373 -0.722 -3.366 -1.348 -1.531 -3.766 -1.706 1.921 -1.003 -0.550 0.568 -0.430 -0.093 0.843 0.283 0.580 0.685 0.271 0.567 0.318 -0.012 0.285 6.530 -7.423 -1.381 1.851 -6.403 -2.667 -1.294 -6.197 -3.192 -1.415 -4.668 -2.079 -2.113 -5.044 -2.430 3.206 -2.349 -1.416 3.562 2.055 2.547 1.722 0.547 1.146 1.585 0.671 1.162 1.784 1.090 1.581 Table 35. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 -0.049 -0.049 -0.049 0.002 0.001 0.001 0.001 0.001 0.001 0.010 0.010 0.010 0.002 0.002 0.002 0.052 0.052 0.052 0.003 0.003 0.003 0.008 0.008 0.007 0.002 0.002 0.002 0.004 0.004 0.004 -0.022 -0.023 -0.022 -0.059 -0.061 -0.073 -0.012 -0.014 -0.013 -0.004 -0.005 -0.007 0.011 0.010 0.006 0.018 0.013 0.045 0.034 0.034 0.033 0.004 0.004 0.017 0.022 0.022 0.030 0.021 0.021 0.034 4.371 -3.293 -0.311 2.127 -1.944 -0.145 0.998 -1.753 -0.116 0.733 -1.319 0.230 0.012 -1.729 -0.130 1.299 -1.276 -0.943 -0.171 -1.122 -0.812 -0.618 -1.213 -0.904 -0.564 -0.987 -0.684 -0.761 -1.094 -0.793 6.981 5.174 5.562 3.688 0.557 1.899 1.241 -1.742 0.095 0.606 -1.958 -0.199 0.435 -1.601 0.188 3.002 -0.877 -0.170 1.048 -1.032 -0.315 0.808 -0.419 0.371 -0.078 -1.069 0.126 0.078 -0.700 0.347 0.002 0.001 0.001 0.000 -0.001 0.000 -0.021 -0.022 -0.021 -0.025 -0.025 -0.025 -0.019 -0.019 -0.019 0.022 0.021 0.021 -0.015 -0.015 -0.015 -0.006 -0.006 -0.006 -0.004 -0.004 -0.004 -0.002 -0.002 -0.002 0.033 0.047 0.033 -0.009 -0.016 -0.014 0.003 0.000 0.019 -0.033 -0.035 -0.004 -0.009 -0.010 0.010 -0.069 -0.066 -0.109 0.024 0.025 0.022 0.025 0.024 0.026 0.017 0.017 0.002 0.012 0.012 0.011 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Balanced 0.091 0.094 0.091 0.037 0.036 0.037 0.003 0.003 0.003 0.032 0.032 0.032 -0.001 -0.001 -0.001 -0.023 -0.023 -0.023 -0.052 -0.052 -0.053 -0.015 -0.015 -0.015 -0.015 -0.015 -0.016 -0.004 -0.004 -0.004 0.057 0.058 0.057 -0.042 -0.043 -0.043 -0.019 -0.019 -0.019 -0.015 -0.015 -0.015 -0.065 -0.065 -0.065 -0.062 -0.063 -0.062 -0.001 -0.001 0.000 -0.030 -0.030 -0.029 -0.006 -0.006 -0.006 -0.009 -0.009 -0.009 5.883 -1.165 1.969 2.313 -1.793 0.060 1.354 -1.355 0.270 0.905 -1.147 0.412 0.344 -1.392 0.204 2.968 0.915 1.178 1.665 0.869 1.131 0.947 0.420 0.696 0.693 0.294 0.578 0.590 0.277 0.559 5.308 3.917 1.448 2.396 -1.309 0.358 3.841 1.996 3.113 2.863 1.249 2.481 2.873 1.704 2.780 4.023 2.469 2.697 2.038 1.221 1.493 2.946 2.544 2.753 1.744 1.392 1.646 1.803 1.526 1.779 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Unbalanced 7.411 4.305 7.803 1.418 -3.238 -0.916 -0.560 -4.234 -1.342 -2.301 -5.415 -1.336 -1.619 -4.106 -0.908 3.167 -0.552 0.348 0.819 -1.207 0.252 -0.567 -2.109 -0.619 -0.761 -1.819 0.092 -1.227 -2.134 -0.379 0.236 0.269 0.237 0.059 0.058 0.017 0.024 0.022 0.090 0.056 0.054 0.053 0.015 0.013 -0.002 -0.121 -0.123 -0.135 -0.059 -0.056 -0.079 -0.023 -0.023 -0.029 -0.024 -0.023 -0.034 -0.034 -0.034 -0.036 4.710 -3.964 -0.969 1.411 -3.185 -0.877 1.100 -2.082 0.062 0.163 -2.279 -0.067 -0.736 -2.928 -0.607 4.397 1.264 1.885 1.650 -0.435 0.031 0.930 -0.553 -0.369 0.731 -0.332 -0.046 0.268 -0.615 -0.388 -0.174 -0.175 -0.174 -0.060 -0.068 -0.009 -0.001 -0.006 0.003 0.064 0.059 0.076 0.026 0.024 0.031 0.014 0.004 -0.012 -0.014 -0.016 0.001 0.021 0.020 0.007 0.002 0.001 -0.014 0.004 0.004 -0.012 158 8.044 6.848 6.323 3.838 -0.386 1.634 3.112 0.518 2.117 1.438 -1.088 0.859 1.956 0.180 1.822 0.982 -2.270 -1.790 0.184 -0.974 -0.584 -0.796 -1.584 -1.166 -0.068 -0.600 -0.218 -0.150 -0.568 -0.187 7.256 -14.058 4.089 6.196 2.998 4.633 4.436 2.614 4.512 4.160 2.448 4.423 3.466 1.914 3.741 5.980 4.160 4.650 5.408 3.876 3.816 1.836 0.198 0.915 2.486 1.392 1.971 1.888 0.968 1.646 -0.156 -0.157 -0.156 -0.075 -0.075 -0.074 0.023 0.023 0.023 -0.011 -0.011 -0.011 0.007 0.007 0.007 -0.049 -0.049 -0.049 0.001 0.001 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.004 0.072 0.046 0.072 0.045 0.041 0.056 -0.012 -0.012 -0.002 -0.019 -0.020 0.004 -0.001 -0.001 0.016 -0.086 -0.086 -0.098 0.010 0.010 0.009 -0.010 -0.010 -0.014 -0.012 -0.012 -0.019 -0.015 -0.015 -0.018 4.888 -0.725 1.855 2.704 -0.467 0.926 1.688 -1.028 0.598 0.849 -1.168 0.361 0.299 -1.434 0.158 2.222 0.196 0.458 0.962 0.096 0.378 0.522 -0.023 0.263 0.394 -0.006 0.278 -0.026 -0.351 -0.058 5.439 -4.402 -1.440 2.865 -1.536 0.288 2.755 -0.186 1.488 1.210 -0.968 0.770 0.570 -1.339 0.613 2.470 -1.722 -0.999 0.711 -1.641 -1.086 -0.252 -1.806 -1.242 -0.014 -1.122 -0.680 -0.677 -1.590 -1.036 Table 35. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.024 0.063 -0.001 -0.064 -0.033 -0.064 -0.085 -0.070 -0.085 -0.073 -0.059 -0.073 -0.047 -0.037 -0.047 0.001 0.005 0.005 0.003 0.003 0.003 -0.014 -0.013 -0.013 0.010 0.010 0.010 0.003 0.003 0.003 0.010 0.077 -0.023 0.071 0.096 0.008 0.013 0.034 -0.024 0.007 0.022 -0.016 0.020 0.033 -0.013 0.041 0.056 0.080 -0.002 0.004 0.020 0.029 0.032 0.065 0.022 0.025 0.040 0.021 0.023 0.035 -0.825 -0.553 -0.363 -0.626 1.121 -0.176 -0.471 2.527 -0.004 -0.270 3.289 0.002 -0.081 4.053 0.004 -0.570 0.548 -0.250 -0.251 0.337 0.060 -0.132 0.288 0.028 -0.073 0.234 -0.003 -0.022 0.226 -0.003 -0.829 -0.437 -0.934 -0.698 0.613 -0.204 -0.519 1.826 0.075 -0.324 2.537 -0.036 -0.178 2.960 -0.083 -0.686 0.633 0.143 -0.435 0.654 0.037 -0.227 0.711 0.091 -0.093 0.729 0.078 0.025 0.721 0.090 -0.008 0.125 -0.017 0.043 0.104 0.039 0.030 0.080 0.029 0.032 0.083 0.032 0.042 0.097 0.042 -0.019 -0.001 -0.020 -0.007 -0.002 -0.007 -0.014 -0.011 -0.014 -0.011 -0.008 -0.010 -0.013 -0.012 -0.013 0.037 0.115 0.087 0.013 0.074 0.008 0.022 0.073 0.024 0.014 0.061 0.011 0.020 0.070 0.023 0.044 0.085 0.051 -0.004 0.016 -0.008 -0.008 0.007 0.010 -0.009 0.003 -0.003 -0.008 0.007 0.000 -0.481 -0.224 -0.453 0.107 0.153 0.135 -0.050 -0.016 -0.051 -0.041 -0.020 -0.041 0.001 -0.008 0.001 0.132 0.151 0.124 -0.001 -0.001 0.000 -0.016 -0.016 -0.016 -0.005 -0.005 -0.005 -0.034 -0.033 -0.034 RX/RM =10, ICCX/ICCM =0.05, g=50, Balanced -0.125 0.007 -0.290 0.089 0.092 0.035 0.047 0.097 0.047 -0.004 -0.033 -0.004 0.023 0.071 0.024 0.082 0.095 0.072 -0.062 -0.062 -0.062 0.048 0.048 0.048 0.039 0.039 0.039 0.028 0.028 0.028 RX/RM =10, ICCX/ICCM =0.05, g=50, Unbalanced 0.221 0.407 0.252 -0.080 -0.077 -0.034 0.101 0.134 0.009 0.059 0.075 -0.018 0.067 0.091 0.030 0.181 0.223 0.208 0.036 0.029 0.092 -0.047 -0.046 -0.021 0.025 0.025 0.023 -0.014 -0.015 -0.013 0.022 0.291 0.069 0.096 0.189 0.179 -0.079 -0.026 0.024 -0.043 -0.010 0.060 -0.042 -0.015 0.003 0.052 0.084 0.066 -0.076 -0.087 -0.149 0.023 0.018 0.020 -0.008 -0.002 -0.013 -0.026 -0.023 -0.036 -0.805 -0.340 -0.794 -0.635 0.230 -0.043 -0.448 0.438 0.123 -0.252 0.405 0.030 -0.074 0.438 0.019 -0.580 0.119 0.452 -0.296 0.212 0.026 -0.131 0.242 0.037 -0.049 0.238 0.028 -0.008 0.240 0.013 -0.829 -0.436 -0.651 -0.657 0.115 0.116 -0.497 0.332 0.044 -0.309 0.434 0.011 -0.117 0.402 -0.019 -0.711 -0.033 -0.172 -0.464 0.209 0.213 -0.295 0.244 0.000 -0.169 0.284 -0.015 -0.048 0.280 -0.011 -0.761 0.083 0.140 -0.585 2.010 0.619 -0.344 4.999 0.330 -0.102 8.022 0.300 0.041 8.931 0.158 -0.671 -0.719 -0.155 -0.340 0.451 -0.126 -0.169 0.242 -0.018 -0.100 0.223 -0.028 -0.055 0.215 -0.035 -0.764 0.049 0.710 -0.586 1.005 -0.533 -0.337 4.116 0.493 -0.168 3.102 0.217 0.010 5.005 0.123 -0.721 0.522 -0.175 -0.362 1.685 0.423 -0.159 1.286 0.219 0.027 1.322 0.180 0.085 1.204 0.113 -0.559 -0.031 -0.043 -0.239 -0.164 -0.157 -0.190 -0.728 -0.190 -0.151 -1.013 -0.152 -0.135 -1.278 -0.138 -0.096 0.366 0.137 -0.042 -0.039 -0.002 -0.016 -0.010 -0.007 -0.012 -0.013 -0.009 -0.010 -0.015 -0.010 -0.559 0.008 -0.234 -0.325 -0.064 -0.190 -0.229 -0.478 -0.203 -0.177 -0.354 -0.167 -0.164 -0.696 -0.157 -0.282 0.033 0.185 -0.167 -0.228 -0.118 -0.103 -0.108 -0.068 -0.083 -0.147 -0.058 -0.085 -0.151 -0.059 0.049 0.181 0.156 0.018 0.059 0.025 -0.048 -0.011 -0.048 0.058 0.086 0.057 0.014 0.028 0.014 0.018 0.028 0.025 0.081 0.082 0.081 0.024 0.024 0.024 -0.006 -0.005 -0.005 0.016 0.016 0.016 -0.106 -0.015 -0.070 -0.134 -0.100 -0.106 -0.007 0.006 -0.002 -0.009 0.004 0.003 -0.034 -0.026 -0.017 0.090 0.103 0.119 0.013 0.019 0.016 -0.001 0.006 -0.014 0.014 0.015 0.008 0.018 0.021 0.022 -0.808 -0.366 -1.072 -0.604 1.037 0.041 -0.403 1.701 0.135 -0.279 2.342 -0.023 -0.073 2.875 0.016 -0.562 0.396 0.150 -0.281 0.315 0.028 -0.146 0.236 0.001 -0.025 0.299 0.050 -0.002 0.245 0.017 -0.812 -0.277 -0.769 -0.654 1.201 -0.132 -0.510 1.505 0.002 -0.341 1.862 -0.048 -0.099 2.346 -0.001 -0.709 0.384 -0.623 -0.479 0.526 0.011 -0.320 0.499 -0.063 -0.146 0.630 -0.016 -0.029 0.609 -0.021 159 Table 35. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.005 0.053 0.005 -0.019 -0.004 -0.019 -0.030 -0.021 -0.030 0.002 0.008 0.002 -0.007 -0.001 -0.007 0.016 0.016 0.016 0.002 0.003 0.002 0.009 0.009 0.009 0.008 0.008 0.009 0.006 0.006 0.006 -0.068 -0.005 -0.073 0.018 0.032 0.028 0.005 0.014 0.002 -0.002 0.010 0.002 -0.007 0.003 -0.018 -0.030 -0.025 0.006 -0.025 -0.021 -0.018 0.002 0.005 0.001 0.000 0.002 -0.002 0.000 0.002 -0.003 -0.829 0.001 -0.396 -0.621 2.685 0.003 -0.452 3.625 -0.048 -0.274 4.369 -0.027 -0.095 5.170 -0.012 -0.549 0.335 0.074 -0.281 0.200 -0.012 -0.134 0.222 0.009 -0.065 0.213 0.002 -0.005 0.229 0.014 -0.805 0.452 -0.445 -0.649 1.824 -0.084 -0.511 2.275 -0.092 -0.311 2.735 -0.037 -0.117 3.099 -0.024 -0.698 0.446 0.116 -0.458 0.535 -0.015 -0.299 0.502 -0.028 -0.152 0.556 -0.010 -0.048 0.550 -0.021 0.047 0.132 0.059 0.021 0.077 0.021 0.019 0.070 0.019 0.014 0.062 0.014 0.010 0.059 0.011 -0.008 -0.001 -0.008 0.002 0.004 0.002 0.008 0.008 0.007 0.002 0.002 0.001 0.002 0.002 0.002 -0.063 0.026 -0.065 -0.003 0.046 -0.010 -0.005 0.041 -0.008 0.005 0.050 0.001 -0.003 0.043 -0.003 0.003 0.030 -0.010 -0.010 0.006 -0.013 -0.004 0.009 -0.007 -0.006 0.006 -0.008 -0.004 0.008 -0.006 RX/RM =10, ICCX/ICCM =0.05, g=200, Balanced -0.345 -0.320 -0.356 -0.090 -0.084 -0.090 0.032 0.038 0.032 -0.020 -0.023 -0.020 -0.034 -0.051 -0.033 0.017 0.017 0.016 -0.051 -0.051 -0.051 0.010 0.010 0.010 -0.003 -0.003 -0.002 -0.005 -0.005 -0.006 -0.746 0.456 0.226 -0.559 3.698 0.107 -0.355 9.584 0.159 -0.185 11.909 0.107 0.075 14.811 0.182 -0.597 0.225 -0.073 -0.253 0.353 0.065 -0.156 0.247 -0.003 -0.087 0.230 -0.015 -0.031 0.234 -0.011 -0.808 -0.148 -0.200 -0.629 0.301 -0.055 -0.443 0.384 -0.009 -0.275 0.400 -0.027 -0.096 0.392 -0.013 -0.576 0.108 0.010 -0.282 0.201 0.004 -0.148 0.208 -0.002 -0.063 0.228 0.008 -0.014 0.228 0.006 -0.167 -0.030 -0.168 0.036 0.062 0.035 0.073 0.094 0.073 0.021 0.011 0.021 0.026 0.034 0.026 -0.023 -0.022 -0.022 0.048 0.048 0.048 0.035 0.035 0.036 0.010 0.010 0.010 0.017 0.017 0.017 RX/RM =10, ICCX/ICCM =0.05, g=200, Unbalanced -0.778 -0.228 -0.433 -0.614 3.186 0.021 -0.424 5.559 0.097 -0.172 8.416 0.202 0.068 8.367 0.197 -0.665 1.496 0.425 -0.437 0.881 0.073 -0.230 0.801 0.089 -0.091 0.758 0.056 0.041 0.819 0.082 -0.112 -0.096 -0.051 0.106 0.082 0.136 -0.052 -0.069 -0.021 -0.031 -0.023 0.020 -0.019 -0.021 -0.001 -0.042 -0.056 -0.076 -0.008 -0.010 -0.006 -0.008 -0.010 -0.024 -0.029 -0.028 -0.050 -0.023 -0.023 -0.030 -0.821 -0.036 -0.209 -0.658 0.301 -0.057 -0.492 0.367 -0.055 -0.315 0.394 -0.037 -0.112 0.389 -0.016 -0.700 0.132 -0.031 -0.456 0.252 0.013 -0.278 0.272 0.005 -0.140 0.285 0.007 -0.033 0.295 0.007 -0.193 -0.066 -0.244 -0.066 -0.028 0.024 0.007 0.054 0.092 -0.021 -0.004 0.020 -0.017 -0.001 0.005 0.024 0.020 0.026 -0.011 -0.009 0.009 -0.037 -0.034 -0.046 -0.021 -0.019 -0.039 -0.007 -0.004 -0.016 160 -0.549 -0.010 -0.120 -0.278 -0.513 -0.160 -0.211 -1.484 -0.186 -0.164 -1.872 -0.158 -0.166 -2.302 -0.169 -0.162 -0.036 -0.020 -0.057 -0.048 -0.037 -0.031 -0.031 -0.024 -0.020 -0.023 -0.018 -0.022 -0.027 -0.021 -0.538 0.228 -0.053 -0.294 -0.410 -0.135 -0.215 -0.814 -0.153 -0.200 -1.311 -0.181 -0.188 -1.283 -0.169 -0.327 -0.225 -0.103 -0.127 -0.093 -0.050 -0.095 -0.105 -0.062 -0.082 -0.104 -0.061 -0.080 -0.113 -0.062 -0.120 -0.109 -0.104 -0.028 -0.006 -0.028 -0.028 -0.016 -0.028 -0.002 0.001 -0.001 -0.008 -0.008 -0.007 -0.003 -0.003 -0.003 -0.004 -0.004 -0.004 0.017 0.017 0.017 0.005 0.005 0.005 0.008 0.008 0.008 -0.059 -0.022 -0.034 0.054 0.067 0.079 -0.013 -0.003 -0.007 0.013 0.021 -0.003 0.019 0.029 0.014 0.014 0.026 0.017 -0.026 -0.022 -0.039 0.000 0.002 -0.003 0.005 0.008 -0.003 0.000 0.002 -0.002 -0.807 0.308 -0.741 -0.638 1.563 -0.144 -0.434 2.775 -0.018 -0.269 3.093 -0.031 -0.089 3.467 -0.009 -0.562 0.279 0.035 -0.285 0.225 -0.003 -0.150 0.216 -0.005 -0.088 0.192 -0.021 -0.037 0.195 -0.018 -0.824 0.269 -0.398 -0.677 1.159 -0.159 -0.511 2.106 -0.039 -0.312 2.520 -0.004 -0.112 2.652 0.003 -0.704 0.610 0.087 -0.486 0.473 -0.054 -0.303 0.496 -0.032 -0.166 0.501 -0.029 -0.051 0.521 -0.027 Table 35. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.039 0.344 0.039 -0.044 0.017 -0.044 -0.019 0.001 -0.018 -0.005 0.004 -0.005 -0.034 -0.032 -0.033 -0.036 -0.031 -0.036 0.048 0.048 0.048 -0.012 -0.012 -0.012 0.005 0.005 0.005 0.004 0.004 0.004 -0.009 0.271 0.014 -0.137 -0.056 -0.070 -0.077 -0.040 -0.020 -0.035 -0.010 0.032 -0.049 -0.028 0.005 0.039 0.078 -0.039 -0.039 -0.033 -0.053 0.063 0.065 0.080 0.004 0.006 0.004 0.026 0.029 0.035 -0.543 0.043 -0.214 -0.231 0.200 0.022 -0.115 0.188 0.007 -0.049 0.191 0.008 -0.016 0.177 -0.001 -0.205 0.024 -0.010 -0.069 0.024 -0.008 -0.029 0.030 -0.001 -0.015 0.027 -0.003 -0.006 0.027 -0.003 -0.539 0.127 0.348 -0.228 0.202 0.033 -0.136 0.171 -0.008 -0.080 0.150 -0.021 -0.022 0.167 -0.010 -0.328 0.062 0.009 -0.127 0.069 0.000 -0.066 0.062 -0.007 -0.031 0.065 -0.004 -0.010 0.065 -0.006 0.228 0.644 0.214 -0.024 0.093 -0.024 0.095 0.152 0.095 0.137 0.173 0.137 0.050 0.077 0.051 -0.022 0.005 -0.023 0.011 0.012 0.011 0.012 0.012 0.011 0.003 0.004 0.003 -0.003 -0.003 -0.003 -0.043 0.379 -0.019 0.059 0.174 -0.001 -0.050 0.026 -0.048 0.015 0.073 0.001 0.005 0.049 -0.009 0.041 0.112 0.007 -0.038 -0.016 -0.059 0.014 0.022 0.004 0.012 0.018 0.002 -0.009 -0.005 -0.017 0.018 0.331 0.046 -0.011 0.163 -0.012 0.065 0.156 0.065 0.114 0.167 0.115 0.058 0.096 0.059 0.226 0.268 0.226 0.092 0.097 0.092 0.052 0.053 0.052 0.039 0.039 0.039 0.048 0.048 0.048 RX/RM =10, ICCX/ICCM =0.25, g=50, Balanced -0.080 0.314 -0.018 0.060 0.156 0.060 -0.043 0.034 -0.043 0.035 0.069 0.035 0.053 0.074 0.055 0.068 0.104 0.068 0.047 0.052 0.047 0.049 0.050 0.049 -0.012 -0.012 -0.012 -0.015 -0.015 -0.014 RX/RM =10, ICCX/ICCM =0.25, g=50, Unbalanced -0.122 0.186 -0.124 -0.163 -0.065 -0.172 0.015 0.078 0.059 0.036 0.056 -0.016 -0.009 -0.004 -0.028 -0.065 0.030 -0.072 0.144 0.158 0.208 0.048 0.051 0.068 0.049 0.051 0.070 0.028 0.030 0.058 0.215 0.378 0.183 -0.063 0.124 -0.048 -0.014 0.046 -0.113 -0.170 -0.055 -0.194 -0.134 -0.061 -0.219 -0.236 -0.109 -0.214 -0.080 -0.054 -0.065 -0.103 -0.090 -0.100 -0.013 -0.003 -0.027 0.016 0.022 0.010 -0.524 -0.015 0.130 -0.237 0.150 0.014 -0.129 0.152 -0.006 -0.053 0.170 0.005 -0.011 0.174 0.005 -0.182 0.059 0.035 -0.057 0.042 0.008 -0.019 0.043 0.010 -0.011 0.033 0.001 0.005 0.040 0.008 -0.542 0.069 0.252 -0.252 0.159 0.012 -0.116 0.181 0.026 -0.055 0.163 0.009 -0.005 0.176 0.013 -0.314 0.049 0.015 -0.118 0.071 0.007 -0.053 0.071 0.004 -0.029 0.062 -0.005 -0.009 0.063 -0.005 -0.385 0.022 0.575 -0.134 0.326 0.161 -0.033 0.283 0.094 0.042 0.343 0.104 0.126 0.419 0.145 -0.117 0.080 0.060 -0.008 0.079 0.048 0.021 0.077 0.047 0.025 0.064 0.036 0.033 0.064 0.036 -0.468 -0.073 0.511 -0.127 0.155 0.140 -0.004 0.309 0.135 0.124 0.366 0.187 0.157 0.348 0.157 -0.148 0.341 0.206 -0.044 0.131 0.071 0.029 0.151 0.085 0.066 0.155 0.084 0.089 0.164 0.091 -0.365 -0.003 -0.307 -0.183 -0.147 -0.142 -0.115 -0.093 -0.091 -0.120 -0.149 -0.115 -0.145 -0.199 -0.146 -0.086 -0.005 -0.008 -0.054 -0.036 -0.040 -0.042 -0.031 -0.036 -0.037 -0.028 -0.034 -0.030 -0.023 -0.029 -0.314 0.042 0.115 -0.181 -0.033 -0.117 -0.176 -0.161 -0.158 -0.184 -0.182 -0.175 -0.171 -0.159 -0.156 -0.223 -0.202 -0.107 -0.082 -0.043 -0.054 -0.077 -0.054 -0.062 -0.076 -0.060 -0.066 -0.078 -0.069 -0.072 -0.288 -0.103 -0.248 0.013 -0.007 0.013 -0.008 -0.023 -0.008 -0.039 -0.048 -0.039 -0.047 -0.053 -0.047 -0.111 -0.119 -0.111 -0.069 -0.070 -0.069 -0.016 -0.016 -0.016 -0.066 -0.066 -0.066 -0.058 -0.058 -0.058 -0.139 -0.098 -0.070 0.103 0.086 0.086 0.070 0.087 0.095 0.067 0.061 0.082 0.038 0.036 0.057 -0.053 -0.070 -0.125 -0.101 -0.105 -0.109 0.006 0.005 0.029 -0.024 -0.025 -0.028 -0.018 -0.019 -0.026 -0.522 0.110 0.070 -0.218 0.234 0.039 -0.103 0.217 0.023 -0.035 0.211 0.022 0.019 0.223 0.035 -0.175 0.081 0.039 -0.028 0.068 0.035 0.013 0.074 0.042 0.024 0.068 0.037 0.033 0.068 0.037 -0.577 0.271 0.169 -0.291 0.191 0.004 -0.136 0.186 0.017 -0.069 0.182 0.017 -0.018 0.185 0.022 -0.348 0.082 0.018 -0.138 0.081 0.011 -0.055 0.089 0.027 -0.027 0.080 0.017 -0.004 0.082 0.018 161 Table 35. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.090 0.237 0.090 0.073 0.081 0.073 0.013 0.013 0.013 0.025 0.026 0.025 0.017 0.017 0.017 -0.044 -0.044 -0.044 -0.015 -0.014 -0.015 -0.001 0.000 0.000 -0.007 -0.007 -0.007 -0.014 -0.014 -0.014 0.059 0.112 0.059 -0.012 0.012 -0.018 -0.011 0.009 -0.026 -0.005 0.013 -0.017 0.003 0.020 -0.011 -0.052 -0.042 -0.141 -0.018 -0.012 -0.030 -0.031 -0.027 -0.040 -0.028 -0.025 -0.044 -0.026 -0.024 -0.039 -0.520 0.029 -0.114 -0.234 0.155 -0.019 -0.120 0.158 -0.010 -0.062 0.151 -0.011 -0.020 0.154 -0.006 -0.199 0.022 -0.009 -0.061 0.027 -0.002 -0.031 0.023 -0.005 -0.018 0.021 -0.007 -0.009 0.022 -0.006 -0.545 0.071 -0.043 -0.264 0.115 -0.040 -0.136 0.144 -0.009 -0.070 0.142 -0.009 -0.020 0.150 -0.001 -0.325 0.043 -0.022 -0.131 0.050 -0.009 -0.057 0.064 0.003 -0.030 0.060 -0.003 -0.012 0.060 -0.003 0.004 0.249 0.004 0.047 0.097 0.047 -0.014 0.010 -0.014 0.059 0.071 0.059 0.031 0.041 0.031 -0.045 -0.045 -0.045 -0.002 -0.002 -0.002 -0.010 -0.010 -0.009 -0.012 -0.012 -0.011 -0.008 -0.008 -0.008 0.079 0.221 0.079 -0.091 -0.030 -0.098 -0.041 -0.005 -0.022 -0.012 0.019 0.013 -0.059 -0.032 -0.045 0.055 0.076 0.055 0.010 0.017 0.016 0.005 0.011 0.017 0.025 0.031 0.027 0.010 0.014 0.014 RX/RM =10, ICCX/ICCM =0.25, g=200, Balanced -0.085 0.054 -0.084 -0.074 -0.029 -0.075 -0.044 -0.031 -0.043 -0.021 -0.010 -0.020 -0.043 -0.037 -0.044 -0.075 -0.067 -0.075 -0.006 -0.007 -0.006 0.001 0.001 0.001 -0.003 -0.003 -0.002 0.008 0.008 0.008 -0.170 0.163 -0.169 -0.001 0.087 -0.001 0.004 0.033 0.004 0.096 0.115 0.096 0.049 0.059 0.050 0.029 0.038 0.028 -0.018 -0.018 -0.018 -0.029 -0.029 -0.029 -0.030 -0.030 -0.030 -0.007 -0.007 -0.006 -0.518 0.064 -0.048 -0.244 0.135 -0.020 -0.117 0.166 0.001 -0.057 0.160 -0.003 -0.014 0.165 0.001 -0.202 0.030 -0.004 -0.070 0.021 -0.009 -0.033 0.024 -0.005 -0.016 0.025 -0.005 -0.009 0.023 -0.006 -0.396 0.263 0.158 -0.100 0.389 0.164 0.000 0.380 0.132 0.078 0.377 0.141 0.121 0.375 0.139 -0.128 0.048 0.023 -0.009 0.070 0.043 0.014 0.063 0.037 0.023 0.059 0.033 0.040 0.069 0.043 RX/RM =10, ICCX/ICCM =0.25, g=200, Unbalanced -0.384 0.229 0.190 -0.118 0.280 0.113 0.024 0.364 0.162 0.096 0.346 0.154 0.153 0.359 0.161 -0.202 0.149 0.090 -0.044 0.122 0.064 0.012 0.120 0.059 0.056 0.139 0.072 0.072 0.138 0.069 0.287 0.370 0.288 0.058 0.062 0.061 0.036 0.037 0.036 -0.040 -0.043 -0.035 0.015 0.014 0.027 -0.038 -0.018 0.009 -0.011 -0.013 -0.006 -0.009 -0.008 -0.002 -0.003 -0.004 0.001 -0.018 -0.017 -0.013 -0.552 0.033 -0.035 -0.262 0.125 -0.024 -0.135 0.147 -0.009 -0.069 0.144 -0.009 -0.023 0.146 -0.007 -0.334 0.035 -0.030 -0.129 0.059 -0.005 -0.067 0.056 -0.009 -0.030 0.061 -0.004 -0.010 0.062 -0.003 0.052 0.307 0.051 0.021 0.101 -0.002 -0.058 -0.014 -0.147 -0.049 -0.007 -0.104 -0.014 0.024 -0.061 -0.079 -0.021 -0.017 -0.028 -0.017 -0.014 -0.029 -0.022 0.018 0.008 0.016 0.059 -0.009 -0.003 0.026 162 -0.386 -0.163 -0.207 -0.210 -0.206 -0.175 -0.174 -0.202 -0.164 -0.158 -0.189 -0.156 -0.151 -0.183 -0.150 -0.101 -0.010 -0.018 -0.053 -0.029 -0.036 -0.040 -0.026 -0.033 -0.035 -0.025 -0.032 -0.035 -0.029 -0.035 -0.381 -0.151 -0.166 -0.213 -0.131 -0.150 -0.183 -0.174 -0.167 -0.169 -0.159 -0.155 -0.171 -0.171 -0.159 -0.210 -0.088 -0.104 -0.101 -0.055 -0.068 -0.077 -0.049 -0.060 -0.082 -0.065 -0.070 -0.075 -0.061 -0.065 0.053 0.088 0.053 0.004 -0.010 0.004 0.011 0.006 0.011 -0.004 -0.007 -0.004 0.022 0.017 0.022 -0.024 -0.027 -0.024 -0.024 -0.024 -0.024 -0.005 -0.005 -0.005 -0.006 -0.006 -0.006 -0.015 -0.015 -0.015 -0.003 0.015 -0.003 0.054 0.065 0.028 0.049 0.058 0.059 0.065 0.070 0.068 0.063 0.067 0.051 0.098 0.102 0.075 0.021 0.024 0.010 0.010 0.011 0.020 0.010 0.010 0.011 0.012 0.013 0.013 -0.511 0.169 -0.043 -0.220 0.203 0.012 -0.106 0.183 0.008 -0.047 0.176 0.007 -0.005 0.175 0.010 -0.193 0.034 0.001 -0.059 0.030 0.000 -0.025 0.031 0.002 -0.010 0.030 0.002 -0.002 0.030 0.001 -0.579 0.089 -0.037 -0.279 0.146 -0.024 -0.152 0.144 -0.019 -0.072 0.152 -0.012 -0.027 0.150 -0.013 -0.342 0.053 -0.018 -0.118 0.086 0.015 -0.062 0.069 -0.001 -0.014 0.085 0.012 0.004 0.082 0.010 Table 36. Absolute relative bias using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 1.169 1.053 1.178 0.502 0.494 0.511 0.427 0.424 0.427 0.386 0.385 0.386 0.321 0.320 0.321 0.363 0.363 0.363 0.211 0.211 0.211 0.158 0.158 0.158 0.133 0.133 0.133 0.127 0.127 0.127 1.168 1.101 1.144 0.477 0.464 0.615 0.346 0.349 0.427 0.286 0.287 0.357 0.241 0.241 0.302 0.613 0.609 0.718 0.284 0.284 0.355 0.229 0.230 0.245 0.196 0.196 0.217 0.162 0.162 0.189 14.895 171.508 165.301 20.637 217.953 89.181 26.557 281.631 53.278 30.978 322.430 44.573 32.464 344.866 35.880 23.276 118.247 99.041 25.443 49.951 38.262 27.445 43.681 33.582 30.198 42.278 32.955 30.726 40.106 31.447 16.189 153.178 187.849 17.867 192.232 108.941 23.901 251.592 52.238 23.659 244.384 40.537 29.201 224.961 36.463 19.355 151.675 158.272 23.922 102.131 55.006 28.575 71.793 43.041 30.810 66.737 40.274 32.548 62.339 38.884 0.542 0.480 0.512 0.210 0.202 0.209 0.187 0.187 0.186 0.138 0.133 0.138 0.132 0.131 0.131 0.170 0.170 0.171 0.084 0.084 0.084 0.063 0.063 0.063 0.057 0.057 0.057 0.048 0.048 0.048 0.571 0.513 0.555 0.245 0.239 0.294 0.176 0.173 0.194 0.155 0.153 0.167 0.127 0.123 0.142 0.255 0.256 0.299 0.128 0.129 0.145 0.112 0.111 0.132 0.087 0.087 0.111 0.080 0.080 0.096 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Balanced 2.359 2.270 2.347 0.980 1.002 0.984 0.776 0.791 0.777 0.607 0.607 0.608 0.560 0.572 0.559 0.754 0.730 0.755 0.380 0.380 0.380 0.319 0.319 0.319 0.275 0.275 0.276 0.237 0.237 0.237 8.686 57.033 55.523 10.281 120.235 39.105 10.705 211.960 23.512 11.071 242.735 16.353 14.011 303.093 15.722 11.278 73.949 54.921 12.098 25.720 18.607 11.317 18.435 13.778 13.468 18.828 14.691 14.126 18.648 14.466 33.039 318.246 289.658 41.211 554.309 225.466 51.373 669.100 108.613 58.946 727.845 86.869 66.031 665.943 73.421 49.019 270.739 358.889 49.668 99.408 75.297 51.947 80.779 62.920 52.104 72.379 56.773 56.877 74.194 58.213 2.455 2.234 2.533 1.169 1.139 1.171 0.815 0.816 0.815 0.623 0.616 0.622 0.608 0.603 0.608 0.706 0.720 0.706 0.494 0.494 0.495 0.328 0.328 0.327 0.264 0.264 0.263 0.235 0.235 0.235 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Unbalanced 8.695 75.665 68.507 10.041 125.133 52.743 11.244 168.091 28.370 13.725 162.481 20.226 14.287 200.538 17.609 8.904 83.091 73.158 9.868 51.036 21.761 12.309 39.136 18.981 12.095 28.477 15.020 12.705 25.984 14.170 2.629 2.191 2.451 1.093 1.052 1.260 0.824 0.826 0.895 0.677 0.686 0.716 0.608 0.603 0.634 1.138 1.131 1.279 0.661 0.655 0.725 0.457 0.457 0.489 0.405 0.405 0.406 0.357 0.357 0.375 29.305 257.306 387.369 39.275 446.323 187.229 52.100 641.026 117.795 50.680 568.553 79.602 61.632 612.886 77.531 34.126 415.246 438.483 46.072 252.334 127.768 50.253 155.576 80.102 55.641 122.601 69.315 60.371 114.569 64.494 1.932 1.819 1.906 0.977 0.985 1.212 0.835 0.867 0.995 0.633 0.638 0.709 0.595 0.582 0.679 1.104 1.082 1.272 0.553 0.552 0.659 0.443 0.444 0.517 0.351 0.350 0.402 0.318 0.319 0.368 163 29.136 141.242 122.039 25.634 89.773 57.604 29.680 91.234 36.622 28.538 75.878 31.504 27.688 61.673 28.357 29.964 61.602 60.681 25.780 30.534 28.542 28.210 30.996 29.488 28.524 30.588 29.083 27.362 28.938 27.517 23.676 138.124 171.069 31.492 95.491 60.276 26.542 96.023 35.135 28.164 67.775 31.048 28.164 77.443 30.464 26.324 88.612 75.590 24.940 41.246 33.776 25.756 34.904 30.317 24.332 29.814 27.844 25.066 28.344 28.163 1.095 0.994 1.109 0.453 0.449 0.455 0.325 0.327 0.325 0.307 0.305 0.307 0.283 0.278 0.283 0.388 0.389 0.388 0.236 0.236 0.236 0.175 0.175 0.175 0.136 0.136 0.136 0.128 0.128 0.128 1.368 1.179 1.305 0.514 0.509 0.580 0.393 0.388 0.487 0.331 0.324 0.397 0.265 0.265 0.339 0.556 0.537 0.627 0.288 0.288 0.339 0.229 0.230 0.261 0.172 0.171 0.203 0.171 0.171 0.198 16.037 100.405 186.806 18.903 161.166 69.950 21.553 173.032 43.856 23.357 204.596 33.865 25.495 211.493 28.232 18.911 78.582 77.033 21.663 42.701 32.494 24.910 37.869 29.922 25.580 34.930 27.719 23.822 30.679 24.347 15.073 92.307 107.687 20.865 163.315 89.172 21.512 197.631 55.654 24.395 173.328 38.634 27.085 202.309 35.625 18.013 124.166 132.954 22.276 81.637 46.684 25.966 65.971 36.457 26.470 51.787 30.278 27.633 49.713 28.295 Table 36. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.605 0.566 0.608 0.261 0.259 0.262 0.211 0.211 0.211 0.157 0.157 0.157 0.142 0.142 0.142 0.185 0.185 0.185 0.105 0.105 0.105 0.074 0.074 0.074 0.073 0.073 0.073 0.062 0.062 0.062 0.492 0.447 0.492 0.241 0.239 0.326 0.219 0.220 0.230 0.167 0.166 0.190 0.155 0.155 0.171 0.262 0.259 0.334 0.159 0.159 0.194 0.110 0.111 0.131 0.082 0.082 0.096 0.085 0.085 0.099 10.370 119.294 98.682 9.556 181.837 37.971 12.537 223.726 23.589 12.715 234.037 17.925 14.113 246.307 15.622 11.971 39.196 30.513 12.739 22.905 18.286 13.009 19.371 15.530 13.486 18.293 14.607 13.697 17.434 13.989 10.477 129.470 110.270 12.119 154.230 33.315 13.650 129.800 24.916 14.222 125.068 21.022 15.182 101.088 17.670 9.250 58.624 36.844 11.965 36.188 22.201 12.824 30.591 19.588 13.821 28.745 18.687 13.953 26.242 17.318 0.261 0.253 0.262 0.120 0.124 0.120 0.079 0.073 0.079 0.078 0.077 0.078 0.063 0.063 0.063 0.099 0.099 0.099 0.046 0.046 0.046 0.041 0.041 0.041 0.034 0.034 0.034 0.030 0.030 0.030 0.253 0.237 0.246 0.114 0.113 0.150 0.073 0.073 0.085 0.064 0.064 0.076 0.059 0.059 0.068 0.130 0.129 0.146 0.069 0.070 0.074 0.057 0.057 0.064 0.045 0.045 0.053 0.039 0.039 0.043 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Balanced 1.163 1.173 1.178 0.520 0.522 0.520 0.403 0.401 0.403 0.314 0.314 0.313 0.306 0.302 0.306 0.393 0.394 0.393 0.225 0.225 0.226 0.152 0.152 0.151 0.144 0.144 0.144 0.119 0.119 0.119 8.704 74.519 30.702 7.154 197.216 13.280 6.921 203.887 10.677 6.314 250.663 8.268 6.404 288.200 6.968 6.741 21.009 14.986 5.787 10.673 8.233 6.767 10.673 8.208 6.813 9.529 7.415 7.176 9.670 7.365 1.201 1.159 1.207 0.515 0.515 0.516 0.366 0.368 0.366 0.294 0.293 0.294 0.265 0.268 0.265 0.413 0.413 0.412 0.203 0.203 0.203 0.160 0.160 0.160 0.145 0.145 0.145 0.123 0.123 0.123 14.650 327.632 193.734 19.088 368.814 57.794 25.068 419.604 46.560 28.683 593.419 40.301 30.634 587.395 33.805 23.121 90.180 68.397 27.849 49.643 39.803 30.974 45.323 36.619 33.729 44.815 36.397 33.000 41.416 33.688 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Unbalanced 8.244 84.324 49.879 6.966 106.303 11.960 6.583 111.191 10.970 6.802 141.885 8.832 6.567 81.159 7.223 7.424 54.665 20.778 6.438 20.626 11.625 6.398 14.883 9.511 6.942 15.492 9.207 7.401 14.858 8.720 1.203 1.080 1.184 0.545 0.554 0.612 0.333 0.333 0.374 0.313 0.311 0.358 0.269 0.269 0.323 0.585 0.571 0.714 0.301 0.304 0.371 0.243 0.243 0.262 0.223 0.223 0.227 0.191 0.190 0.204 15.549 342.832 215.865 17.792 280.991 50.659 25.377 293.418 50.518 26.331 257.288 41.575 31.422 254.393 38.283 18.186 156.341 105.029 22.049 72.051 45.295 23.621 58.969 37.575 29.521 57.027 36.228 28.488 50.164 31.599 1.105 1.047 1.112 0.449 0.453 0.578 0.390 0.395 0.495 0.299 0.302 0.353 0.270 0.272 0.333 0.548 0.540 0.616 0.305 0.305 0.332 0.202 0.203 0.246 0.185 0.184 0.221 0.158 0.157 0.185 164 13.228 78.813 54.335 14.720 40.002 20.505 14.502 29.406 16.559 15.588 36.292 16.476 15.162 32.891 15.405 10.798 16.150 14.989 11.434 12.974 12.395 13.026 14.076 13.524 12.140 12.856 12.343 12.344 12.874 12.389 12.276 87.379 51.430 12.598 38.310 19.919 12.720 28.166 16.040 12.138 22.978 15.343 12.094 21.176 14.536 11.948 33.670 24.622 15.380 20.878 18.764 13.976 17.418 16.006 15.440 17.842 16.411 15.522 17.382 16.210 0.627 0.569 0.621 0.281 0.282 0.281 0.193 0.194 0.194 0.173 0.172 0.173 0.140 0.141 0.141 0.196 0.196 0.196 0.104 0.104 0.104 0.088 0.088 0.088 0.065 0.065 0.065 0.059 0.059 0.059 0.600 0.551 0.597 0.280 0.273 0.314 0.181 0.182 0.224 0.168 0.169 0.188 0.136 0.137 0.149 0.255 0.248 0.305 0.169 0.170 0.173 0.115 0.115 0.135 0.109 0.108 0.126 0.094 0.094 0.117 9.722 95.853 66.194 9.835 139.283 26.628 11.145 179.607 21.396 12.613 192.086 17.041 13.632 235.402 15.083 11.269 36.629 28.618 12.955 22.625 18.123 10.177 14.829 11.945 12.568 16.723 13.512 12.123 15.325 12.371 9.286 119.220 158.059 11.645 99.871 26.378 11.657 118.547 23.658 12.792 115.764 19.903 13.986 95.501 17.132 8.839 66.046 43.172 11.049 30.695 19.312 11.137 25.459 14.962 13.053 25.308 15.375 12.453 22.421 13.588 Table 36. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 1.410 1.342 1.409 0.634 0.634 0.634 0.477 0.477 0.477 0.377 0.377 0.377 0.315 0.315 0.315 0.404 0.404 0.404 0.244 0.244 0.244 0.205 0.205 0.205 0.140 0.140 0.140 0.119 0.119 0.119 1.308 1.228 1.308 0.626 0.611 0.767 0.398 0.396 0.445 0.319 0.317 0.411 0.285 0.286 0.314 0.690 0.686 0.741 0.347 0.346 0.388 0.230 0.229 0.269 0.190 0.189 0.240 0.155 0.155 0.190 14.811 64.057 45.804 13.943 24.019 19.084 14.568 19.824 16.571 14.277 18.229 15.169 13.931 17.068 14.177 13.993 18.187 17.594 12.949 14.124 13.718 13.149 14.019 13.561 13.036 13.621 13.198 13.018 13.451 13.060 13.904 62.776 59.187 13.437 24.647 19.325 14.372 21.219 16.712 13.648 18.333 15.170 14.408 18.163 14.604 14.329 23.990 21.625 13.809 17.637 16.873 13.703 15.972 16.125 13.997 15.791 16.180 13.369 14.671 15.190 1.181 1.147 1.180 0.685 0.685 0.685 0.373 0.373 0.373 0.326 0.326 0.327 0.262 0.262 0.262 0.426 0.426 0.426 0.270 0.270 0.270 0.194 0.194 0.194 0.174 0.174 0.174 0.145 0.145 0.145 1.249 1.220 1.269 0.568 0.573 0.707 0.358 0.361 0.449 0.311 0.312 0.403 0.273 0.274 0.338 0.562 0.564 0.652 0.370 0.369 0.419 0.251 0.250 0.275 0.219 0.219 0.239 0.199 0.199 0.208 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Balanced 2.416 2.328 2.432 1.017 1.018 1.017 0.819 0.820 0.820 0.608 0.607 0.607 0.586 0.586 0.586 0.635 0.635 0.634 0.427 0.427 0.428 0.261 0.261 0.260 0.249 0.249 0.249 0.230 0.230 0.230 29.739 132.532 124.012 26.698 49.400 37.587 29.206 42.724 34.146 28.644 38.373 30.833 28.237 35.284 28.785 29.624 40.224 38.534 30.635 34.257 33.029 28.631 30.688 29.608 29.523 31.119 29.980 30.227 31.472 30.350 13.625 59.671 36.586 13.291 25.199 19.031 14.410 20.968 16.823 15.517 20.741 16.691 15.301 19.258 15.601 12.528 17.654 16.851 13.221 14.942 14.369 12.279 13.120 12.674 12.433 13.101 12.622 12.384 12.918 12.443 2.889 2.713 2.884 1.118 1.117 1.118 0.894 0.893 0.894 0.813 0.813 0.813 0.637 0.637 0.637 0.976 0.975 0.975 0.467 0.467 0.467 0.365 0.365 0.366 0.295 0.295 0.295 0.261 0.261 0.261 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Unbalanced 14.654 67.162 52.857 14.312 26.652 21.166 15.296 23.234 19.108 14.249 19.987 16.995 15.216 19.558 16.402 13.682 26.761 21.990 12.470 15.712 15.221 12.703 14.899 14.387 11.268 12.751 13.075 12.268 13.391 13.375 2.377 2.329 2.443 1.391 1.402 1.595 0.799 0.798 0.902 0.686 0.689 0.803 0.581 0.583 0.660 1.150 1.153 1.564 0.692 0.693 0.844 0.542 0.542 0.671 0.447 0.448 0.538 0.388 0.389 0.483 30.794 138.703 143.777 30.214 64.159 46.598 30.643 47.832 40.235 30.955 44.250 37.366 31.618 42.180 36.276 28.260 60.832 52.361 27.737 35.889 34.026 26.256 31.142 32.259 29.366 33.494 33.573 28.238 31.338 31.885 2.320 2.148 2.321 1.038 1.041 1.237 0.781 0.788 0.882 0.673 0.673 0.756 0.566 0.567 0.656 1.144 1.150 1.226 0.615 0.617 0.692 0.443 0.443 0.541 0.379 0.380 0.408 0.358 0.358 0.398 165 26.276 122.046 120.298 23.986 47.606 35.105 24.952 37.886 29.539 22.870 31.344 24.753 21.980 28.320 22.462 29.034 42.752 40.361 29.146 33.280 31.876 30.266 32.890 31.497 31.130 33.044 31.670 31.458 32.964 31.605 23.660 125.527 101.230 23.474 53.668 35.276 23.314 37.038 29.688 26.754 37.118 30.003 27.442 35.784 29.629 28.042 63.758 53.455 26.794 35.838 33.032 28.176 34.698 34.289 28.326 32.896 31.466 27.330 30.718 29.512 1.326 1.228 1.341 0.623 0.623 0.623 0.442 0.441 0.442 0.388 0.388 0.388 0.352 0.352 0.352 0.479 0.479 0.479 0.262 0.262 0.262 0.217 0.217 0.217 0.152 0.152 0.152 0.141 0.141 0.141 1.424 1.296 1.420 0.590 0.589 0.747 0.469 0.465 0.558 0.392 0.391 0.459 0.313 0.311 0.372 0.610 0.603 0.708 0.351 0.351 0.382 0.237 0.237 0.296 0.218 0.218 0.242 0.193 0.193 0.221 13.096 40.610 34.000 13.744 23.424 18.779 14.621 21.013 17.049 13.584 17.669 14.520 13.506 16.712 13.754 12.357 16.202 15.616 11.854 13.308 12.824 11.377 12.141 11.739 10.753 11.280 10.902 10.687 11.098 10.728 15.697 43.879 38.903 13.971 25.803 19.757 13.614 20.776 16.854 12.710 17.621 15.413 12.673 16.784 14.541 14.448 21.874 19.840 13.375 16.793 15.788 12.875 14.688 13.782 12.744 14.030 13.672 13.425 14.487 13.662 Table 36. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.683 0.682 0.683 0.267 0.267 0.267 0.216 0.216 0.216 0.179 0.178 0.178 0.143 0.143 0.143 0.250 0.250 0.250 0.117 0.117 0.116 0.096 0.096 0.096 0.087 0.087 0.087 0.074 0.074 0.073 0.685 0.583 0.684 0.300 0.301 0.345 0.224 0.223 0.268 0.197 0.197 0.244 0.178 0.178 0.206 0.312 0.311 0.363 0.155 0.155 0.195 0.140 0.139 0.162 0.114 0.113 0.135 0.108 0.108 0.127 7.683 22.296 15.992 7.261 11.149 9.358 6.764 9.150 7.702 6.678 8.230 7.006 6.616 8.014 6.723 6.240 8.174 7.889 6.999 7.797 7.525 6.709 7.138 6.906 6.664 7.021 6.764 6.655 6.919 6.680 9.324 22.978 19.590 7.355 12.320 10.221 7.807 11.245 9.392 7.583 10.252 8.736 7.560 9.588 8.339 7.234 11.321 10.737 7.055 8.724 8.597 7.003 8.089 8.031 7.421 8.368 8.304 7.662 8.332 8.382 0.711 0.711 0.711 0.291 0.290 0.291 0.223 0.223 0.222 0.165 0.165 0.165 0.154 0.154 0.154 0.226 0.226 0.226 0.132 0.132 0.132 0.093 0.093 0.093 0.084 0.084 0.085 0.067 0.067 0.067 0.645 0.599 0.645 0.243 0.245 0.310 0.201 0.199 0.267 0.162 0.162 0.187 0.141 0.140 0.163 0.294 0.289 0.326 0.184 0.182 0.206 0.124 0.123 0.155 0.111 0.111 0.132 0.102 0.102 0.111 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Balanced 1.095 1.092 1.095 0.528 0.528 0.527 0.369 0.369 0.370 0.314 0.314 0.314 0.269 0.269 0.269 0.322 0.322 0.321 0.200 0.200 0.200 0.177 0.177 0.178 0.125 0.125 0.125 0.123 0.123 0.123 8.075 27.133 16.783 8.059 12.765 10.503 7.112 9.949 8.150 7.511 9.415 7.947 7.210 8.876 7.343 6.248 8.197 7.914 6.493 6.989 6.821 6.067 6.412 6.228 6.395 6.670 6.474 6.034 6.227 6.052 1.319 1.319 1.319 0.633 0.633 0.633 0.445 0.445 0.445 0.366 0.366 0.366 0.332 0.332 0.332 0.407 0.407 0.407 0.227 0.227 0.227 0.188 0.188 0.188 0.176 0.176 0.176 0.142 0.142 0.142 16.096 65.489 39.171 15.462 24.367 20.296 13.875 18.584 15.748 13.593 16.797 14.362 13.409 15.886 13.615 13.499 17.703 17.151 13.518 14.955 14.484 14.558 15.396 14.965 14.090 14.690 14.269 14.035 14.484 14.087 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Unbalanced 15.055 53.093 38.638 14.974 25.418 21.165 14.508 20.926 17.019 15.853 20.835 16.540 13.939 17.542 14.632 14.379 22.454 20.751 13.519 16.995 16.886 13.705 15.887 15.475 13.697 15.203 15.019 14.103 15.364 15.455 1.165 1.066 1.165 0.540 0.529 0.655 0.363 0.360 0.470 0.291 0.288 0.337 0.277 0.275 0.316 0.508 0.507 0.643 0.311 0.308 0.323 0.249 0.247 0.271 0.177 0.177 0.203 0.178 0.178 0.212 7.934 23.192 19.916 6.577 12.115 9.829 7.442 11.052 9.129 7.089 9.625 8.027 7.254 9.490 7.833 6.897 10.112 9.282 7.350 9.001 8.496 6.410 7.417 7.012 6.821 7.564 6.872 6.722 7.339 7.014 1.400 1.339 1.400 0.553 0.552 0.698 0.442 0.442 0.481 0.387 0.384 0.508 0.325 0.325 0.385 0.592 0.594 0.717 0.350 0.348 0.415 0.238 0.238 0.288 0.218 0.218 0.265 0.162 0.162 0.201 166 13.172 62.836 35.465 11.842 19.710 15.949 12.112 16.846 13.893 11.054 14.264 11.763 11.464 13.904 11.650 11.126 16.150 15.420 11.924 13.742 13.139 11.952 12.956 12.425 12.708 13.440 12.916 12.386 12.996 12.449 12.540 56.186 33.477 12.380 22.482 18.002 12.804 19.034 16.340 13.712 18.404 15.873 14.234 18.130 15.722 14.256 24.212 22.062 14.592 18.492 17.266 12.428 14.754 14.701 13.014 14.896 15.085 13.788 15.164 15.046 0.629 0.629 0.629 0.295 0.294 0.295 0.239 0.239 0.239 0.199 0.199 0.199 0.176 0.176 0.176 0.206 0.206 0.206 0.139 0.139 0.139 0.105 0.105 0.105 0.085 0.085 0.085 0.079 0.079 0.080 0.630 0.612 0.630 0.286 0.286 0.356 0.229 0.230 0.257 0.174 0.175 0.197 0.147 0.147 0.178 0.281 0.280 0.301 0.173 0.171 0.196 0.130 0.131 0.158 0.114 0.114 0.138 0.102 0.102 0.118 8.030 23.739 16.484 6.025 9.150 7.613 6.910 9.237 7.756 6.242 7.922 6.632 6.435 7.788 6.532 6.329 8.150 7.911 5.627 6.084 5.925 5.403 5.744 5.564 5.631 5.880 5.703 5.482 5.669 5.500 8.845 22.338 17.717 6.968 11.677 9.997 7.746 9.987 8.378 7.061 8.796 7.493 6.854 8.126 7.109 5.686 9.311 8.603 6.571 8.094 7.675 5.592 6.536 6.699 5.820 6.530 6.412 5.880 6.406 6.541 Table 36. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 1.106 0.989 1.113 0.515 0.507 0.514 0.411 0.406 0.411 0.347 0.344 0.347 0.299 0.297 0.299 0.347 0.348 0.346 0.210 0.210 0.210 0.150 0.150 0.150 0.138 0.138 0.137 0.120 0.120 0.120 1.144 1.039 1.122 0.539 0.529 0.628 0.453 0.449 0.566 0.335 0.336 0.386 0.295 0.292 0.328 0.559 0.556 0.669 0.308 0.310 0.341 0.264 0.264 0.304 0.213 0.212 0.251 0.195 0.194 0.230 0.825 1.725 2.111 0.629 1.857 1.035 0.479 2.941 0.502 0.344 3.500 0.392 0.284 4.123 0.301 0.570 1.033 1.246 0.394 0.707 0.499 0.325 0.540 0.379 0.301 0.441 0.314 0.336 0.453 0.342 0.829 1.580 1.529 0.698 1.764 1.485 0.538 2.438 0.691 0.388 2.882 0.444 0.356 3.107 0.370 0.686 1.315 1.259 0.446 0.955 0.483 0.283 0.764 0.335 0.256 0.883 0.365 0.267 0.790 0.321 0.448 0.434 0.441 0.214 0.223 0.213 0.170 0.178 0.169 0.142 0.156 0.142 0.130 0.148 0.129 0.169 0.165 0.166 0.083 0.083 0.084 0.083 0.082 0.082 0.061 0.061 0.061 0.054 0.053 0.054 0.537 0.454 0.543 0.221 0.229 0.286 0.188 0.193 0.215 0.148 0.156 0.193 0.121 0.132 0.156 0.235 0.252 0.278 0.144 0.142 0.154 0.104 0.104 0.116 0.092 0.091 0.099 0.080 0.080 0.083 2.333 2.197 2.329 0.985 0.994 0.990 0.738 0.735 0.737 0.585 0.570 0.586 0.549 0.523 0.550 0.753 0.746 0.751 0.418 0.417 0.418 0.321 0.321 0.321 0.294 0.294 0.294 0.267 0.267 0.266 0.773 2.135 3.547 0.642 4.862 2.265 0.517 7.243 1.105 0.428 9.618 0.710 0.530 10.430 0.624 0.706 3.224 2.071 0.514 1.618 0.733 0.457 0.744 0.542 0.481 0.742 0.530 0.476 0.666 0.487 RX/RM =10, ICCX/ICCM =0.05, g=50, Balanced 2.446 2.289 2.518 1.035 1.046 1.023 0.741 0.738 0.741 0.644 0.650 0.644 0.527 0.536 0.527 0.839 0.836 0.836 0.420 0.420 0.421 0.334 0.334 0.334 0.253 0.253 0.252 0.219 0.219 0.219 RX/RM =10, ICCX/ICCM =0.05, g=50, Unbalanced 2.372 2.284 2.368 1.024 1.031 1.270 0.669 0.664 0.839 0.616 0.610 0.733 0.556 0.555 0.622 1.172 1.185 1.503 0.611 0.611 0.688 0.381 0.380 0.446 0.404 0.404 0.464 0.328 0.328 0.373 0.805 0.524 1.017 0.635 0.344 0.561 0.448 0.443 0.314 0.253 0.405 0.149 0.105 0.438 0.100 0.580 0.259 0.695 0.296 0.249 0.154 0.137 0.246 0.097 0.075 0.239 0.077 0.062 0.240 0.065 0.829 0.580 0.853 0.657 0.266 0.577 0.497 0.347 0.306 0.309 0.434 0.154 0.134 0.402 0.097 0.711 0.282 0.938 0.464 0.266 0.399 0.295 0.251 0.133 0.171 0.286 0.094 0.083 0.282 0.080 2.614 2.294 2.641 1.015 1.009 1.182 0.772 0.774 0.949 0.710 0.720 0.770 0.618 0.648 0.658 1.053 1.045 1.224 0.534 0.526 0.580 0.433 0.429 0.536 0.408 0.408 0.452 0.373 0.374 0.417 0.773 2.799 3.594 0.672 4.933 2.960 0.493 6.277 1.202 0.596 6.482 0.977 0.631 7.596 0.776 0.733 3.259 3.427 0.536 2.861 1.181 0.493 2.138 0.814 0.587 1.832 0.757 0.593 1.648 0.679 0.560 0.913 1.099 0.316 0.985 0.537 0.274 1.158 0.351 0.244 1.340 0.268 0.214 1.502 0.220 0.253 0.780 0.554 0.241 0.404 0.284 0.215 0.254 0.229 0.235 0.268 0.243 0.234 0.257 0.235 0.564 1.293 1.198 0.365 1.013 0.601 0.289 1.054 0.365 0.286 1.020 0.348 0.261 1.196 0.303 0.322 0.817 0.851 0.260 0.592 0.326 0.218 0.385 0.255 0.228 0.345 0.256 0.236 0.348 0.254 1.136 1.021 1.134 0.483 0.486 0.472 0.374 0.363 0.374 0.300 0.308 0.300 0.271 0.271 0.271 0.366 0.365 0.364 0.259 0.259 0.259 0.189 0.189 0.189 0.151 0.151 0.151 0.136 0.136 0.136 0.989 0.884 0.972 0.519 0.483 0.762 0.359 0.363 0.417 0.303 0.297 0.354 0.273 0.270 0.327 0.571 0.535 0.669 0.291 0.294 0.366 0.244 0.241 0.278 0.171 0.171 0.207 0.167 0.167 0.193 0.808 1.103 1.824 0.604 1.559 0.891 0.425 2.068 0.534 0.355 2.486 0.371 0.282 2.920 0.308 0.566 0.945 0.777 0.338 0.557 0.372 0.264 0.394 0.284 0.250 0.436 0.277 0.252 0.380 0.258 0.812 0.916 1.293 0.654 1.536 0.795 0.517 1.829 0.462 0.379 1.955 0.347 0.248 2.374 0.278 0.709 1.254 1.400 0.495 0.956 0.541 0.365 0.745 0.367 0.299 0.801 0.336 0.297 0.729 0.306 167 Table 36. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.638 0.613 0.638 0.253 0.250 0.253 0.205 0.206 0.205 0.146 0.145 0.146 0.133 0.132 0.133 0.182 0.182 0.182 0.108 0.109 0.108 0.094 0.094 0.094 0.076 0.076 0.076 0.066 0.066 0.066 0.556 0.509 0.545 0.282 0.286 0.327 0.168 0.168 0.217 0.149 0.151 0.175 0.135 0.136 0.154 0.269 0.267 0.338 0.153 0.152 0.182 0.104 0.104 0.135 0.095 0.095 0.114 0.083 0.083 0.097 0.829 1.227 1.059 0.621 2.819 0.369 0.452 3.776 0.224 0.278 4.369 0.194 0.142 5.170 0.140 0.549 0.484 0.323 0.288 0.292 0.195 0.178 0.289 0.176 0.151 0.271 0.156 0.152 0.269 0.156 0.805 1.138 1.256 0.649 1.893 0.328 0.511 2.375 0.270 0.317 2.742 0.213 0.171 3.109 0.176 0.698 0.731 0.791 0.458 0.583 0.204 0.306 0.546 0.199 0.179 0.576 0.163 0.150 0.568 0.158 0.295 0.285 0.295 0.113 0.133 0.113 0.091 0.105 0.091 0.069 0.086 0.069 0.063 0.083 0.063 0.090 0.090 0.090 0.048 0.048 0.048 0.036 0.036 0.036 0.033 0.034 0.033 0.026 0.026 0.026 0.226 0.209 0.229 0.103 0.108 0.128 0.077 0.084 0.086 0.068 0.079 0.075 0.055 0.066 0.060 0.126 0.121 0.137 0.061 0.062 0.071 0.053 0.053 0.059 0.045 0.045 0.048 0.035 0.035 0.040 RX/RM =10, ICCX/ICCM =0.05, g=200, Balanced 1.082 1.093 1.092 0.468 0.459 0.468 0.340 0.343 0.341 0.290 0.280 0.289 0.222 0.215 0.223 0.368 0.368 0.368 0.214 0.214 0.215 0.169 0.169 0.170 0.131 0.131 0.132 0.109 0.109 0.109 0.746 2.626 2.050 0.559 6.019 0.691 0.386 9.819 0.436 0.291 12.151 0.371 0.284 14.827 0.341 0.600 0.934 0.649 0.318 0.531 0.356 0.273 0.453 0.304 0.266 0.399 0.284 0.252 0.391 0.258 0.808 0.324 0.714 0.629 0.304 0.234 0.443 0.384 0.122 0.275 0.400 0.076 0.099 0.392 0.050 0.576 0.161 0.189 0.282 0.201 0.066 0.148 0.208 0.046 0.065 0.228 0.039 0.031 0.228 0.029 1.304 1.186 1.324 0.438 0.446 0.438 0.390 0.399 0.390 0.328 0.327 0.328 0.270 0.274 0.270 0.416 0.416 0.417 0.231 0.231 0.231 0.181 0.181 0.181 0.138 0.138 0.138 0.126 0.126 0.126 RX/RM =10, ICCX/ICCM =0.05, g=200, Unbalanced 0.778 3.521 4.225 0.614 3.877 0.637 0.433 6.525 0.419 0.306 8.496 0.436 0.276 8.375 0.376 0.665 2.226 1.224 0.449 1.233 0.501 0.324 0.977 0.421 0.276 0.875 0.334 0.276 0.902 0.314 1.254 1.176 1.253 0.483 0.461 0.677 0.378 0.384 0.507 0.325 0.325 0.352 0.260 0.261 0.296 0.525 0.528 0.683 0.288 0.284 0.355 0.199 0.200 0.238 0.204 0.203 0.237 0.182 0.182 0.201 0.821 0.382 0.691 0.658 0.307 0.255 0.492 0.367 0.119 0.315 0.394 0.074 0.114 0.389 0.049 0.700 0.208 0.353 0.456 0.252 0.098 0.278 0.272 0.056 0.140 0.285 0.048 0.045 0.295 0.037 1.048 0.963 1.061 0.491 0.490 0.648 0.401 0.390 0.476 0.303 0.301 0.348 0.283 0.280 0.307 0.613 0.607 0.670 0.335 0.337 0.392 0.223 0.224 0.263 0.204 0.205 0.215 0.186 0.187 0.207 168 0.549 0.590 0.415 0.283 0.959 0.220 0.212 1.524 0.197 0.176 1.896 0.177 0.174 2.305 0.177 0.188 0.253 0.198 0.125 0.143 0.130 0.120 0.135 0.124 0.117 0.129 0.119 0.116 0.127 0.116 0.538 1.019 0.766 0.297 0.650 0.227 0.217 1.060 0.177 0.206 1.327 0.196 0.191 1.290 0.177 0.328 0.451 0.277 0.157 0.234 0.155 0.134 0.178 0.137 0.127 0.165 0.129 0.130 0.168 0.130 0.569 0.510 0.565 0.268 0.260 0.268 0.176 0.179 0.176 0.163 0.170 0.163 0.133 0.134 0.133 0.209 0.209 0.209 0.092 0.092 0.092 0.090 0.090 0.090 0.075 0.075 0.075 0.063 0.063 0.063 0.527 0.518 0.525 0.253 0.261 0.303 0.187 0.185 0.251 0.158 0.154 0.183 0.163 0.164 0.173 0.272 0.272 0.310 0.162 0.162 0.184 0.115 0.114 0.144 0.110 0.110 0.124 0.091 0.091 0.102 0.807 0.849 1.032 0.638 1.643 0.354 0.434 2.775 0.220 0.284 3.093 0.188 0.149 3.467 0.142 0.562 0.447 0.302 0.287 0.299 0.176 0.190 0.272 0.168 0.172 0.251 0.167 0.148 0.248 0.148 0.824 0.857 1.497 0.677 1.333 0.350 0.511 2.106 0.221 0.312 2.520 0.169 0.153 2.652 0.144 0.704 0.822 0.458 0.486 0.525 0.210 0.305 0.518 0.170 0.192 0.503 0.153 0.136 0.527 0.146 Table 36. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 1.171 1.106 1.171 0.672 0.650 0.672 0.431 0.427 0.431 0.348 0.347 0.348 0.324 0.323 0.324 0.448 0.446 0.448 0.256 0.256 0.255 0.187 0.187 0.187 0.158 0.158 0.158 0.149 0.149 0.149 1.469 1.358 1.460 0.597 0.581 0.693 0.418 0.415 0.502 0.352 0.354 0.425 0.310 0.308 0.361 0.541 0.529 0.692 0.333 0.327 0.398 0.271 0.271 0.311 0.197 0.196 0.211 0.173 0.173 0.185 0.543 0.381 0.546 0.236 0.251 0.166 0.143 0.211 0.121 0.101 0.196 0.097 0.082 0.188 0.083 0.208 0.124 0.115 0.110 0.104 0.099 0.083 0.085 0.080 0.080 0.085 0.080 0.076 0.080 0.076 0.539 0.476 1.148 0.238 0.253 0.171 0.159 0.213 0.126 0.121 0.192 0.112 0.099 0.193 0.101 0.332 0.215 0.216 0.139 0.125 0.104 0.097 0.104 0.089 0.078 0.103 0.084 0.075 0.099 0.083 1.249 1.247 1.245 0.579 0.569 0.579 0.397 0.393 0.398 0.363 0.365 0.364 0.346 0.344 0.345 0.396 0.393 0.396 0.233 0.232 0.233 0.190 0.190 0.189 0.174 0.174 0.174 0.144 0.144 0.144 1.242 1.060 1.241 0.592 0.602 0.676 0.397 0.406 0.419 0.331 0.337 0.415 0.320 0.318 0.356 0.661 0.627 0.750 0.317 0.317 0.373 0.226 0.226 0.266 0.199 0.199 0.236 0.155 0.155 0.190 2.878 2.524 2.791 1.095 1.082 1.095 0.868 0.865 0.868 0.678 0.682 0.678 0.609 0.613 0.609 0.915 0.909 0.915 0.521 0.523 0.521 0.398 0.398 0.398 0.291 0.291 0.291 0.286 0.286 0.287 RX/RM =10, ICCX/ICCM =0.25, g=50, Balanced 2.300 2.031 2.330 0.967 0.956 0.967 0.678 0.651 0.679 0.533 0.534 0.533 0.472 0.475 0.473 0.764 0.770 0.763 0.500 0.499 0.500 0.303 0.303 0.303 0.280 0.280 0.280 0.230 0.230 0.230 RX/RM =10, ICCX/ICCM =0.25, g=50, Unbalanced 2.636 2.633 2.784 1.005 0.976 1.192 0.708 0.685 0.906 0.616 0.608 0.622 0.557 0.562 0.598 1.010 0.976 1.213 0.661 0.662 0.862 0.513 0.514 0.582 0.432 0.433 0.535 0.364 0.362 0.434 2.617 2.273 2.569 1.108 1.058 1.359 0.744 0.724 1.024 0.706 0.696 0.850 0.570 0.577 0.709 1.337 1.310 1.494 0.783 0.773 0.870 0.543 0.537 0.669 0.454 0.450 0.563 0.385 0.386 0.451 0.524 0.378 0.554 0.239 0.183 0.133 0.142 0.177 0.110 0.091 0.181 0.086 0.084 0.179 0.086 0.185 0.138 0.132 0.083 0.089 0.079 0.071 0.080 0.071 0.058 0.066 0.057 0.062 0.072 0.063 0.542 0.376 0.800 0.254 0.208 0.157 0.130 0.191 0.103 0.095 0.176 0.091 0.075 0.183 0.075 0.314 0.158 0.177 0.131 0.125 0.107 0.089 0.101 0.079 0.067 0.085 0.068 0.066 0.081 0.069 0.401 0.901 1.981 0.289 0.719 0.524 0.206 0.494 0.299 0.203 0.477 0.250 0.201 0.467 0.216 0.224 0.434 0.399 0.138 0.188 0.166 0.154 0.192 0.169 0.151 0.179 0.159 0.138 0.158 0.140 0.489 0.840 2.405 0.273 0.687 0.538 0.234 0.524 0.327 0.223 0.472 0.274 0.250 0.452 0.261 0.250 0.642 0.671 0.234 0.387 0.314 0.181 0.270 0.210 0.170 0.247 0.187 0.181 0.241 0.181 0.374 0.511 1.023 0.218 0.387 0.294 0.168 0.274 0.195 0.168 0.257 0.178 0.173 0.248 0.175 0.159 0.282 0.260 0.107 0.126 0.117 0.109 0.125 0.115 0.100 0.111 0.103 0.097 0.104 0.098 0.334 0.571 0.979 0.218 0.399 0.324 0.208 0.309 0.230 0.204 0.269 0.211 0.204 0.248 0.199 0.260 0.410 0.448 0.149 0.216 0.186 0.138 0.178 0.152 0.129 0.153 0.134 0.132 0.152 0.132 1.488 1.396 1.519 0.512 0.509 0.512 0.399 0.402 0.399 0.360 0.365 0.360 0.326 0.328 0.326 0.434 0.434 0.434 0.219 0.219 0.219 0.191 0.191 0.191 0.168 0.168 0.168 0.136 0.136 0.136 1.287 1.182 1.310 0.504 0.506 0.624 0.468 0.474 0.491 0.314 0.312 0.397 0.301 0.305 0.357 0.638 0.634 0.772 0.363 0.360 0.412 0.285 0.283 0.334 0.247 0.247 0.305 0.215 0.214 0.259 0.522 0.449 0.496 0.238 0.302 0.212 0.158 0.262 0.154 0.136 0.240 0.141 0.132 0.239 0.134 0.184 0.186 0.174 0.142 0.168 0.155 0.133 0.153 0.140 0.139 0.154 0.143 0.139 0.151 0.140 0.577 0.512 0.534 0.295 0.283 0.209 0.179 0.250 0.175 0.142 0.232 0.158 0.127 0.224 0.149 0.349 0.224 0.210 0.163 0.176 0.151 0.123 0.163 0.147 0.117 0.150 0.139 0.116 0.148 0.139 169 Table 36. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.698 0.648 0.698 0.288 0.292 0.288 0.219 0.218 0.219 0.190 0.190 0.191 0.159 0.159 0.159 0.227 0.227 0.227 0.125 0.125 0.125 0.096 0.095 0.095 0.090 0.090 0.090 0.068 0.068 0.068 0.642 0.550 0.642 0.278 0.271 0.347 0.220 0.221 0.254 0.192 0.195 0.214 0.169 0.173 0.194 0.325 0.310 0.375 0.168 0.167 0.182 0.138 0.138 0.152 0.109 0.109 0.122 0.096 0.096 0.110 0.520 0.210 0.218 0.234 0.161 0.065 0.121 0.160 0.046 0.068 0.152 0.043 0.040 0.154 0.038 0.199 0.073 0.064 0.065 0.050 0.044 0.047 0.040 0.037 0.038 0.039 0.035 0.035 0.039 0.035 0.545 0.215 0.234 0.264 0.127 0.075 0.136 0.149 0.064 0.078 0.143 0.052 0.050 0.151 0.049 0.325 0.112 0.105 0.131 0.070 0.054 0.065 0.075 0.046 0.046 0.065 0.042 0.039 0.066 0.042 0.728 0.657 0.728 0.285 0.285 0.285 0.212 0.206 0.212 0.185 0.186 0.185 0.164 0.164 0.164 0.231 0.230 0.230 0.132 0.131 0.131 0.097 0.097 0.097 0.085 0.085 0.085 0.068 0.068 0.068 0.769 0.648 0.768 0.321 0.295 0.347 0.202 0.200 0.229 0.152 0.154 0.172 0.145 0.138 0.159 0.289 0.282 0.340 0.164 0.161 0.178 0.134 0.133 0.153 0.120 0.121 0.138 0.105 0.106 0.112 RX/RM =10, ICCX/ICCM =0.25, g=200, Balanced 1.140 1.063 1.141 0.504 0.487 0.502 0.407 0.405 0.407 0.346 0.346 0.346 0.285 0.287 0.285 0.379 0.377 0.378 0.205 0.204 0.205 0.140 0.140 0.140 0.126 0.126 0.127 0.119 0.119 0.119 1.350 1.183 1.349 0.614 0.598 0.614 0.469 0.459 0.469 0.362 0.360 0.362 0.334 0.336 0.335 0.411 0.405 0.410 0.246 0.246 0.246 0.197 0.197 0.197 0.165 0.165 0.165 0.126 0.126 0.127 0.518 0.204 0.222 0.244 0.149 0.076 0.118 0.166 0.049 0.062 0.160 0.038 0.036 0.165 0.033 0.202 0.066 0.059 0.072 0.045 0.040 0.042 0.038 0.033 0.034 0.037 0.031 0.030 0.036 0.029 0.396 0.632 0.663 0.144 0.472 0.253 0.108 0.399 0.181 0.110 0.382 0.155 0.137 0.376 0.151 0.151 0.190 0.171 0.064 0.100 0.083 0.068 0.096 0.079 0.064 0.084 0.069 0.068 0.087 0.070 RX/RM =10, ICCX/ICCM =0.25, g=200, Unbalanced 0.384 0.531 1.065 0.152 0.356 0.201 0.109 0.399 0.197 0.131 0.360 0.180 0.166 0.370 0.175 0.213 0.324 0.275 0.096 0.156 0.116 0.092 0.154 0.114 0.088 0.153 0.101 0.097 0.152 0.097 1.130 0.998 1.130 0.540 0.527 0.742 0.392 0.389 0.430 0.296 0.295 0.343 0.276 0.274 0.291 0.573 0.583 0.600 0.290 0.292 0.337 0.239 0.238 0.276 0.195 0.195 0.218 0.177 0.177 0.197 0.552 0.205 0.250 0.262 0.140 0.079 0.136 0.150 0.061 0.073 0.145 0.050 0.044 0.147 0.043 0.334 0.095 0.098 0.129 0.075 0.052 0.071 0.065 0.043 0.046 0.067 0.040 0.037 0.067 0.038 1.239 1.087 1.239 0.533 0.520 0.636 0.389 0.379 0.455 0.322 0.316 0.388 0.305 0.301 0.343 0.644 0.641 0.797 0.330 0.326 0.380 0.290 0.289 0.326 0.224 0.226 0.252 0.198 0.198 0.224 170 0.387 0.425 0.437 0.213 0.275 0.208 0.175 0.222 0.171 0.159 0.193 0.158 0.152 0.189 0.152 0.110 0.124 0.114 0.067 0.067 0.065 0.059 0.059 0.058 0.055 0.055 0.055 0.055 0.055 0.055 0.381 0.355 0.705 0.217 0.215 0.183 0.185 0.202 0.174 0.169 0.173 0.156 0.173 0.184 0.163 0.212 0.193 0.167 0.105 0.096 0.091 0.088 0.086 0.082 0.086 0.079 0.078 0.082 0.077 0.075 0.638 0.626 0.638 0.298 0.299 0.299 0.224 0.225 0.224 0.168 0.168 0.168 0.174 0.172 0.174 0.213 0.212 0.213 0.119 0.119 0.119 0.105 0.105 0.105 0.091 0.091 0.091 0.083 0.083 0.083 0.723 0.677 0.723 0.351 0.347 0.418 0.243 0.240 0.296 0.228 0.229 0.248 0.193 0.193 0.194 0.316 0.311 0.373 0.189 0.189 0.219 0.135 0.135 0.147 0.100 0.101 0.114 0.102 0.102 0.108 0.511 0.292 0.216 0.220 0.216 0.102 0.114 0.190 0.069 0.067 0.178 0.057 0.054 0.176 0.053 0.193 0.097 0.090 0.078 0.075 0.067 0.063 0.069 0.062 0.060 0.067 0.061 0.060 0.066 0.060 0.579 0.247 0.223 0.279 0.175 0.106 0.158 0.155 0.079 0.091 0.156 0.071 0.071 0.153 0.070 0.342 0.124 0.114 0.121 0.109 0.072 0.079 0.084 0.062 0.056 0.093 0.064 0.052 0.091 0.059 Table 37. RMSE using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.293 0.263 0.294 0.127 0.125 0.128 0.108 0.107 0.108 0.096 0.096 0.096 0.078 0.078 0.078 0.089 0.089 0.089 0.053 0.053 0.053 0.040 0.040 0.040 0.033 0.033 0.033 0.030 0.030 0.030 0.306 0.286 0.301 0.125 0.122 0.154 0.088 0.088 0.105 0.070 0.070 0.092 0.061 0.061 0.076 0.152 0.152 0.178 0.071 0.071 0.091 0.059 0.059 0.063 0.048 0.048 0.053 0.041 0.041 0.047 0.394 4.573 6.594 0.525 5.628 2.761 0.660 7.115 1.379 0.783 8.534 1.150 0.822 9.211 0.911 0.570 3.514 3.703 0.648 1.337 0.996 0.707 1.150 0.870 0.709 1.013 0.777 0.758 1.011 0.777 0.402 3.822 6.246 0.451 5.077 6.067 0.589 7.022 1.303 0.634 6.848 1.066 0.755 6.515 0.951 0.488 4.730 9.022 0.625 3.114 1.488 0.713 1.910 1.104 0.765 1.682 0.983 0.810 1.610 0.956 0.070 0.062 0.066 0.027 0.026 0.027 0.023 0.023 0.023 0.018 0.017 0.018 0.017 0.017 0.017 0.022 0.021 0.022 0.011 0.011 0.011 0.008 0.008 0.008 0.007 0.007 0.007 0.006 0.006 0.006 0.072 0.065 0.070 0.030 0.030 0.035 0.022 0.022 0.025 0.019 0.019 0.021 0.016 0.016 0.018 0.031 0.031 0.036 0.016 0.016 0.018 0.014 0.014 0.017 0.011 0.011 0.014 0.010 0.010 0.012 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Balanced 0.145 0.140 0.146 0.059 0.060 0.060 0.048 0.049 0.048 0.037 0.037 0.037 0.034 0.034 0.034 0.049 0.046 0.049 0.024 0.024 0.024 0.021 0.021 0.021 0.017 0.017 0.017 0.015 0.015 0.015 0.406 5.216 5.749 0.518 7.266 4.199 0.620 9.900 1.373 0.766 10.875 1.176 0.836 8.749 0.932 0.578 3.956 10.858 0.636 1.316 0.977 0.680 1.068 0.826 0.688 0.968 0.751 0.750 0.986 0.768 0.306 0.284 0.311 0.146 0.144 0.147 0.101 0.099 0.101 0.078 0.078 0.078 0.073 0.072 0.073 0.089 0.090 0.089 0.059 0.059 0.059 0.041 0.041 0.041 0.033 0.033 0.033 0.030 0.030 0.030 0.100 0.801 1.359 0.126 1.761 0.715 0.134 3.028 0.299 0.144 3.257 0.209 0.177 4.233 0.197 0.139 1.282 1.254 0.145 0.322 0.223 0.141 0.234 0.173 0.162 0.227 0.176 0.169 0.223 0.173 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Unbalanced 0.373 3.588 7.228 0.493 6.191 3.474 0.661 9.424 1.609 0.636 8.463 1.031 0.776 9.062 0.973 0.456 6.985 11.791 0.566 4.156 2.060 0.661 2.293 1.061 0.694 1.586 0.855 0.745 1.470 0.798 0.124 0.118 0.123 0.061 0.061 0.079 0.052 0.054 0.061 0.038 0.039 0.044 0.036 0.035 0.041 0.068 0.068 0.078 0.035 0.035 0.043 0.027 0.027 0.033 0.021 0.021 0.025 0.019 0.019 0.022 0.108 1.026 1.357 0.132 1.748 1.038 0.144 2.503 0.444 0.165 2.462 0.249 0.181 3.091 0.215 0.114 1.250 1.568 0.127 0.912 0.283 0.161 0.588 0.242 0.155 0.409 0.193 0.159 0.329 0.172 0.330 0.274 0.309 0.140 0.138 0.161 0.105 0.105 0.115 0.088 0.089 0.090 0.077 0.077 0.079 0.141 0.140 0.165 0.085 0.085 0.094 0.059 0.059 0.062 0.051 0.051 0.052 0.046 0.046 0.048 171 0.177 0.932 0.946 0.162 0.680 0.721 0.186 0.757 0.230 0.183 0.691 0.202 0.180 0.440 0.184 0.181 0.470 0.532 0.160 0.190 0.178 0.176 0.194 0.184 0.171 0.183 0.174 0.167 0.177 0.168 0.156 1.072 1.782 0.192 0.656 0.510 0.171 0.694 0.228 0.179 0.581 0.213 0.182 0.772 0.200 0.166 0.714 0.571 0.157 0.285 0.206 0.159 0.230 0.188 0.153 0.191 0.175 0.159 0.186 0.175 0.289 0.267 0.293 0.116 0.115 0.116 0.084 0.084 0.084 0.076 0.075 0.076 0.067 0.066 0.067 0.095 0.095 0.095 0.058 0.058 0.058 0.042 0.042 0.042 0.035 0.035 0.035 0.032 0.032 0.032 0.342 0.296 0.329 0.128 0.128 0.143 0.097 0.096 0.121 0.079 0.078 0.095 0.066 0.066 0.082 0.143 0.139 0.169 0.071 0.071 0.085 0.057 0.057 0.065 0.044 0.044 0.053 0.042 0.042 0.049 0.418 2.770 8.395 0.481 4.210 1.960 0.533 4.505 1.170 0.596 5.154 0.881 0.659 5.518 0.731 0.475 2.103 2.686 0.553 1.167 0.847 0.623 0.943 0.746 0.649 0.891 0.705 0.625 0.802 0.639 0.383 2.453 3.543 0.547 4.422 3.728 0.566 5.341 1.541 0.612 4.674 0.999 0.691 5.296 0.877 0.479 3.408 5.473 0.553 2.203 1.203 0.668 1.742 0.921 0.653 1.363 0.781 0.708 1.329 0.741 Table 37. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.147 0.141 0.148 0.065 0.065 0.065 0.053 0.053 0.053 0.039 0.039 0.039 0.035 0.035 0.035 0.047 0.047 0.047 0.027 0.027 0.027 0.019 0.019 0.019 0.017 0.017 0.017 0.015 0.015 0.015 0.120 0.109 0.120 0.063 0.062 0.079 0.054 0.054 0.058 0.042 0.042 0.048 0.038 0.037 0.042 0.064 0.064 0.083 0.038 0.038 0.049 0.027 0.027 0.031 0.020 0.020 0.025 0.022 0.022 0.026 0.248 3.365 5.256 0.237 5.235 1.711 0.314 6.143 0.583 0.318 6.519 0.440 0.349 6.802 0.385 0.286 1.093 0.806 0.314 0.569 0.453 0.314 0.472 0.375 0.351 0.483 0.382 0.341 0.439 0.349 0.257 3.577 4.545 0.297 4.702 0.928 0.335 3.709 0.647 0.370 3.970 0.544 0.398 2.770 0.447 0.238 1.768 1.049 0.283 0.917 0.542 0.323 0.787 0.493 0.345 0.717 0.461 0.354 0.654 0.428 0.031 0.030 0.031 0.015 0.015 0.015 0.010 0.009 0.010 0.010 0.009 0.010 0.008 0.008 0.008 0.012 0.012 0.012 0.006 0.006 0.006 0.005 0.005 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.031 0.029 0.031 0.014 0.014 0.018 0.009 0.009 0.011 0.008 0.008 0.010 0.007 0.007 0.009 0.016 0.016 0.019 0.008 0.009 0.009 0.007 0.007 0.008 0.006 0.006 0.007 0.005 0.005 0.006 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Balanced 0.074 0.075 0.075 0.034 0.034 0.034 0.025 0.025 0.025 0.019 0.019 0.019 0.019 0.019 0.019 0.025 0.025 0.025 0.014 0.014 0.014 0.009 0.009 0.009 0.009 0.009 0.009 0.007 0.007 0.007 0.149 0.144 0.150 0.064 0.064 0.064 0.046 0.046 0.046 0.037 0.037 0.037 0.034 0.034 0.034 0.049 0.049 0.049 0.026 0.026 0.026 0.020 0.020 0.020 0.018 0.018 0.018 0.015 0.015 0.015 0.180 4.430 3.365 0.242 5.889 0.785 0.321 5.675 0.592 0.356 9.639 0.499 0.387 7.933 0.427 0.290 1.231 0.894 0.342 0.615 0.492 0.397 0.582 0.470 0.418 0.559 0.451 0.418 0.527 0.427 0.097 1.209 0.535 0.084 3.017 0.174 0.086 3.143 0.140 0.079 3.903 0.103 0.078 4.508 0.085 0.084 0.269 0.186 0.071 0.129 0.099 0.085 0.129 0.100 0.086 0.119 0.093 0.088 0.117 0.091 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Unbalanced 0.201 5.778 6.616 0.217 5.664 0.682 0.313 4.473 0.610 0.332 3.782 0.502 0.394 4.177 0.456 0.220 2.877 2.858 0.277 0.916 0.564 0.304 0.760 0.479 0.377 0.769 0.481 0.379 0.700 0.442 0.068 0.067 0.069 0.029 0.029 0.036 0.025 0.025 0.030 0.019 0.019 0.023 0.017 0.017 0.021 0.035 0.035 0.040 0.019 0.019 0.021 0.013 0.013 0.015 0.011 0.011 0.014 0.010 0.010 0.011 0.090 1.312 1.131 0.082 1.847 0.154 0.085 2.180 0.139 0.086 2.981 0.111 0.080 1.826 0.090 0.086 1.109 0.288 0.081 0.258 0.142 0.080 0.189 0.119 0.086 0.183 0.113 0.091 0.181 0.109 0.144 0.131 0.141 0.071 0.072 0.078 0.041 0.041 0.048 0.040 0.040 0.045 0.034 0.034 0.041 0.074 0.073 0.090 0.036 0.037 0.044 0.030 0.030 0.033 0.026 0.026 0.028 0.023 0.023 0.025 172 0.084 0.585 0.555 0.092 0.330 0.126 0.094 0.207 0.108 0.095 0.427 0.101 0.096 0.274 0.098 0.070 0.102 0.094 0.073 0.083 0.079 0.081 0.087 0.084 0.076 0.080 0.077 0.077 0.080 0.077 0.079 0.821 0.540 0.080 0.345 0.122 0.079 0.202 0.102 0.079 0.184 0.097 0.077 0.152 0.092 0.075 0.294 0.172 0.093 0.126 0.114 0.087 0.108 0.100 0.093 0.108 0.100 0.094 0.106 0.100 0.161 0.150 0.160 0.070 0.070 0.070 0.050 0.050 0.050 0.043 0.043 0.043 0.036 0.037 0.037 0.049 0.049 0.049 0.026 0.026 0.026 0.023 0.023 0.023 0.017 0.017 0.017 0.015 0.015 0.015 0.147 0.136 0.148 0.068 0.066 0.080 0.044 0.044 0.054 0.040 0.040 0.045 0.033 0.033 0.036 0.065 0.064 0.079 0.042 0.042 0.044 0.028 0.028 0.033 0.026 0.026 0.030 0.023 0.023 0.028 0.233 2.374 2.001 0.241 3.733 0.677 0.267 4.858 0.512 0.317 5.533 0.441 0.348 6.876 0.384 0.278 0.954 0.728 0.313 0.575 0.452 0.264 0.399 0.316 0.314 0.424 0.340 0.314 0.398 0.320 0.232 3.244 8.753 0.280 3.199 0.686 0.292 3.499 0.585 0.313 3.818 0.502 0.354 3.241 0.437 0.226 1.890 1.330 0.277 0.784 0.492 0.287 0.638 0.380 0.330 0.637 0.393 0.339 0.608 0.368 Table 37. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.359 0.344 0.359 0.157 0.156 0.157 0.116 0.116 0.116 0.093 0.093 0.093 0.077 0.077 0.077 0.103 0.103 0.103 0.062 0.062 0.062 0.050 0.050 0.050 0.035 0.035 0.035 0.029 0.029 0.029 0.327 0.299 0.327 0.154 0.151 0.184 0.105 0.105 0.117 0.080 0.080 0.103 0.073 0.073 0.082 0.168 0.167 0.181 0.085 0.085 0.097 0.059 0.059 0.069 0.048 0.048 0.060 0.039 0.039 0.047 0.376 1.963 1.500 0.354 0.611 0.482 0.351 0.477 0.398 0.356 0.453 0.378 0.346 0.422 0.352 0.355 0.473 0.456 0.347 0.383 0.371 0.349 0.372 0.360 0.345 0.361 0.350 0.350 0.363 0.351 0.353 1.973 2.244 0.341 0.678 0.515 0.360 0.555 0.429 0.347 0.477 0.395 0.350 0.446 0.372 0.356 0.619 0.559 0.351 0.453 0.428 0.358 0.418 0.407 0.358 0.404 0.393 0.345 0.380 0.374 0.159 0.155 0.159 0.085 0.085 0.085 0.049 0.049 0.049 0.043 0.043 0.043 0.033 0.033 0.033 0.051 0.051 0.051 0.033 0.033 0.033 0.025 0.025 0.025 0.021 0.021 0.021 0.018 0.018 0.018 0.162 0.158 0.163 0.072 0.071 0.087 0.047 0.047 0.055 0.040 0.040 0.049 0.035 0.035 0.042 0.072 0.073 0.083 0.046 0.046 0.052 0.031 0.031 0.034 0.027 0.027 0.030 0.025 0.025 0.026 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Balanced 0.155 0.149 0.155 0.062 0.062 0.062 0.051 0.051 0.051 0.039 0.039 0.039 0.036 0.036 0.036 0.042 0.042 0.042 0.027 0.027 0.027 0.016 0.016 0.016 0.016 0.016 0.016 0.015 0.015 0.014 0.360 0.342 0.359 0.137 0.137 0.137 0.108 0.107 0.107 0.097 0.097 0.097 0.075 0.075 0.075 0.117 0.117 0.117 0.062 0.062 0.061 0.046 0.046 0.046 0.037 0.037 0.037 0.033 0.033 0.033 0.167 0.865 0.491 0.168 0.336 0.246 0.182 0.280 0.216 0.196 0.266 0.212 0.192 0.245 0.196 0.151 0.213 0.203 0.166 0.188 0.180 0.149 0.159 0.153 0.151 0.159 0.153 0.151 0.157 0.151 0.366 1.861 3.341 0.344 0.634 0.480 0.353 0.530 0.416 0.349 0.464 0.374 0.348 0.439 0.355 0.385 0.531 0.508 0.403 0.456 0.438 0.367 0.395 0.380 0.387 0.408 0.393 0.388 0.404 0.390 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Unbalanced 0.391 2.019 3.099 0.367 0.829 0.566 0.377 0.590 0.487 0.377 0.546 0.462 0.379 0.508 0.436 0.370 0.834 0.692 0.339 0.451 0.442 0.327 0.391 0.404 0.358 0.411 0.422 0.354 0.396 0.415 0.147 0.136 0.148 0.065 0.065 0.079 0.048 0.049 0.054 0.041 0.041 0.046 0.034 0.034 0.041 0.071 0.070 0.078 0.039 0.040 0.044 0.028 0.028 0.034 0.023 0.023 0.027 0.023 0.023 0.026 0.198 0.946 0.760 0.192 0.347 0.274 0.196 0.303 0.246 0.179 0.252 0.210 0.195 0.254 0.213 0.169 0.347 0.275 0.156 0.199 0.194 0.160 0.188 0.183 0.150 0.168 0.170 0.160 0.175 0.172 0.305 0.300 0.310 0.168 0.169 0.203 0.103 0.103 0.114 0.087 0.088 0.101 0.074 0.074 0.086 0.154 0.154 0.198 0.090 0.090 0.107 0.069 0.069 0.085 0.056 0.057 0.067 0.051 0.051 0.062 173 0.160 0.857 1.050 0.149 0.310 0.221 0.161 0.254 0.192 0.145 0.200 0.157 0.144 0.190 0.147 0.181 0.284 0.264 0.176 0.202 0.193 0.186 0.202 0.193 0.184 0.195 0.187 0.188 0.197 0.189 0.150 0.980 0.778 0.150 0.384 0.229 0.146 0.239 0.186 0.162 0.239 0.189 0.167 0.223 0.183 0.178 0.410 0.335 0.168 0.228 0.212 0.179 0.222 0.216 0.181 0.213 0.206 0.177 0.201 0.199 0.317 0.296 0.320 0.160 0.160 0.160 0.109 0.109 0.109 0.098 0.098 0.098 0.085 0.085 0.085 0.117 0.117 0.118 0.065 0.065 0.065 0.056 0.056 0.056 0.039 0.039 0.039 0.035 0.035 0.035 0.355 0.324 0.357 0.148 0.147 0.188 0.120 0.119 0.143 0.102 0.101 0.117 0.080 0.080 0.094 0.154 0.152 0.174 0.087 0.087 0.097 0.058 0.058 0.073 0.054 0.054 0.061 0.047 0.047 0.054 0.342 1.062 0.988 0.332 0.575 0.455 0.371 0.541 0.433 0.339 0.444 0.363 0.340 0.422 0.346 0.314 0.422 0.404 0.296 0.332 0.320 0.280 0.301 0.290 0.278 0.292 0.282 0.270 0.280 0.271 0.369 1.136 1.080 0.327 0.660 0.481 0.333 0.518 0.422 0.318 0.449 0.391 0.314 0.420 0.371 0.361 0.570 0.513 0.346 0.441 0.415 0.325 0.373 0.346 0.335 0.374 0.352 0.346 0.376 0.354 Table 37. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.167 0.167 0.167 0.066 0.066 0.066 0.054 0.054 0.054 0.044 0.044 0.044 0.035 0.035 0.035 0.062 0.063 0.063 0.030 0.030 0.030 0.023 0.023 0.023 0.022 0.022 0.022 0.018 0.018 0.018 0.169 0.148 0.168 0.076 0.076 0.088 0.055 0.055 0.068 0.048 0.048 0.061 0.045 0.045 0.054 0.079 0.079 0.091 0.038 0.038 0.048 0.033 0.033 0.040 0.028 0.027 0.034 0.027 0.027 0.032 0.189 0.588 0.413 0.185 0.292 0.240 0.165 0.225 0.187 0.167 0.209 0.176 0.164 0.199 0.167 0.157 0.208 0.200 0.180 0.200 0.193 0.170 0.182 0.176 0.171 0.179 0.173 0.173 0.180 0.174 0.222 0.609 0.517 0.193 0.317 0.262 0.191 0.276 0.238 0.184 0.247 0.215 0.188 0.237 0.207 0.183 0.285 0.269 0.170 0.214 0.212 0.179 0.207 0.207 0.186 0.209 0.209 0.190 0.208 0.211 0.085 0.085 0.085 0.036 0.036 0.036 0.028 0.028 0.028 0.021 0.021 0.021 0.019 0.019 0.019 0.028 0.028 0.028 0.016 0.016 0.016 0.011 0.011 0.011 0.010 0.010 0.010 0.009 0.009 0.009 0.079 0.074 0.079 0.030 0.030 0.040 0.026 0.026 0.034 0.020 0.020 0.024 0.017 0.017 0.021 0.036 0.036 0.041 0.023 0.022 0.026 0.016 0.015 0.019 0.014 0.014 0.016 0.013 0.013 0.014 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Balanced 0.068 0.068 0.068 0.033 0.033 0.033 0.023 0.023 0.024 0.020 0.020 0.020 0.017 0.016 0.017 0.021 0.021 0.021 0.013 0.013 0.013 0.011 0.011 0.011 0.008 0.008 0.008 0.008 0.008 0.008 0.165 0.165 0.165 0.078 0.078 0.078 0.054 0.054 0.054 0.044 0.044 0.044 0.041 0.041 0.041 0.052 0.052 0.052 0.029 0.029 0.029 0.024 0.024 0.024 0.022 0.022 0.022 0.017 0.017 0.017 0.096 0.360 0.212 0.103 0.166 0.135 0.091 0.125 0.103 0.094 0.119 0.100 0.094 0.115 0.096 0.083 0.105 0.102 0.080 0.087 0.084 0.076 0.081 0.078 0.080 0.084 0.081 0.076 0.078 0.076 0.194 1.029 0.474 0.189 0.310 0.254 0.172 0.230 0.195 0.170 0.211 0.180 0.166 0.197 0.169 0.171 0.222 0.215 0.163 0.180 0.174 0.176 0.186 0.181 0.173 0.181 0.175 0.171 0.177 0.172 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Unbalanced 0.186 0.760 0.490 0.187 0.329 0.270 0.185 0.271 0.224 0.191 0.253 0.209 0.176 0.225 0.195 0.179 0.279 0.261 0.174 0.217 0.211 0.174 0.201 0.196 0.173 0.193 0.191 0.178 0.194 0.194 0.077 0.070 0.077 0.033 0.033 0.041 0.023 0.022 0.029 0.019 0.018 0.021 0.018 0.018 0.020 0.032 0.032 0.039 0.018 0.018 0.020 0.016 0.016 0.017 0.011 0.011 0.013 0.011 0.011 0.013 0.101 0.309 0.257 0.082 0.150 0.123 0.095 0.140 0.115 0.088 0.120 0.101 0.091 0.118 0.098 0.091 0.134 0.125 0.092 0.115 0.108 0.080 0.093 0.090 0.086 0.095 0.090 0.086 0.094 0.091 0.174 0.168 0.174 0.070 0.070 0.088 0.054 0.054 0.060 0.049 0.049 0.062 0.041 0.041 0.048 0.072 0.072 0.088 0.042 0.042 0.051 0.031 0.031 0.037 0.027 0.027 0.033 0.020 0.020 0.026 174 0.085 0.446 0.246 0.074 0.123 0.099 0.078 0.108 0.089 0.071 0.091 0.075 0.074 0.090 0.075 0.072 0.103 0.099 0.075 0.086 0.083 0.075 0.081 0.078 0.078 0.083 0.080 0.077 0.081 0.078 0.080 0.625 0.215 0.080 0.145 0.115 0.081 0.120 0.103 0.087 0.118 0.102 0.089 0.114 0.098 0.086 0.150 0.138 0.089 0.113 0.104 0.078 0.093 0.090 0.084 0.096 0.095 0.086 0.095 0.094 0.158 0.158 0.158 0.075 0.075 0.075 0.060 0.059 0.060 0.049 0.049 0.049 0.043 0.043 0.043 0.054 0.054 0.054 0.033 0.033 0.033 0.026 0.026 0.026 0.022 0.022 0.022 0.020 0.020 0.020 0.156 0.152 0.157 0.071 0.071 0.093 0.057 0.057 0.066 0.043 0.043 0.049 0.036 0.036 0.044 0.067 0.067 0.073 0.043 0.042 0.050 0.032 0.032 0.039 0.028 0.028 0.033 0.027 0.027 0.029 0.205 0.740 0.453 0.149 0.231 0.190 0.173 0.233 0.195 0.155 0.197 0.164 0.157 0.191 0.159 0.153 0.194 0.188 0.140 0.153 0.149 0.140 0.148 0.144 0.140 0.146 0.142 0.137 0.142 0.138 0.218 0.654 0.478 0.174 0.289 0.242 0.186 0.243 0.204 0.171 0.214 0.184 0.168 0.204 0.175 0.148 0.238 0.220 0.164 0.200 0.188 0.140 0.166 0.164 0.144 0.160 0.159 0.147 0.162 0.161 Table 37. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.280 0.247 0.282 0.126 0.125 0.126 0.103 0.101 0.103 0.085 0.084 0.085 0.074 0.073 0.074 0.088 0.088 0.088 0.052 0.052 0.052 0.038 0.038 0.038 0.035 0.035 0.034 0.030 0.030 0.030 0.291 0.261 0.285 0.135 0.133 0.154 0.112 0.112 0.135 0.081 0.081 0.094 0.071 0.071 0.081 0.141 0.140 0.161 0.078 0.079 0.085 0.066 0.066 0.075 0.053 0.053 0.062 0.050 0.050 0.058 1.697 4.561 7.894 1.352 4.882 5.842 1.085 7.370 1.625 0.853 8.162 0.996 0.685 9.144 0.745 1.236 3.629 7.230 0.968 1.967 1.314 0.796 1.378 0.929 0.732 1.125 0.783 0.817 1.177 0.836 1.700 4.277 4.376 1.479 4.615 6.870 1.222 6.151 1.947 0.951 6.857 1.077 0.871 7.658 0.914 1.437 3.573 4.418 1.039 2.703 1.261 0.682 2.173 0.867 0.663 2.422 0.919 0.656 2.048 0.814 0.057 0.053 0.056 0.027 0.028 0.026 0.021 0.022 0.021 0.018 0.019 0.018 0.017 0.019 0.016 0.020 0.020 0.020 0.011 0.011 0.011 0.010 0.010 0.010 0.008 0.008 0.008 0.007 0.007 0.007 0.066 0.060 0.067 0.028 0.029 0.037 0.023 0.024 0.028 0.019 0.020 0.024 0.016 0.017 0.020 0.030 0.031 0.034 0.018 0.018 0.018 0.013 0.013 0.014 0.011 0.011 0.012 0.010 0.010 0.011 0.841 2.887 7.875 0.763 7.200 4.341 0.670 8.656 1.457 0.561 12.708 0.927 0.681 13.739 0.797 0.814 5.192 3.039 0.659 5.393 1.135 0.557 0.944 0.663 0.603 0.916 0.657 0.593 0.843 0.607 RX/RM =10, ICCX/ICCM =0.05, g=50, Balanced 0.155 0.144 0.158 0.067 0.067 0.066 0.046 0.046 0.046 0.040 0.040 0.040 0.034 0.034 0.034 0.052 0.052 0.052 0.027 0.027 0.027 0.021 0.021 0.021 0.015 0.015 0.015 0.014 0.014 0.014 RX/RM =10, ICCX/ICCM =0.05, g=50, Unbalanced 0.143 0.137 0.143 0.061 0.061 0.080 0.042 0.042 0.053 0.039 0.039 0.048 0.034 0.034 0.038 0.074 0.075 0.095 0.038 0.038 0.042 0.025 0.025 0.029 0.025 0.025 0.028 0.020 0.020 0.023 0.859 4.616 7.522 0.801 6.506 6.165 0.625 9.380 1.837 0.740 8.634 1.239 0.791 10.559 0.978 0.824 4.132 6.536 0.656 4.578 1.820 0.619 3.372 1.043 0.697 2.790 0.946 0.755 2.671 0.864 0.809 0.752 1.516 0.644 0.520 0.936 0.460 0.559 0.528 0.277 0.454 0.198 0.132 0.470 0.129 0.588 0.358 1.609 0.312 0.326 0.213 0.158 0.285 0.123 0.098 0.266 0.099 0.081 0.264 0.083 0.833 0.745 1.021 0.664 0.351 1.182 0.508 0.440 0.423 0.324 0.497 0.194 0.157 0.435 0.124 0.717 0.369 2.011 0.475 0.349 1.104 0.310 0.306 0.179 0.196 0.333 0.114 0.104 0.317 0.101 0.287 0.269 0.289 0.125 0.126 0.126 0.092 0.092 0.092 0.069 0.069 0.069 0.067 0.063 0.067 0.094 0.093 0.094 0.052 0.052 0.052 0.039 0.039 0.040 0.038 0.038 0.038 0.033 0.033 0.033 0.324 0.287 0.330 0.129 0.127 0.149 0.098 0.098 0.117 0.087 0.088 0.094 0.078 0.081 0.084 0.133 0.130 0.153 0.066 0.065 0.073 0.055 0.055 0.067 0.050 0.050 0.057 0.044 0.044 0.050 0.311 0.576 0.859 0.186 0.764 0.408 0.170 0.756 0.222 0.151 0.926 0.170 0.144 1.040 0.149 0.155 0.590 0.382 0.150 0.392 0.177 0.134 0.158 0.143 0.143 0.162 0.147 0.142 0.156 0.143 0.312 1.294 1.307 0.215 0.720 0.494 0.179 0.822 0.238 0.182 0.707 0.217 0.165 0.858 0.185 0.193 0.525 0.892 0.169 0.484 0.212 0.136 0.252 0.161 0.146 0.233 0.159 0.143 0.236 0.154 0.290 0.255 0.288 0.124 0.124 0.122 0.097 0.094 0.097 0.078 0.080 0.078 0.067 0.067 0.067 0.092 0.092 0.092 0.065 0.065 0.065 0.046 0.046 0.046 0.037 0.037 0.037 0.033 0.033 0.033 0.261 0.235 0.260 0.126 0.118 0.181 0.090 0.091 0.109 0.077 0.076 0.087 0.068 0.067 0.079 0.137 0.130 0.167 0.070 0.070 0.089 0.061 0.060 0.072 0.046 0.046 0.054 0.043 0.043 0.049 1.657 2.900 7.118 1.310 3.979 2.762 0.995 4.849 1.519 0.853 6.003 0.956 0.693 6.615 0.757 1.237 2.689 2.674 0.812 2.007 1.137 0.643 1.088 0.710 0.634 1.115 0.702 0.625 0.964 0.640 1.662 2.524 3.858 1.377 4.058 2.563 1.133 4.619 1.250 0.904 4.928 0.925 0.647 5.611 0.707 1.474 3.249 5.946 1.142 2.534 1.415 0.890 2.202 1.063 0.752 2.173 0.893 0.723 1.885 0.796 175 Table 37. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.155 0.150 0.155 0.061 0.060 0.061 0.051 0.051 0.051 0.037 0.036 0.037 0.034 0.034 0.034 0.044 0.044 0.044 0.028 0.028 0.028 0.023 0.023 0.023 0.018 0.018 0.018 0.017 0.017 0.017 0.138 0.125 0.136 0.071 0.071 0.080 0.044 0.044 0.054 0.039 0.039 0.046 0.035 0.035 0.039 0.068 0.067 0.084 0.037 0.037 0.045 0.027 0.027 0.034 0.023 0.023 0.028 0.021 0.021 0.025 1.669 3.252 3.191 1.266 6.749 0.944 0.954 8.180 0.620 0.642 9.280 0.497 0.386 10.818 0.374 1.127 1.340 0.840 0.648 0.763 0.491 0.449 0.705 0.433 0.387 0.650 0.394 0.372 0.663 0.381 1.619 3.088 4.544 1.318 4.882 0.871 1.075 5.691 0.712 0.721 6.204 0.538 0.432 6.976 0.438 1.416 2.168 6.640 0.949 1.463 0.517 0.696 1.320 0.509 0.448 1.326 0.421 0.387 1.267 0.416 0.035 0.034 0.035 0.015 0.016 0.015 0.011 0.013 0.011 0.009 0.011 0.009 0.008 0.010 0.008 0.011 0.011 0.011 0.006 0.006 0.006 0.004 0.004 0.004 0.004 0.004 0.004 0.003 0.003 0.003 0.028 0.026 0.028 0.013 0.014 0.016 0.010 0.010 0.011 0.009 0.010 0.010 0.007 0.008 0.008 0.015 0.015 0.017 0.007 0.007 0.009 0.007 0.007 0.008 0.006 0.006 0.006 0.004 0.005 0.005 RX/RM =10, ICCX/ICCM =0.05, g=200, Balanced 0.066 0.065 0.067 0.031 0.030 0.031 0.020 0.020 0.020 0.018 0.018 0.018 0.014 0.013 0.014 0.023 0.023 0.023 0.013 0.013 0.013 0.011 0.011 0.011 0.008 0.008 0.009 0.007 0.007 0.007 0.772 4.039 3.327 0.614 8.956 0.940 0.439 12.692 0.581 0.357 15.107 0.460 0.355 17.552 0.428 0.646 1.377 0.848 0.375 0.703 0.447 0.352 0.560 0.386 0.329 0.512 0.349 0.321 0.491 0.328 0.809 0.429 1.013 0.632 0.363 0.294 0.446 0.412 0.153 0.281 0.413 0.095 0.113 0.400 0.066 0.578 0.213 0.261 0.286 0.219 0.085 0.155 0.220 0.056 0.075 0.237 0.047 0.038 0.233 0.037 0.157 0.146 0.158 0.059 0.059 0.059 0.051 0.052 0.051 0.040 0.040 0.040 0.035 0.035 0.035 0.051 0.051 0.051 0.028 0.028 0.028 0.023 0.023 0.023 0.017 0.017 0.017 0.016 0.016 0.016 RX/RM =10, ICCX/ICCM =0.05, g=200, Unbalanced 0.805 6.892 10.325 0.653 5.631 0.818 0.491 8.862 0.545 0.385 11.646 0.553 0.338 10.108 0.451 0.695 3.470 2.479 0.526 1.673 0.641 0.406 1.212 0.523 0.346 1.050 0.413 0.341 1.059 0.387 0.075 0.071 0.075 0.032 0.031 0.043 0.024 0.024 0.032 0.020 0.020 0.022 0.017 0.017 0.020 0.033 0.033 0.041 0.018 0.018 0.022 0.013 0.013 0.014 0.012 0.012 0.014 0.011 0.011 0.013 0.822 0.606 0.854 0.660 0.385 0.348 0.494 0.399 0.146 0.319 0.413 0.094 0.124 0.400 0.065 0.701 0.274 0.709 0.459 0.277 0.136 0.281 0.286 0.070 0.147 0.295 0.059 0.054 0.303 0.048 0.137 0.121 0.137 0.064 0.063 0.078 0.049 0.048 0.058 0.038 0.038 0.043 0.035 0.035 0.039 0.076 0.076 0.083 0.041 0.041 0.047 0.028 0.028 0.034 0.025 0.025 0.028 0.023 0.023 0.026 176 0.281 0.468 0.292 0.156 0.786 0.139 0.122 1.006 0.124 0.103 1.265 0.105 0.104 1.338 0.107 0.111 0.174 0.128 0.077 0.092 0.082 0.074 0.083 0.077 0.071 0.078 0.072 0.074 0.080 0.075 0.278 1.141 0.722 0.165 0.568 0.142 0.123 0.795 0.112 0.120 0.942 0.119 0.112 0.809 0.107 0.178 0.353 0.189 0.093 0.151 0.096 0.080 0.110 0.085 0.075 0.100 0.080 0.077 0.101 0.079 0.142 0.129 0.141 0.066 0.064 0.066 0.045 0.045 0.045 0.040 0.041 0.040 0.033 0.033 0.033 0.053 0.053 0.053 0.023 0.023 0.023 0.022 0.022 0.022 0.019 0.019 0.019 0.016 0.016 0.016 0.134 0.127 0.133 0.062 0.064 0.075 0.047 0.046 0.062 0.039 0.039 0.044 0.040 0.040 0.043 0.066 0.067 0.076 0.040 0.040 0.047 0.030 0.030 0.035 0.029 0.029 0.032 0.023 0.023 0.026 1.621 2.249 4.096 1.300 4.201 0.849 0.908 6.048 0.556 0.630 6.536 0.463 0.366 7.197 0.350 1.148 1.302 0.843 0.645 0.736 0.444 0.459 0.690 0.419 0.410 0.633 0.403 0.372 0.609 0.374 1.658 2.353 5.745 1.379 3.310 0.911 1.056 4.867 0.544 0.689 5.487 0.416 0.389 5.609 0.374 1.422 2.226 1.347 1.005 1.321 0.516 0.675 1.232 0.417 0.460 1.156 0.380 0.347 1.182 0.367 Table 37. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.289 0.280 0.289 0.171 0.165 0.171 0.110 0.109 0.110 0.087 0.086 0.087 0.081 0.080 0.081 0.108 0.108 0.108 0.062 0.062 0.062 0.048 0.048 0.048 0.038 0.038 0.038 0.036 0.036 0.036 0.382 0.333 0.381 0.150 0.145 0.176 0.105 0.104 0.127 0.090 0.089 0.106 0.076 0.075 0.093 0.138 0.134 0.177 0.083 0.082 0.096 0.069 0.068 0.079 0.050 0.050 0.054 0.045 0.045 0.048 1.128 1.112 2.852 0.534 0.701 0.441 0.344 0.534 0.305 0.250 0.503 0.251 0.210 0.448 0.212 0.465 0.309 0.295 0.269 0.267 0.251 0.204 0.221 0.204 0.204 0.222 0.205 0.199 0.215 0.200 1.125 1.329 6.394 0.549 0.634 0.453 0.380 0.541 0.329 0.317 0.481 0.293 0.252 0.471 0.255 0.715 0.608 0.610 0.324 0.318 0.258 0.239 0.275 0.230 0.202 0.264 0.214 0.190 0.252 0.207 0.161 0.162 0.161 0.072 0.071 0.072 0.050 0.050 0.050 0.046 0.047 0.046 0.042 0.042 0.042 0.049 0.049 0.049 0.029 0.029 0.029 0.023 0.023 0.023 0.021 0.021 0.021 0.018 0.018 0.018 0.165 0.140 0.166 0.075 0.076 0.085 0.049 0.049 0.053 0.042 0.042 0.051 0.040 0.040 0.045 0.085 0.083 0.094 0.040 0.039 0.046 0.029 0.029 0.034 0.025 0.025 0.029 0.019 0.019 0.023 0.368 0.327 0.360 0.137 0.134 0.137 0.109 0.109 0.109 0.085 0.086 0.085 0.074 0.075 0.074 0.116 0.115 0.116 0.063 0.063 0.063 0.049 0.049 0.049 0.037 0.037 0.037 0.037 0.037 0.037 RX/RM =10, ICCX/ICCM =0.25, g=50, Balanced 0.142 0.132 0.144 0.061 0.060 0.061 0.045 0.042 0.045 0.036 0.036 0.036 0.031 0.031 0.031 0.048 0.048 0.048 0.031 0.031 0.031 0.019 0.019 0.019 0.018 0.018 0.018 0.015 0.015 0.015 RX/RM =10, ICCX/ICCM =0.25, g=50, Unbalanced 0.168 0.168 0.175 0.062 0.060 0.075 0.046 0.045 0.057 0.039 0.038 0.040 0.035 0.035 0.038 0.063 0.061 0.075 0.042 0.042 0.053 0.030 0.030 0.035 0.027 0.027 0.032 0.023 0.023 0.026 0.345 0.293 0.344 0.140 0.138 0.177 0.096 0.092 0.128 0.090 0.089 0.104 0.075 0.074 0.090 0.169 0.166 0.191 0.094 0.093 0.111 0.067 0.066 0.080 0.055 0.055 0.070 0.047 0.047 0.057 0.547 0.550 1.010 0.264 0.230 0.164 0.173 0.216 0.138 0.112 0.216 0.109 0.101 0.213 0.103 0.214 0.170 0.163 0.104 0.112 0.098 0.085 0.101 0.087 0.072 0.084 0.073 0.076 0.090 0.077 0.564 0.510 1.587 0.284 0.276 0.202 0.157 0.238 0.127 0.118 0.210 0.111 0.092 0.210 0.094 0.340 0.197 0.234 0.166 0.151 0.133 0.111 0.127 0.102 0.085 0.105 0.085 0.080 0.104 0.084 0.475 1.294 3.849 0.358 1.105 0.733 0.261 0.611 0.371 0.255 0.598 0.310 0.254 0.567 0.271 0.286 0.580 0.506 0.177 0.246 0.215 0.190 0.243 0.211 0.182 0.216 0.191 0.166 0.192 0.169 0.574 1.259 7.070 0.341 0.927 0.767 0.295 0.726 0.445 0.275 0.587 0.344 0.311 0.570 0.323 0.313 0.913 1.011 0.285 0.489 0.401 0.226 0.353 0.272 0.216 0.308 0.236 0.225 0.308 0.232 0.215 0.369 0.920 0.136 0.298 0.197 0.103 0.167 0.119 0.101 0.161 0.108 0.102 0.153 0.104 0.098 0.182 0.160 0.068 0.081 0.075 0.068 0.078 0.072 0.063 0.070 0.065 0.060 0.064 0.060 0.193 0.440 0.716 0.138 0.257 0.211 0.126 0.204 0.143 0.122 0.169 0.127 0.122 0.154 0.119 0.155 0.310 0.332 0.091 0.138 0.117 0.086 0.112 0.095 0.081 0.096 0.083 0.081 0.093 0.081 0.385 0.363 0.393 0.128 0.127 0.128 0.099 0.100 0.099 0.090 0.091 0.090 0.082 0.083 0.082 0.106 0.106 0.106 0.056 0.055 0.056 0.048 0.048 0.048 0.043 0.043 0.043 0.034 0.034 0.034 0.312 0.291 0.314 0.126 0.125 0.156 0.114 0.115 0.124 0.080 0.079 0.097 0.073 0.074 0.088 0.160 0.158 0.197 0.091 0.091 0.106 0.073 0.072 0.086 0.061 0.060 0.072 0.055 0.055 0.065 1.105 1.184 1.341 0.563 0.891 0.628 0.386 0.658 0.389 0.339 0.608 0.356 0.320 0.599 0.331 0.465 0.481 0.434 0.344 0.408 0.375 0.324 0.385 0.347 0.333 0.375 0.343 0.339 0.374 0.342 1.201 1.412 1.908 0.668 0.750 0.550 0.454 0.693 0.467 0.361 0.588 0.394 0.329 0.572 0.372 0.759 0.607 0.574 0.413 0.426 0.377 0.317 0.398 0.362 0.295 0.370 0.336 0.292 0.367 0.342 177 Table 37. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.168 0.158 0.168 0.075 0.076 0.075 0.055 0.055 0.055 0.047 0.047 0.047 0.041 0.041 0.041 0.057 0.057 0.057 0.031 0.031 0.031 0.023 0.023 0.023 0.022 0.022 0.022 0.016 0.016 0.016 0.162 0.149 0.162 0.071 0.069 0.084 0.053 0.053 0.062 0.048 0.048 0.055 0.042 0.042 0.048 0.080 0.077 0.098 0.042 0.042 0.045 0.035 0.034 0.038 0.027 0.027 0.031 0.025 0.025 0.029 1.055 0.504 0.531 0.481 0.386 0.164 0.262 0.353 0.121 0.161 0.333 0.111 0.103 0.330 0.097 0.418 0.183 0.170 0.157 0.122 0.106 0.110 0.108 0.095 0.093 0.098 0.087 0.089 0.100 0.088 1.101 0.542 0.584 0.542 0.318 0.186 0.300 0.348 0.156 0.183 0.323 0.130 0.124 0.334 0.122 0.665 0.273 0.245 0.284 0.175 0.134 0.155 0.180 0.118 0.111 0.162 0.106 0.096 0.159 0.104 0.090 0.082 0.090 0.036 0.036 0.036 0.026 0.025 0.026 0.023 0.023 0.023 0.021 0.021 0.021 0.028 0.028 0.028 0.015 0.015 0.015 0.012 0.012 0.012 0.010 0.010 0.010 0.008 0.008 0.008 0.096 0.081 0.096 0.042 0.039 0.045 0.027 0.027 0.031 0.019 0.019 0.023 0.019 0.018 0.021 0.037 0.036 0.043 0.021 0.021 0.023 0.018 0.018 0.020 0.014 0.015 0.017 0.013 0.013 0.014 RX/RM =10, ICCX/ICCM =0.25, g=200, Balanced 0.071 0.066 0.071 0.032 0.031 0.032 0.025 0.025 0.025 0.021 0.021 0.021 0.019 0.019 0.018 0.024 0.024 0.024 0.013 0.013 0.013 0.009 0.009 0.009 0.008 0.008 0.008 0.007 0.007 0.007 0.172 0.148 0.172 0.077 0.075 0.077 0.056 0.056 0.056 0.046 0.046 0.046 0.042 0.042 0.042 0.052 0.052 0.052 0.031 0.031 0.031 0.024 0.024 0.024 0.020 0.020 0.020 0.016 0.016 0.016 0.523 0.283 0.311 0.253 0.175 0.098 0.128 0.180 0.061 0.072 0.170 0.048 0.044 0.172 0.042 0.210 0.080 0.072 0.083 0.055 0.049 0.050 0.049 0.040 0.042 0.047 0.039 0.037 0.045 0.037 0.423 0.939 0.959 0.180 0.597 0.326 0.133 0.472 0.221 0.139 0.425 0.192 0.162 0.413 0.178 0.176 0.243 0.218 0.080 0.123 0.102 0.083 0.115 0.095 0.078 0.103 0.084 0.081 0.103 0.083 RX/RM =10, ICCX/ICCM =0.25, g=200, Unbalanced 0.410 0.753 3.646 0.186 0.439 0.266 0.138 0.453 0.241 0.159 0.413 0.212 0.194 0.408 0.204 0.255 0.415 0.345 0.117 0.201 0.150 0.110 0.190 0.137 0.108 0.182 0.123 0.116 0.176 0.117 0.071 0.064 0.071 0.034 0.033 0.046 0.025 0.024 0.027 0.019 0.019 0.021 0.017 0.017 0.018 0.034 0.035 0.039 0.018 0.018 0.021 0.015 0.015 0.017 0.012 0.012 0.014 0.011 0.011 0.013 0.556 0.257 0.345 0.270 0.170 0.097 0.150 0.172 0.077 0.089 0.161 0.062 0.056 0.158 0.053 0.342 0.121 0.119 0.141 0.091 0.064 0.083 0.079 0.054 0.056 0.079 0.049 0.046 0.079 0.047 0.161 0.146 0.161 0.065 0.064 0.078 0.049 0.049 0.058 0.041 0.041 0.050 0.037 0.037 0.042 0.081 0.079 0.096 0.041 0.041 0.047 0.037 0.037 0.041 0.028 0.028 0.031 0.025 0.025 0.028 178 0.204 0.316 0.288 0.117 0.169 0.124 0.099 0.134 0.101 0.088 0.112 0.089 0.085 0.108 0.086 0.067 0.075 0.070 0.042 0.041 0.040 0.037 0.037 0.036 0.033 0.033 0.033 0.033 0.033 0.033 0.197 0.238 1.181 0.118 0.136 0.112 0.103 0.123 0.102 0.094 0.105 0.090 0.095 0.106 0.090 0.114 0.124 0.106 0.062 0.059 0.056 0.054 0.054 0.051 0.051 0.049 0.047 0.048 0.046 0.044 0.159 0.156 0.159 0.075 0.075 0.075 0.054 0.054 0.054 0.043 0.043 0.043 0.044 0.044 0.044 0.053 0.053 0.053 0.031 0.031 0.030 0.027 0.027 0.027 0.023 0.023 0.023 0.020 0.020 0.020 0.179 0.167 0.179 0.086 0.084 0.103 0.061 0.060 0.073 0.055 0.055 0.060 0.049 0.049 0.050 0.080 0.079 0.089 0.046 0.046 0.053 0.033 0.033 0.037 0.025 0.025 0.029 0.025 0.025 0.028 1.034 0.813 0.562 0.476 0.514 0.252 0.262 0.433 0.180 0.168 0.400 0.150 0.136 0.388 0.140 0.417 0.244 0.222 0.192 0.181 0.164 0.157 0.170 0.154 0.149 0.166 0.150 0.147 0.163 0.147 1.171 0.613 0.563 0.587 0.428 0.264 0.352 0.371 0.200 0.220 0.363 0.172 0.177 0.356 0.171 0.706 0.308 0.270 0.279 0.260 0.178 0.185 0.210 0.152 0.138 0.228 0.153 0.128 0.217 0.146 Table 38. Coverage rate using the manifest aggregation approach, latent aggregation approach, and the new approach with within-group fpc under different design conditions 1-1-1 mediation βmw βmb 2-1-1 mediation βmw βmb MLM βxw N r βxw βxb βxb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.950 0.960 0.960 0.960 0.960 0.959 0.930 0.930 0.930 0.900 0.910 0.910 0.930 0.930 0.930 0.960 0.960 0.960 0.930 0.930 0.930 0.960 0.960 0.960 0.960 0.960 0.960 0.970 0.970 0.960 0.920 0.920 0.918 0.960 0.960 0.848 0.980 0.980 0.940 0.970 0.970 0.900 0.980 0.980 0.920 0.950 0.950 0.677 0.970 0.970 0.810 0.960 0.960 0.780 0.960 0.960 0.810 0.960 0.960 0.810 0.910 0.940 0.980 0.920 0.940 1.000 0.940 0.950 1.000 0.890 0.930 0.990 0.920 0.920 1.000 0.930 0.960 0.990 0.940 0.950 0.950 0.900 0.900 0.970 0.960 0.960 0.990 0.930 0.930 0.980 0.930 0.970 0.990 0.980 0.950 0.980 0.950 0.950 0.990 0.930 0.900 0.970 0.940 0.920 0.960 0.900 0.930 0.970 0.940 0.920 0.960 0.950 0.920 0.950 0.940 0.950 0.960 0.950 0.940 0.930 0.920 0.940 0.947 0.940 0.948 0.939 0.930 0.926 0.940 0.930 0.946 0.930 0.950 0.957 0.950 0.930 0.930 0.930 0.960 0.960 0.970 0.970 0.970 0.980 0.970 0.970 0.970 0.950 0.950 0.950 0.940 0.929 0.948 0.930 0.928 0.887 0.950 0.937 0.890 0.940 0.939 0.910 0.950 0.947 0.900 0.950 0.959 0.722 0.960 0.959 0.840 0.930 0.930 0.720 0.950 0.950 0.740 0.940 0.940 0.710 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Balanced 0.940 0.958 0.935 0.960 0.959 0.960 0.960 0.958 0.960 0.990 0.990 0.990 0.970 0.969 0.970 0.930 0.938 0.930 0.970 0.970 0.970 0.920 0.920 0.910 0.950 0.950 0.950 0.970 0.970 0.970 0.940 0.947 0.935 0.960 0.959 0.960 0.940 0.947 0.940 0.960 0.959 0.960 0.950 0.948 0.950 0.970 0.969 0.970 0.940 0.940 0.940 0.940 0.940 0.940 0.980 0.980 0.980 0.930 0.930 0.940 0.900 1.000 0.989 0.920 0.990 1.000 0.930 1.000 1.000 0.950 0.989 1.000 0.940 0.979 0.990 0.880 0.990 0.990 0.960 0.980 0.980 0.980 0.980 0.990 0.970 0.950 0.970 0.960 0.960 0.990 0.940 0.989 1.000 0.930 0.979 1.000 0.940 0.989 1.000 0.910 0.980 1.000 0.930 0.979 1.000 0.920 0.948 0.970 0.940 0.940 0.990 0.950 0.950 0.970 0.960 0.950 0.970 0.960 0.950 0.970 RX/RM =0.1, ICCX/ICCM =0.05, g=50, Unbalanced 0.950 0.970 0.989 0.930 0.958 1.000 0.930 0.979 1.000 0.940 0.958 0.990 0.920 0.939 0.990 0.930 0.960 0.969 0.920 0.950 1.000 0.940 0.940 0.980 0.940 0.930 0.980 0.970 0.960 0.990 0.980 0.960 0.978 0.960 0.947 0.900 0.920 0.916 0.850 0.960 0.968 0.930 0.940 0.949 0.900 0.950 0.950 0.724 0.960 0.950 0.850 0.950 0.950 0.770 0.990 0.990 0.790 0.990 0.990 0.800 0.870 1.000 1.000 0.890 1.000 0.990 0.940 1.000 1.000 0.950 0.990 1.000 0.910 0.989 1.000 0.870 1.000 1.000 0.930 1.000 1.000 0.920 0.960 0.970 0.940 0.950 0.970 0.940 0.950 0.980 0.890 0.950 0.901 0.940 0.937 0.890 0.940 0.937 0.900 0.910 0.916 0.910 0.910 0.918 0.920 0.930 0.930 0.806 0.910 0.910 0.740 0.940 0.950 0.820 0.940 0.940 0.840 0.940 0.940 0.770 179 0.880 0.989 1.000 0.950 0.990 0.990 0.920 0.989 0.950 0.880 0.980 0.920 0.900 0.969 0.950 0.920 0.959 0.950 0.950 0.950 0.950 0.920 0.920 0.920 0.960 0.960 0.960 0.950 0.950 0.950 0.930 1.000 1.000 0.880 0.989 0.950 0.920 0.979 0.960 0.900 0.968 0.930 0.920 0.990 0.950 0.890 0.980 0.980 0.960 0.950 0.960 0.940 0.950 0.960 0.970 0.970 0.950 0.950 0.930 0.950 0.840 0.926 0.946 0.940 0.969 0.960 0.960 0.968 0.970 0.960 0.969 0.970 0.980 1.000 0.990 0.930 0.938 0.930 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.940 0.770 0.930 0.956 0.930 0.968 0.910 0.970 0.968 0.860 0.970 0.979 0.910 0.980 0.980 0.910 0.920 0.940 0.735 0.970 0.970 0.760 0.960 0.960 0.750 0.970 0.970 0.800 0.960 0.960 0.790 0.910 0.958 1.000 0.910 0.928 1.000 0.960 0.916 0.990 0.960 0.929 0.990 0.920 0.948 1.000 0.950 0.948 0.990 0.960 0.940 0.970 0.950 0.950 0.960 0.930 0.930 0.980 0.940 0.940 0.990 0.900 0.940 1.000 0.890 0.926 0.990 0.920 0.926 1.000 0.920 0.926 0.990 0.920 0.898 0.990 0.890 0.900 0.949 0.930 0.950 1.000 0.880 0.940 0.990 0.900 0.930 0.960 0.900 0.950 0.970 Table 38. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.970 0.980 0.970 0.940 0.940 0.940 0.930 0.930 0.930 0.940 0.940 0.930 0.930 0.930 0.930 0.940 0.940 0.940 0.950 0.950 0.950 0.960 0.960 0.960 0.970 0.970 0.960 0.970 0.970 0.980 1.000 1.000 1.000 0.950 0.940 0.890 0.910 0.900 0.890 0.940 0.950 0.890 0.940 0.940 0.930 0.970 0.990 0.720 0.970 0.970 0.760 0.980 0.980 0.820 1.000 1.000 0.830 0.950 0.950 0.780 0.880 0.980 0.990 0.970 0.980 1.000 0.930 0.970 0.990 0.970 0.980 1.000 0.960 0.940 0.990 0.950 0.930 0.970 0.970 0.950 0.980 1.000 0.990 1.000 0.960 0.960 0.970 0.990 0.990 0.990 0.870 0.980 1.000 0.890 0.950 1.000 0.930 0.930 0.990 0.920 0.930 0.980 0.930 0.940 0.990 0.930 0.970 0.990 0.950 0.950 0.960 0.960 0.960 0.970 0.960 0.950 0.940 0.970 0.960 0.950 0.980 0.979 0.980 0.930 0.926 0.930 0.990 1.000 0.990 0.930 0.924 0.920 0.950 0.948 0.950 0.880 0.880 0.890 0.970 0.970 0.950 0.940 0.940 0.910 0.910 0.910 0.900 0.850 0.850 0.900 0.970 0.969 0.968 0.960 0.958 0.890 0.980 0.990 0.910 0.990 0.990 0.950 0.980 0.990 0.920 0.910 0.900 0.770 0.930 0.940 0.790 0.930 0.920 0.760 0.890 0.890 0.750 0.920 0.920 0.770 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Balanced 0.910 0.915 0.908 0.900 0.899 0.900 0.900 0.899 0.900 0.950 0.950 0.950 0.960 0.960 0.950 0.920 0.920 0.920 0.920 0.920 0.910 0.980 0.980 0.980 0.940 0.940 0.940 0.950 0.950 0.950 0.920 0.915 0.918 0.960 0.960 0.950 0.950 0.949 0.950 0.950 0.950 0.950 0.960 0.960 0.960 0.980 0.980 0.980 0.940 0.940 0.940 0.950 0.950 0.950 0.990 0.990 0.990 0.960 0.960 0.970 0.440 1.000 0.980 0.810 0.989 1.000 0.890 0.989 0.980 0.930 0.967 1.000 0.970 0.969 1.000 0.790 0.970 0.970 0.950 0.990 1.000 0.960 0.960 0.980 0.940 0.910 0.950 0.950 0.930 0.960 0.960 1.000 1.000 0.960 0.980 1.000 0.920 0.939 0.970 0.920 0.960 1.000 0.940 0.939 0.990 0.880 0.900 0.940 0.930 0.940 0.960 0.920 0.920 0.950 0.910 0.930 0.930 0.880 0.890 0.940 RX/RM =0.1, ICCX/ICCM =0.05, g=200, Unbalanced 0.900 0.969 0.989 0.970 1.000 1.000 0.920 0.970 1.000 0.930 0.950 0.990 0.940 0.990 0.990 0.960 0.970 1.000 0.950 0.960 0.970 0.960 0.950 0.960 0.940 0.960 0.970 0.930 0.950 0.960 0.960 0.928 0.957 0.950 0.950 0.900 0.940 0.940 0.860 0.940 0.930 0.900 0.930 0.930 0.940 0.930 0.940 0.790 0.950 0.950 0.820 0.970 0.970 0.760 0.950 0.950 0.820 0.980 0.980 0.770 0.490 1.000 1.000 0.770 1.000 1.000 0.870 1.000 0.990 0.900 1.000 1.000 0.980 0.980 1.000 0.700 1.000 0.970 0.880 0.970 0.950 0.920 0.920 0.930 0.940 0.930 0.940 0.890 0.910 0.950 0.970 0.959 0.968 0.900 0.900 0.930 1.000 1.000 0.950 0.950 0.950 0.910 0.970 0.980 0.930 0.910 0.920 0.720 0.970 0.970 0.730 0.940 0.940 0.790 0.950 0.950 0.790 0.940 0.940 0.790 180 0.900 1.000 1.000 0.890 1.000 0.940 0.870 0.990 0.960 0.880 0.960 0.950 0.880 0.970 0.930 0.950 0.980 0.980 0.960 0.960 0.960 0.950 0.950 0.950 0.950 0.960 0.970 0.950 0.950 0.960 0.930 0.990 1.000 0.930 0.980 0.970 0.910 0.950 0.930 0.940 0.960 0.960 0.940 0.960 0.940 0.940 0.940 0.920 0.870 0.870 0.900 0.920 0.920 0.910 0.900 0.910 0.930 0.900 0.900 0.930 0.800 0.904 0.908 0.910 0.929 0.930 0.910 0.929 0.920 0.930 0.950 0.960 0.920 0.919 0.920 0.940 0.940 0.940 0.940 0.960 0.960 0.930 0.930 0.930 0.960 0.960 0.960 0.920 0.920 0.940 0.840 0.959 0.968 0.920 0.980 0.880 0.980 0.980 0.950 0.950 0.950 0.910 0.970 0.970 0.950 0.950 0.960 0.820 0.910 0.940 0.770 0.970 0.970 0.750 0.950 0.950 0.750 0.950 0.960 0.640 0.880 0.968 0.990 0.960 0.960 0.980 0.970 0.980 1.000 0.940 0.940 0.980 0.920 0.949 0.990 0.920 0.940 0.950 0.930 0.930 0.950 0.970 0.970 0.970 0.950 0.940 0.970 0.970 0.950 0.980 0.830 0.948 0.979 0.900 0.970 0.990 0.920 0.930 0.980 0.940 0.900 0.990 0.980 0.910 0.990 0.910 0.950 0.960 0.930 0.940 0.980 0.950 0.970 0.980 0.940 0.950 0.950 0.930 0.940 0.950 Table 38. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.910 0.920 0.909 0.940 0.940 0.940 0.930 0.930 0.930 0.920 0.920 0.930 0.960 0.960 0.960 0.960 0.960 0.960 0.950 0.950 0.950 0.940 0.940 0.940 0.970 0.970 0.970 0.970 0.970 0.980 0.950 0.960 0.950 0.960 0.960 0.900 0.930 0.930 0.930 0.950 0.950 0.920 0.940 0.940 0.930 0.940 0.930 0.770 0.950 0.950 0.760 0.960 0.960 0.840 0.960 0.960 0.810 1.000 1.000 0.810 0.880 0.970 0.990 0.920 0.930 0.970 0.960 0.970 0.980 0.950 0.970 0.980 0.950 0.950 0.990 0.930 0.950 0.960 0.920 0.910 0.940 0.920 0.920 0.940 0.910 0.910 0.920 0.920 0.920 0.930 0.940 0.950 1.000 0.940 0.940 0.990 0.930 0.950 0.960 0.950 0.960 0.950 0.970 0.960 0.950 0.950 0.940 0.960 0.970 0.940 0.940 0.950 0.950 0.930 0.940 0.930 0.940 0.940 0.940 0.950 0.980 0.980 0.980 0.900 0.900 0.900 0.960 0.960 0.970 0.940 0.940 0.940 0.970 0.970 0.960 0.970 0.970 0.970 0.940 0.940 0.940 0.940 0.940 0.940 0.930 0.930 0.930 0.940 0.940 0.940 0.950 0.940 0.959 0.960 0.950 0.840 0.970 0.980 0.970 0.970 0.970 0.920 0.970 0.970 0.900 0.960 0.960 0.790 0.920 0.920 0.720 0.980 0.980 0.780 0.970 0.970 0.760 0.950 0.950 0.790 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Balanced 0.870 0.889 0.876 0.960 0.960 0.960 0.960 0.960 0.960 0.940 0.940 0.940 0.930 0.940 0.940 0.960 0.960 0.960 0.910 0.910 0.910 0.970 0.970 0.970 0.970 0.970 0.970 0.970 0.970 0.970 0.890 0.889 0.897 0.960 0.960 0.960 0.950 0.950 0.950 0.950 0.950 0.950 0.980 0.980 0.980 0.940 0.940 0.940 0.920 0.920 0.930 0.960 0.960 0.960 0.950 0.950 0.950 0.960 0.960 0.960 0.940 0.980 0.990 0.940 0.960 0.960 0.970 0.980 0.980 0.990 0.990 0.990 0.970 0.970 1.000 0.940 0.950 0.950 0.900 0.900 0.900 0.900 0.920 0.930 0.890 0.890 0.890 0.910 0.910 0.920 0.950 1.000 1.000 0.960 0.960 0.990 0.960 0.980 0.980 0.930 0.950 0.950 0.930 0.930 0.980 0.960 0.940 0.950 0.940 0.920 0.940 0.960 0.960 0.970 0.970 0.970 0.990 0.970 0.970 0.980 RX/RM =0.1, ICCX/ICCM =0.25, g=50, Unbalanced 0.920 0.969 1.000 0.930 0.940 0.980 0.950 0.920 0.960 0.930 0.910 0.930 0.950 0.960 0.960 0.910 0.900 0.930 0.950 0.940 0.900 0.960 0.960 0.910 0.940 0.940 0.890 0.940 0.920 0.850 0.960 0.980 0.958 0.950 0.950 0.890 0.920 0.920 0.920 0.960 0.960 0.920 0.960 0.970 0.910 0.940 0.930 0.790 0.920 0.920 0.750 0.950 0.950 0.750 0.960 0.960 0.800 0.920 0.920 0.750 0.870 0.970 0.979 0.900 0.920 0.940 0.910 0.910 0.930 0.950 0.940 0.960 0.930 0.920 0.950 0.920 0.940 0.950 0.940 0.940 0.940 0.970 0.980 0.960 0.980 0.980 0.970 0.970 0.970 0.960 0.970 0.969 0.958 0.900 0.910 0.820 0.950 0.950 0.940 0.960 0.960 0.930 0.960 0.960 0.900 0.930 0.920 0.690 0.930 0.940 0.790 0.930 0.930 0.720 0.950 0.950 0.740 0.950 0.950 0.730 181 0.940 1.000 1.000 0.950 0.960 0.960 0.960 0.970 0.980 0.960 0.970 0.990 0.960 0.960 0.990 0.930 0.930 0.930 0.920 0.920 0.940 0.930 0.910 0.930 0.910 0.910 0.930 0.910 0.920 0.930 0.940 1.000 1.000 0.950 0.970 0.990 0.960 0.930 0.980 0.940 0.950 0.970 0.930 0.920 0.960 0.910 0.930 0.940 0.930 0.930 0.930 0.920 0.920 0.900 0.910 0.910 0.920 0.920 0.920 0.930 0.840 0.960 0.948 0.900 0.950 0.950 0.940 0.950 0.950 0.920 0.950 0.950 0.940 0.950 0.950 0.930 0.940 0.940 0.950 0.960 0.960 0.880 0.880 0.880 0.940 0.940 0.940 0.920 0.930 0.930 0.800 0.918 0.926 0.950 0.960 0.890 0.920 0.920 0.870 0.900 0.920 0.880 0.910 0.940 0.940 0.910 0.930 0.790 0.950 0.950 0.790 0.990 0.990 0.770 0.940 0.950 0.750 0.990 0.990 0.770 0.920 0.960 0.979 0.930 0.950 0.970 0.890 0.890 0.910 0.930 0.920 0.950 0.930 0.920 0.960 0.940 0.960 0.970 0.950 0.950 0.960 0.950 0.960 0.970 0.950 0.950 0.970 0.970 0.960 0.970 0.940 0.959 0.979 0.950 0.960 0.980 0.940 0.920 0.940 0.930 0.910 0.930 0.940 0.920 0.940 0.890 0.920 0.930 0.930 0.920 0.910 0.930 0.940 0.960 0.930 0.920 0.920 0.910 0.890 0.930 Table 38. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.940 0.960 0.940 0.980 0.980 0.980 0.940 0.940 0.940 0.960 0.960 0.960 0.970 0.970 0.970 0.940 0.940 0.940 0.940 0.940 0.940 0.960 0.960 0.960 0.950 0.950 0.940 0.960 0.960 0.960 0.950 0.960 0.950 0.950 0.950 0.900 0.940 0.940 0.880 0.930 0.930 0.850 0.930 0.930 0.890 0.940 0.940 0.780 0.990 0.990 0.760 0.950 0.950 0.740 0.970 0.970 0.750 0.910 0.910 0.700 0.930 0.910 0.970 0.900 0.950 0.950 0.980 0.980 0.990 0.980 0.960 0.980 0.970 0.970 1.000 0.950 0.970 0.970 0.900 0.920 0.920 0.910 0.910 0.920 0.910 0.920 0.930 0.920 0.940 0.940 0.870 0.910 0.970 0.910 0.920 0.920 0.920 0.930 0.920 0.930 0.930 0.960 0.930 0.940 0.970 0.910 0.970 0.940 0.960 0.930 0.950 0.940 0.920 0.920 0.930 0.910 0.940 0.910 0.910 0.920 0.950 0.950 0.950 0.970 0.970 0.970 0.950 0.950 0.950 0.940 0.940 0.940 0.950 0.950 0.950 0.940 0.940 0.940 0.910 0.910 0.910 0.980 0.980 0.950 0.940 0.940 0.920 0.960 0.960 0.950 0.950 0.950 0.950 0.990 0.990 0.900 0.950 0.950 0.880 0.970 0.970 0.910 0.990 0.990 0.930 0.980 0.980 0.830 0.950 0.950 0.740 0.960 0.960 0.740 0.920 0.920 0.760 0.920 0.920 0.790 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Balanced 0.960 0.960 0.960 0.960 0.960 0.960 0.940 0.940 0.930 0.970 0.970 0.970 0.980 0.980 0.980 0.970 0.970 0.970 0.940 0.940 0.940 0.950 0.950 0.950 0.990 0.990 0.990 0.940 0.940 0.950 0.960 0.960 0.960 0.930 0.930 0.930 0.960 0.960 0.960 0.960 0.960 0.960 0.950 0.950 0.940 0.960 0.960 0.970 0.950 0.950 0.950 0.940 0.940 0.940 0.940 0.940 0.930 0.950 0.950 0.960 0.900 1.000 1.000 0.900 0.920 0.920 0.910 0.930 0.960 0.920 0.920 0.950 0.910 0.890 0.940 0.910 0.940 0.950 0.950 0.950 0.950 0.950 0.970 0.970 0.940 0.940 0.940 0.950 0.950 0.950 0.930 0.960 0.980 0.920 0.910 0.940 0.930 0.930 0.990 0.940 0.950 0.990 0.960 0.970 1.000 0.920 0.930 0.950 0.970 0.970 0.980 0.940 0.940 0.940 0.940 0.930 0.960 0.940 0.940 0.940 RX/RM =0.1, ICCX/ICCM =0.25, g=200, Unbalanced 0.900 0.930 0.960 0.920 0.920 0.940 0.920 0.900 0.910 0.930 0.940 0.930 0.940 0.950 0.950 0.930 0.930 0.950 0.930 0.930 0.910 0.920 0.920 0.940 0.930 0.940 0.930 0.900 0.910 0.920 0.940 0.950 0.940 0.950 0.950 0.850 0.970 0.960 0.890 0.950 0.950 0.950 0.910 0.930 0.860 0.930 0.930 0.760 0.980 0.980 0.810 0.910 0.910 0.720 0.990 0.990 0.800 0.930 0.930 0.710 0.920 0.970 0.990 0.960 0.950 0.960 0.910 0.930 0.950 0.950 0.940 0.950 0.940 0.930 0.930 0.930 0.950 0.940 0.920 0.930 0.930 0.980 0.970 0.960 0.940 0.940 0.940 0.960 0.960 0.960 0.930 0.940 0.930 0.990 0.980 0.900 0.970 0.980 0.940 0.920 0.920 0.840 0.920 0.920 0.910 0.970 0.980 0.750 0.960 0.960 0.700 0.960 0.960 0.770 0.950 0.950 0.780 0.990 0.990 0.810 182 0.890 1.000 1.000 0.970 0.960 0.970 0.930 0.920 0.950 0.960 0.970 0.970 0.960 0.950 0.980 0.950 0.980 0.980 0.950 0.950 0.950 0.960 0.960 0.960 0.960 0.950 0.960 0.950 0.950 0.960 0.930 1.000 1.000 0.950 0.950 0.970 0.940 0.960 0.950 0.930 0.940 0.940 0.910 0.930 0.930 0.910 0.950 0.940 0.950 0.950 0.960 0.960 0.970 0.980 0.940 0.930 0.940 0.940 0.940 0.950 0.860 0.960 0.960 0.940 0.960 0.960 0.920 0.940 0.950 0.940 0.950 0.940 0.940 0.940 0.940 0.900 0.950 0.950 0.900 0.910 0.910 0.940 0.940 0.940 0.920 0.930 0.920 0.910 0.910 0.910 0.810 0.950 0.960 0.940 0.950 0.900 0.900 0.940 0.870 0.940 0.940 0.930 0.950 0.960 0.930 0.970 0.980 0.820 0.940 0.960 0.750 0.960 0.960 0.750 0.940 0.940 0.760 0.930 0.920 0.790 0.880 0.940 0.970 0.970 0.980 0.990 0.930 0.930 0.980 0.950 0.960 0.980 0.960 0.960 0.990 0.970 0.990 0.990 0.940 0.950 0.950 0.950 0.930 0.950 0.960 0.950 0.980 0.970 0.970 0.980 0.870 0.920 0.950 0.920 0.920 0.960 0.910 0.960 0.940 0.950 0.960 0.960 0.940 0.960 0.960 0.950 0.950 0.940 0.930 0.940 0.920 0.960 0.960 0.940 0.960 0.970 0.960 0.960 0.960 0.930 Table 38. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.950 0.970 0.948 0.970 0.970 0.970 0.950 0.950 0.950 0.950 0.950 0.950 0.920 0.930 0.920 0.970 0.970 0.970 0.970 0.970 0.970 0.960 0.960 0.960 0.970 0.970 0.960 0.970 0.970 0.970 0.940 0.970 0.939 0.960 0.960 0.900 0.910 0.920 0.880 0.980 0.970 0.920 0.960 0.950 0.910 0.930 0.940 0.750 0.940 0.930 0.790 0.910 0.910 0.700 0.900 0.900 0.760 0.900 0.920 0.730 0.030 0.970 0.959 0.230 0.990 0.980 0.630 0.960 0.990 0.860 0.830 0.990 0.950 0.630 1.000 0.390 0.950 0.960 0.760 0.820 0.890 0.880 0.880 0.950 0.930 0.920 0.970 0.930 0.860 0.950 0.010 0.920 0.929 0.200 0.980 0.960 0.530 0.920 0.970 0.810 0.790 0.990 0.880 0.740 0.970 0.110 0.980 0.960 0.620 0.850 0.990 0.950 0.910 1.000 0.950 0.800 0.960 0.970 0.810 0.970 0.960 0.960 0.966 0.970 0.940 0.979 0.930 0.940 0.930 0.940 0.919 0.940 0.910 0.899 0.910 0.980 0.970 0.989 0.960 0.960 0.960 0.940 0.950 0.950 0.960 0.960 0.960 0.960 0.960 0.950 0.910 0.929 0.931 0.940 0.929 0.876 0.930 0.910 0.850 0.930 0.928 0.850 0.930 0.911 0.840 0.940 0.940 0.783 0.900 0.910 0.758 0.940 0.950 0.750 0.960 0.980 0.740 0.930 0.939 0.780 0.940 0.949 0.934 0.960 0.980 0.958 0.960 0.957 0.960 0.980 0.976 0.980 0.980 0.989 0.970 0.950 0.949 0.949 0.960 0.960 0.960 0.950 0.950 0.950 0.910 0.910 0.920 0.940 0.940 0.950 RX/RM =10, ICCX/ICCM =0.05, g=50, Balanced 0.910 0.929 0.912 0.930 0.929 0.926 0.960 0.957 0.950 0.940 0.928 0.940 0.930 0.933 0.930 0.940 0.939 0.939 0.960 0.960 0.970 0.960 0.960 0.960 0.970 0.970 0.970 0.970 0.970 0.980 RX/RM =10, ICCX/ICCM =0.05, g=50, Unbalanced 0.930 0.938 0.935 0.970 0.958 0.876 0.970 0.968 0.920 0.950 0.949 0.880 0.930 0.925 0.910 0.940 0.929 0.680 0.930 0.929 0.740 0.970 0.970 0.840 0.930 0.930 0.750 0.960 0.960 0.840 0.920 0.918 0.913 0.950 0.969 0.928 0.970 0.968 0.900 0.930 0.929 0.940 0.910 0.892 0.910 0.930 0.929 0.773 0.970 0.969 0.810 0.950 0.949 0.810 0.980 0.980 0.800 0.960 0.960 0.730 0.000 0.920 0.705 0.000 1.000 0.917 0.040 1.000 0.980 0.310 1.000 1.000 0.860 1.000 1.000 0.000 0.980 0.947 0.150 1.000 0.960 0.660 0.920 0.990 0.870 0.740 0.990 0.920 0.500 1.000 0.000 0.878 0.713 0.000 1.000 0.887 0.000 1.000 0.970 0.190 1.000 1.000 0.760 1.000 1.000 0.000 0.980 0.837 0.000 0.990 0.939 0.130 1.000 0.990 0.560 0.990 0.990 0.910 0.898 0.990 0.440 0.990 1.000 0.760 0.980 0.989 0.860 0.957 1.000 0.940 0.988 1.000 0.970 0.978 1.000 0.670 0.980 0.990 0.900 0.940 0.980 0.960 0.980 0.990 0.960 0.950 0.980 0.950 0.980 1.000 0.420 0.969 0.989 0.690 0.979 1.000 0.890 0.968 0.990 0.880 0.959 0.980 0.920 0.935 0.990 0.620 0.980 1.000 0.890 0.908 0.990 0.900 0.879 0.980 0.930 0.890 0.940 0.910 0.870 0.950 0.520 1.000 0.989 0.820 0.990 1.000 0.880 1.000 0.990 0.920 1.000 0.990 0.920 1.000 0.980 0.910 0.980 0.980 0.930 0.940 0.940 0.920 0.920 0.930 0.930 0.920 0.930 0.950 0.950 0.950 0.400 1.000 0.978 0.780 0.979 0.969 0.850 1.000 0.970 0.830 0.969 0.910 0.890 0.989 0.950 0.790 1.000 0.990 0.850 0.949 0.950 0.940 0.970 0.960 0.930 0.940 0.960 0.970 0.970 0.960 0.820 0.949 0.945 0.930 0.970 0.968 0.960 0.967 0.970 0.930 0.940 0.940 0.960 0.956 0.960 0.940 0.939 0.939 0.880 0.890 0.890 0.900 0.910 0.900 0.910 0.920 0.920 0.920 0.920 0.920 0.900 0.959 0.957 0.940 0.990 0.856 0.930 0.957 0.920 0.950 0.980 0.960 0.960 0.957 0.920 0.910 0.939 0.722 0.960 0.990 0.730 0.940 0.960 0.770 0.950 0.950 0.780 0.940 0.940 0.820 0.010 0.929 0.923 0.280 0.879 0.968 0.690 0.728 1.000 0.870 0.687 1.000 0.950 0.522 1.000 0.320 0.909 0.949 0.820 0.900 0.970 0.920 0.920 0.970 0.940 0.890 0.960 0.960 0.910 0.990 0.000 0.928 0.946 0.180 0.927 0.979 0.480 0.883 0.990 0.790 0.796 0.990 0.940 0.613 1.000 0.110 0.908 0.938 0.550 0.867 0.930 0.770 0.879 0.950 0.880 0.830 0.930 0.930 0.840 0.960 183 Table 38. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.940 0.940 0.940 0.980 0.980 0.980 0.950 0.950 0.950 0.970 0.970 0.970 0.940 0.930 0.940 0.980 0.980 0.980 0.930 0.930 0.930 0.930 0.930 0.930 0.960 0.960 0.950 0.950 0.950 0.960 0.980 0.990 0.979 0.920 0.910 0.900 0.970 0.960 0.920 0.950 0.950 0.930 0.950 0.950 0.920 0.940 0.940 0.730 0.970 0.980 0.750 0.970 0.970 0.790 0.980 0.980 0.760 0.960 0.960 0.760 0.000 0.930 0.820 0.000 0.710 0.960 0.110 0.340 0.980 0.620 0.120 0.990 0.920 0.050 0.990 0.000 0.860 1.000 0.520 0.900 0.960 0.830 0.790 0.960 0.920 0.810 0.960 0.940 0.770 0.950 0.000 0.960 0.907 0.000 0.800 0.960 0.090 0.450 0.930 0.520 0.290 0.980 0.890 0.120 0.980 0.000 0.890 0.960 0.060 0.780 0.980 0.470 0.700 0.970 0.860 0.620 0.960 0.890 0.580 0.970 0.960 0.960 0.968 0.950 0.910 0.949 0.950 0.910 0.950 0.940 0.890 0.940 0.960 0.900 0.950 0.950 0.940 0.950 0.980 0.980 0.980 0.990 0.990 0.970 0.950 0.950 0.950 0.900 0.900 0.970 0.970 0.990 0.969 0.950 0.960 0.930 0.940 0.940 0.940 0.960 0.940 0.920 0.960 0.929 0.930 0.970 0.950 0.758 0.990 0.980 0.820 0.920 0.920 0.760 0.890 0.880 0.740 0.940 0.960 0.800 RX/RM =10, ICCX/ICCM =0.05, g=200, Balanced 0.940 0.958 0.938 0.930 0.939 0.930 0.990 0.989 0.980 0.970 0.967 0.970 1.000 1.000 1.000 0.930 0.930 0.930 0.960 0.960 0.950 0.900 0.900 0.910 0.920 0.920 0.910 0.940 0.940 0.960 0.950 0.958 0.948 0.940 0.939 0.940 0.920 0.904 0.920 0.960 0.956 0.960 0.950 0.943 0.940 0.950 0.950 0.950 0.940 0.940 0.940 0.920 0.920 0.920 0.960 0.960 0.950 0.930 0.930 0.940 0.000 0.940 0.606 0.000 1.000 0.939 0.000 1.000 1.000 0.000 0.660 0.990 0.520 0.200 1.000 0.000 0.980 0.940 0.000 0.720 0.990 0.090 0.260 1.000 0.630 0.020 0.980 0.940 0.000 1.000 0.030 1.000 1.000 0.370 0.929 1.000 0.760 0.904 1.000 0.930 0.857 1.000 0.940 0.770 1.000 0.290 0.960 0.970 0.860 0.880 0.980 0.910 0.940 0.980 0.960 0.940 0.970 0.960 0.920 0.990 RX/RM =10, ICCX/ICCM =0.05, g=200, Unbalanced 0.030 1.000 0.979 0.230 0.948 0.980 0.660 0.894 0.990 0.900 0.784 1.000 0.980 0.760 1.000 0.080 0.950 0.969 0.590 0.890 0.970 0.840 0.850 0.940 0.920 0.860 0.970 0.960 0.800 0.970 0.950 0.927 0.947 0.950 0.948 0.870 0.940 0.926 0.820 0.940 0.938 0.900 0.970 0.970 0.920 0.950 0.960 0.735 0.960 0.960 0.760 0.970 0.970 0.850 0.970 0.970 0.750 0.920 0.920 0.730 0.000 0.917 0.663 0.000 1.000 0.850 0.000 0.990 0.980 0.000 0.610 0.980 0.410 0.286 1.000 0.000 0.990 0.889 0.000 0.990 0.980 0.000 0.420 1.000 0.170 0.090 1.000 0.870 0.000 1.000 0.920 0.969 0.926 0.950 0.948 0.890 0.960 0.957 0.920 0.960 0.969 0.940 0.960 0.960 0.920 0.930 0.880 0.735 0.970 0.960 0.680 0.930 0.930 0.760 0.950 0.950 0.810 0.930 0.930 0.750 184 0.010 1.000 1.000 0.450 1.000 0.960 0.700 1.000 0.920 0.810 1.000 0.960 0.800 0.966 0.920 0.760 0.940 0.940 0.930 0.920 0.940 0.970 0.960 0.980 0.950 0.960 0.970 0.960 0.960 0.970 0.030 1.000 0.989 0.380 0.990 0.950 0.680 0.989 0.940 0.710 1.000 0.910 0.710 0.950 0.890 0.360 0.990 0.980 0.820 0.950 0.950 0.910 0.960 0.930 0.950 0.970 0.930 0.950 0.960 0.960 0.840 0.958 0.959 0.930 0.959 0.960 0.950 0.957 0.960 0.970 0.967 0.970 0.950 0.966 0.950 0.910 0.930 0.930 0.980 0.980 0.980 0.960 0.960 0.960 0.940 0.950 0.950 0.950 0.950 0.950 0.850 0.969 0.968 0.940 0.969 0.930 0.960 0.968 0.840 0.960 0.959 0.930 0.930 0.930 0.900 0.920 0.980 0.765 0.950 0.950 0.720 0.940 0.940 0.750 0.920 0.910 0.720 0.920 0.920 0.740 0.000 0.979 0.959 0.000 0.755 0.960 0.150 0.255 1.000 0.530 0.099 1.000 0.920 0.000 1.000 0.000 0.880 0.970 0.520 0.840 0.960 0.800 0.820 0.920 0.870 0.770 0.940 0.900 0.780 0.950 0.000 0.948 0.937 0.000 0.753 0.950 0.020 0.404 1.000 0.490 0.134 0.990 0.880 0.050 0.990 0.000 0.880 0.929 0.030 0.800 0.970 0.470 0.720 0.980 0.780 0.610 0.980 0.930 0.530 0.970 Table 38. (cont’d) M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb 0.980 0.970 0.980 0.920 0.920 0.920 0.950 0.950 0.950 0.960 0.960 0.960 0.940 0.940 0.940 0.980 0.980 0.980 0.940 0.940 0.940 0.940 0.940 0.940 0.980 0.980 0.980 0.950 0.950 0.940 0.890 0.930 0.899 0.940 0.940 0.880 0.960 0.950 0.900 0.940 0.940 0.930 0.960 0.960 0.880 0.950 0.950 0.770 0.960 0.960 0.810 0.930 0.930 0.730 0.960 0.960 0.830 0.970 0.960 0.830 0.030 0.950 0.930 0.570 0.940 0.980 0.810 0.820 1.000 0.900 0.780 1.000 0.960 0.770 0.990 0.630 0.980 0.970 0.860 0.910 0.940 0.940 0.940 0.980 0.890 0.920 0.920 0.920 0.900 0.940 0.150 0.970 0.949 0.610 0.910 0.980 0.780 0.850 0.990 0.850 0.840 0.970 0.900 0.740 0.990 0.290 0.940 0.920 0.830 0.950 0.980 0.910 0.910 0.950 0.930 0.910 0.970 0.940 0.920 0.990 0.910 0.920 0.919 0.960 0.950 0.960 0.950 0.950 0.950 0.950 0.960 0.940 0.930 0.930 0.930 0.990 0.980 0.990 0.960 0.960 0.960 0.960 0.960 0.960 0.940 0.940 0.940 0.930 0.930 0.920 0.950 0.960 0.949 0.940 0.919 0.890 0.960 0.979 0.940 0.950 0.950 0.920 0.940 0.950 0.940 0.920 0.920 0.770 0.980 0.970 0.800 0.970 0.970 0.840 0.960 0.970 0.770 0.980 0.980 0.870 0.910 0.931 0.914 0.950 0.970 0.950 0.960 0.949 0.960 0.970 0.970 0.970 0.960 0.960 0.960 0.900 0.910 0.900 0.970 0.970 0.970 0.950 0.950 0.950 0.960 0.960 0.960 0.950 0.950 0.950 RX/RM =10, ICCX/ICCM =0.25, g=50, Balanced 0.950 0.954 0.957 0.960 0.960 0.960 0.970 0.960 0.960 0.940 0.939 0.940 0.970 0.980 0.970 0.960 0.960 0.950 0.900 0.890 0.900 0.950 0.950 0.960 0.930 0.930 0.930 0.940 0.940 0.930 RX/RM =10, ICCX/ICCM =0.25, g=50, Unbalanced 0.920 0.868 0.909 0.960 0.970 0.890 0.950 0.947 0.880 0.940 0.950 0.950 0.930 0.939 0.930 0.970 0.970 0.770 0.910 0.910 0.650 0.960 0.960 0.670 0.920 0.910 0.690 0.970 0.970 0.710 0.950 0.912 0.932 0.970 0.949 0.890 0.970 0.979 0.900 0.930 0.940 0.910 0.940 0.969 0.910 0.930 0.909 0.720 0.940 0.930 0.700 0.950 0.950 0.670 0.970 0.970 0.730 0.980 0.970 0.780 0.100 0.980 0.970 0.550 1.000 0.990 0.750 0.980 0.990 0.910 0.900 1.000 0.940 0.750 1.000 0.640 0.970 0.970 0.880 0.920 0.950 0.930 0.850 0.980 0.920 0.930 0.970 0.940 0.870 0.970 0.050 0.950 0.929 0.490 0.990 0.980 0.780 0.948 1.000 0.910 0.890 0.990 0.950 0.760 0.990 0.280 1.000 0.960 0.750 0.970 0.960 0.880 0.870 0.930 0.950 0.900 0.990 0.960 0.830 0.980 0.770 1.000 1.000 0.880 0.970 0.980 0.940 0.990 0.990 0.950 0.909 0.990 0.910 0.880 0.990 0.870 0.960 0.960 0.980 0.960 0.990 0.930 0.910 0.980 0.920 0.920 0.990 0.930 0.930 0.990 0.680 1.000 1.000 0.900 0.990 1.000 0.920 0.926 0.990 0.900 0.890 1.000 0.840 0.837 0.970 0.900 1.000 1.000 0.840 0.900 0.930 0.920 0.890 0.950 0.890 0.890 0.970 0.900 0.820 0.970 0.670 1.000 1.000 0.880 0.980 0.980 0.880 0.970 0.980 0.880 0.970 0.990 0.840 0.930 0.990 0.930 0.980 0.980 0.950 0.970 0.990 0.920 0.930 0.960 0.900 0.910 0.950 0.910 0.890 0.980 0.760 1.000 1.000 0.810 0.990 0.970 0.810 0.958 0.970 0.790 0.980 0.990 0.720 0.888 0.950 0.800 0.980 0.970 0.910 0.950 0.990 0.900 0.930 0.980 0.910 0.920 0.950 0.870 0.860 0.960 0.810 0.897 0.871 0.960 0.970 0.970 0.970 0.970 0.970 0.950 0.949 0.960 0.920 0.920 0.920 0.950 0.950 0.950 0.950 0.950 0.950 0.940 0.940 0.940 0.900 0.900 0.900 0.960 0.960 0.970 0.820 0.934 0.943 0.960 0.970 0.940 0.930 0.947 0.920 0.980 0.990 0.950 0.960 0.969 0.920 0.880 0.919 0.730 0.920 0.930 0.730 0.930 0.930 0.740 0.930 0.930 0.670 0.890 0.890 0.670 0.160 0.966 0.925 0.740 0.879 0.970 0.920 0.828 0.980 0.960 0.788 0.980 0.950 0.760 0.990 0.780 0.950 0.970 0.890 0.900 0.910 0.920 0.890 0.930 0.900 0.860 0.900 0.880 0.850 0.920 0.090 0.956 0.932 0.600 0.939 0.950 0.840 0.853 0.950 0.920 0.830 0.940 0.940 0.837 0.960 0.460 0.929 0.960 0.870 0.930 0.930 0.920 0.940 0.950 0.950 0.940 0.970 0.950 0.900 0.930 185 Table 38. (cont’d) MLM βxw βxb 2-1-1 mediation βmw βmb βxw βxb 1-1-1 mediation βmw βmb αw αb M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N M L N N r 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 20 100 20 100 0.950 0.950 0.950 0.950 0.950 0.950 0.930 0.930 0.940 0.940 0.930 0.930 0.940 0.940 0.940 0.970 0.970 0.970 0.950 0.950 0.950 0.960 0.960 0.950 0.980 0.980 0.970 0.970 0.970 0.980 0.950 0.930 0.950 0.950 0.950 0.880 0.960 0.970 0.900 0.950 0.930 0.950 0.920 0.950 0.910 0.930 0.950 0.750 0.950 0.950 0.850 0.900 0.900 0.780 0.980 0.980 0.790 0.940 0.950 0.720 0.000 0.930 0.870 0.010 0.730 0.990 0.570 0.550 0.990 0.820 0.430 0.990 0.940 0.300 1.000 0.100 0.880 0.930 0.730 0.900 0.960 0.930 0.900 0.970 0.970 0.910 0.980 0.960 0.900 0.970 0.000 0.950 0.920 0.020 0.840 0.960 0.410 0.580 0.980 0.740 0.560 0.990 0.920 0.380 0.990 0.010 0.890 0.970 0.350 0.880 0.980 0.800 0.820 0.960 0.920 0.810 0.970 0.970 0.820 0.960 0.910 0.899 0.920 0.950 0.930 0.960 0.960 0.960 0.960 0.940 0.940 0.940 0.950 0.940 0.940 0.940 0.940 0.940 0.950 0.950 0.950 0.950 0.950 0.960 0.940 0.940 0.930 0.970 0.970 0.960 0.880 0.939 0.860 0.920 0.920 0.900 0.920 0.930 0.920 0.970 0.960 0.950 0.930 0.930 0.900 0.940 0.950 0.770 0.950 0.950 0.830 0.920 0.910 0.760 0.980 0.970 0.770 0.930 0.930 0.780 RX/RM =10, ICCX/ICCM =0.25, g=200, Balanced 0.940 0.958 0.940 0.940 0.940 0.940 0.920 0.920 0.920 0.940 0.940 0.940 0.920 0.919 0.920 0.940 0.930 0.940 0.940 0.940 0.940 0.960 0.960 0.960 0.930 0.930 0.940 0.950 0.950 0.940 0.920 0.968 0.920 0.940 0.930 0.940 0.950 0.950 0.950 0.950 0.950 0.950 0.950 0.949 0.950 0.970 0.970 0.970 0.960 0.960 0.960 0.940 0.940 0.940 0.980 0.980 0.980 0.960 0.960 0.960 0.000 0.980 0.900 0.040 0.890 0.960 0.400 0.560 1.000 0.820 0.380 1.000 0.960 0.140 0.990 0.030 0.960 0.970 0.610 0.930 0.960 0.840 0.890 0.970 0.890 0.900 0.960 0.910 0.890 0.980 0.300 1.000 1.000 0.920 0.910 0.960 0.940 0.740 0.960 0.870 0.600 0.980 0.820 0.475 0.980 0.820 0.930 0.960 0.990 0.930 1.000 0.920 0.930 0.970 0.950 0.910 0.990 0.920 0.870 0.990 RX/RM =10, ICCX/ICCM =0.25, g=200, Unbalanced 0.270 0.990 0.980 0.840 0.950 0.980 0.910 0.690 0.940 0.870 0.610 0.940 0.740 0.480 0.950 0.690 0.980 0.980 0.960 0.910 0.990 0.930 0.840 0.950 0.910 0.780 0.990 0.900 0.730 0.970 0.940 0.929 0.940 0.900 0.910 0.820 0.930 0.930 0.910 0.950 0.950 0.930 0.950 0.950 0.910 0.960 0.950 0.770 0.940 0.930 0.780 0.960 0.960 0.760 0.930 0.940 0.780 0.920 0.920 0.790 0.000 0.960 0.920 0.010 0.920 0.970 0.390 0.590 0.980 0.720 0.500 0.980 0.900 0.340 0.980 0.000 0.940 0.960 0.320 0.860 0.990 0.690 0.820 0.960 0.780 0.700 0.950 0.920 0.660 0.960 0.960 0.959 0.960 0.970 0.980 0.930 0.940 0.950 0.950 0.950 0.950 0.930 0.980 0.980 0.940 0.940 0.930 0.760 0.950 0.950 0.830 0.910 0.920 0.740 0.960 0.970 0.780 0.940 0.940 0.770 186 0.150 1.000 0.990 0.440 0.940 0.920 0.530 0.810 0.860 0.520 0.760 0.860 0.470 0.636 0.860 0.760 0.960 0.970 0.860 0.940 0.960 0.870 0.920 0.940 0.920 0.940 0.980 0.860 0.880 0.950 0.110 0.980 0.980 0.450 0.960 0.920 0.450 0.820 0.850 0.470 0.750 0.810 0.380 0.700 0.830 0.470 0.960 0.970 0.740 0.970 0.980 0.810 0.860 0.920 0.780 0.910 0.940 0.770 0.860 0.930 0.850 0.968 0.970 0.910 0.960 0.960 0.930 0.940 0.940 0.960 0.960 0.960 0.920 0.929 0.930 0.930 0.950 0.940 0.980 0.980 0.980 0.910 0.910 0.910 0.880 0.880 0.880 0.920 0.920 0.920 0.780 0.949 0.940 0.850 0.890 0.870 0.910 0.950 0.850 0.890 0.900 0.840 0.900 0.900 0.860 0.920 0.950 0.780 0.970 0.970 0.710 0.960 0.960 0.760 0.970 0.970 0.830 0.950 0.960 0.790 0.000 0.937 0.940 0.310 0.680 0.960 0.770 0.630 0.960 0.920 0.590 0.980 0.960 0.576 0.970 0.310 0.930 0.920 0.840 0.950 0.970 0.940 0.930 0.970 0.950 0.950 0.950 0.960 0.940 0.970 0.000 0.929 0.960 0.130 0.790 0.940 0.540 0.780 0.940 0.850 0.670 0.970 0.930 0.670 0.970 0.030 0.910 0.940 0.710 0.900 0.980 0.910 0.890 0.970 0.990 0.820 0.970 0.980 0.870 0.990 REFERENCES 187 REFERENCES Asparouhov, T. 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