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D . , Measurement , Evaluation , degree In and Research Design (JQPNTA Major professor Date May 12, 1982 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 ¥i—-\—— q “— MSU LIBRARIES “ f e. TL"; 1 r~ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ——_—— THE ESTIMATION AND HYPOTHESIS TESTING OF TREATMENT EFFECTS IN NONEQUIVALENT CONTROL GROUP DESIGNS WHEN CONTINUOUS GROWTH MODELS ARE ASSUMED BY Carol Joyce Blumberg A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Counseling, Educational Psychology, and Special Education 1982 Glnmg ABSTRACT THE ESTIMATION AND HYPOTHESIS TESTING OF TREATMENT EFFECTS IN NONEQUIVALENT CONTROL GROUP DESIGNS WHEN CONTINUOUS GROWTH MODELS ARE ASSUMED BY CAROL JOYCE BLUMBERG The class of continuous growth models, where it is assumed there is a correlation of +1 between true scores at any two time points within each group, serves as the basis of discussion for the dissertation. This class can be expressed symbolically Yij(t2) = gj(t2)°Y:j(tl) + hj and Y..(t) = Yf.(t) + e..(t), 13 1] 13 where t1 and t2 are any two time points: Y:j(t) and Yij(t) represent the true and observed scores, respectively, on the measure of interest, for the ith individual in the jEE group; eij(t) represents the errors of measurement; gj(t) and hj(t) are continuous functions; and aj(t) represents the population treatment effect. CAROL JOYCE BLUMBERG The set of designs considered is that where there is one or more treatment groups with or without the presence of one or more control groups. The various approaches that have been suggested for data analysis for these designs were examined under the given class of growth models. Appropriate procedures for the estimation and hypothesis testing of differences in treatment effects exist under these approaches when (i) gj(t) is identical for all j and hj(t) is identical for all j; (ii) no errors of measurement are present and h.t 50; J( ) r "' . t = b.’ t4— t + l d h. t = .° t - t o (111) 93‘ ) 3 ( 1) an 3( ) cJ ( l) for bj, cj real-valued constants. New methods of data analysis are developed which pro- vide consistent estimates of treatment effects and differ- ences in effects and appropriate procedures for the hypothesis testing of nonzero treatment effects and nonzero differences in effects under the entire class of growth models. These methods require only that either the func- tional forms of the hj(t)'s are known or that hj(t) is the same, but unknown, for all j. When no errors of measure- ment exist, algebraic and numerical analysis techniques CAROL JOYCE BLUMBERG are used. When errors of measurement exist, the methods developed are several-stage procedures using numerical analysis techniques and the statistical techniques of maximum likelihood estimation and jackknifing. The ad- vantage of these new methods is that they are applicable under a wider class of growth models than are the existing approaches. Copyright by CAROL JOYCE BLUMBERG 1982 ACKNOWLEDGEMENTS I wish to thank Andrew C. Porter for serving as my academic advisor and dissertation chairman. He has given generously of his time and has always provided the guid- ance needed, in all areas, in order for me to complete the Ph.D. degree in Educational Psychology. I would also like to thank James Stapleton for serving as my advisor for my master's degree in Statistics and Probability and as a member of my dissertation committee, and for the generous help he has given me with the dissertation. I would like to thank Robert Floden and William Schmidt for serving on my committee. I would further like to thank the Department of Math- ematics at Michigan State University for five years of financial support. Various arms of the College of Educa- tion also provided financial support during the 1979 and 1980 calendar years. The Department of Educational Studies at the University of Delaware provided both financial and moral support during the writing of this dissertation. I would also like to acknowledge the help of Berle Reiter, Diane Murray, Arthur Hoerl, and Victor Martuza. Finally, I wish to acknowledge the moral support of my parents and brother. iii LIST OF Chapter 1. 2. 3. TABLE OF CONTENTS FIGURES................. INTRO DUCT I ON C O O O O O O O O O O O O O 0 Continuous Growth Models . . . . . . . . Overview . . . . . . . . . . . . . . . . EXAMPLES OF NATURAL GROWTH . . . . . . . . REVIEW OF THE LITERATURE . . . . . . . . . Relationship Between the Fan Spread Hypothesis and Natural Growth Models . . Relationship Between the Fan Spread Hypothesis and Differential Linear Growth Analysis Strategies. . . . . . . . . . . ANOVA of Index of Response . . . . . . . ANOVA of Gain Scores . . . . . . . . . ANOVA of Standardized Change Scores. . . ANOVA of Standardized Change Scores with Reliability Correction . . . . . . . ANOVA of Observed Standardized Change Scores . . . . . . . . . . . . . ANOVA of Residual Gain Scores. . . ANOVA of True Residual Gain Scores ANOVA of Raw Residual Gain Scores. True Difference Scores . . . . . ANOVA of Estimated Standardized Change Scores . . . . . . . . . . . . . . . . ANOVA of Estimated Residual Gain Scores. Analysis of Covariance . . . . . . . . . Estimated True Scores Analysis of Covariance . . . . . . . . . . . Rogosa's Method. . . . . . . . . Adjusted Gain Scores . . . . . . Empirical Bayes Estimation . . . iv Page vii Chapter 4. RESULTS FOR CASES WHEN NO MEASUREMENT ARE PRESENT . Case 1. . . . . . . . . Cases 3 and S . . . . . Cases 7, 9, and 11. . . Polynomial Form . . . . Exponential Form. . . . Cases 13, 15, and 17. . Case 13 . . . . . . . . Cases 15 and 17 . . . . Contradictory Solutions ERRORS OF 5. POINT ESTIMATION WHEN ERRORS OF ARE PRESENT O O O O O O 0 Case 2 Estimators . . . Overview of the Procedures for the Remaining Cases . . . Stage 1: andhj(t)'s...... Substage 2: Parameters in g(t) and h(t) Substage l. . andh()'s.... . Q Maximum.Likelihood Approach . Other Approaches. . . . Stage 2: Effects Case 4. Case 6. Case 8. Case 10 Case 12 Case 14 Case 16 Case 18 Bias of P-oooooooo flooooooooo (1". 5‘ 0 "do 0 o o o o o o o O D 6. INTERVAL ESTIMATION AND HYPOTHESIS TESTING P ROCE DUE S O O C O O O O O 7. DISCUSSION Summary . . . . . . . Estimato IS Point Estimation of MEASUREMENT Estimation of the gj (t)'s Estimation of the Unknown Estimation of the g(tk)'s Treatment Directions for Further Research . Page 75 78 80 82 84 85 89 89 93 96 96 97 98 99 102 104 113 118 118 123 123 124 124 125 127 128 128 135 145 148 Implications for Data Analysis and Collection . Appendix A. IDENTIFIABILITY. B. ESTIMATES UNDER ASSUMPTIONS 4, BIBLIOGRAPHY. vi DERIVATION OF MAXIMUM LIKELIHOOD 5, and 6. Page 149 152 158 166 Figure l 2 10 11 12 13 14 15 16 LIST OF FIGURES An example of parallel growth . . Differential linear growth when b c 1 o O O O I O O O O O O O O O O Differential linear growth when b c a 0 . . . Differential linear growth when b c < 0 . . . Exponential Exponential Exponential O < c < l . Exponential Logarithmic Logarithmic Logarithmic Logarithmic growth when b growth when growth when growth when growth when growth when growth when growth when 0 0 0 < < < -tr o' re P‘ be To C C Cumulative Normal or Logistic and < 1 < l and < O and < O and > 0 and c > l . and c > and c > 1 . and b > 0 . and b < 0 . < 1 < 1 and b V and b A growth. . . An example of polynomial growth when * 2 Yi(t) = (oz-t + c with c2 > 0, C1 > O, and d 1 Olejnik's example . . . . 1 < 0 . * -t + l)°Yi(0) + dl-t Exponential group growth under the fan spread hypothesis . . . vii 15 16 17 18 19 20 24 25 26 27 28 29 38 39 Figure 17 18 19 Olejnik's model . . . . . . . . . . . . . Subcases of the general growth model. . . Pictorial representation of the structural relation. . . . . . . . . . . . . . . . . viii 104 CHAPTER 1 INTRODUCTION In many educational settings, a true experimental design is not possible when a researcher wants to evaluate the effects of different treatments. Thus, quasi-experi- mental designs are employed. One of the more commonly used quasi-experimental designs is the nonequivalent control group design (Campbell & Stanley, 1966; Campbell, 1969). Campbell and Stanely define this design as having two groups, a control group and a treatment group, which are formed by some method other than random assignment. They then require that both a preobservation and a postobser- vation be made on each individual on some measure of interest. Campbell (1969) extends the definition of a nonequivalent control group design to include any number of observations occurring at times before the treatment begins, during the treatment and after the treatment. More generally a nonequivalent control group design can include multiple-group designs, with or without the presence of one or more control groups, and can involve the investigation of interactions through the use of crossed factors. There has been much discussion in the literature of the analysis strategies that are appropriate for use in connection with nonequivalent control group designs. The basic problem is to identify analysis strategies which will provide unbiased estimates of the treatment effects. This problem has come to be known as the problem of measuring change. Both Lord (1967) and Cronbach and Furby (1970) have argued that unless some assumptions are made, there is no way of knowing which analysis strategy is appropriate for use with any particular application of a nonequivalent control group design. While the literature on the problem of measuring change is voluminous, considerable confusion remains as to appropriate solutions. One of the reasons for this confusion is that different authors have made different assumptions when making recommendations for the methods of analysis to be used with data arising from nonequivalent control group designs. Some of the more popular assump- tions will now be discussed. Kenny and Cohen (Kenny, 1975; Kenny & Cohen, 1980) and others (e.g., Cochran & Rubin, 1973) have taken as their assumption that the method of selection of the individuals into the treatment and control groups is known. They discuss various methods of selection and based on each method of selection they recommend a specific method of data analysis. Another popular assumption is the fan spread hypothesis (Bryk & Weisberg, 1977; Campbell & Erlebacher, 1970; Kenny, 1975; Olejnik, 1977). The fan spread hypothesis states that the ratio of the differences of population means to the standard deviation common to the populations of interest is constant over time. The fan spread hypothesis will be discussed further in Chapter 3. There it is described under what conditions data collected using a nonequivalent control group design will conform to the fan spread hypothesis. Campbell and Boruch (1975), Kenny (1975), and Olejnik and Porter (1981) have given examples of data sets which seem to conform to the fan spread hypothesis. Hence, evidence exists that the fan spread hypothesis may occur for some data sets arising from educational research settings. A third group of assumptions are stated in terms of continuous growth models. The idea of continuous growth models dates back to at least 1964 (Potthoff & Roy, 1964). But, it was not until 1976 that the idea of applying continuous growth models to aid in the analysis of data from nonequivalent control group designs was born (Bryk & Weisberg, 1976). Continuous Growth Models Let J represent the number of groups in a particular design. Let qj + 1 represent the number of time points at which observations for group j are made on the measure of interest. Let Yij(t1j), Yij(t2j), ... , Yij(tqj), and Yij(tqj+l) represent the observations for the ith individual in the jth group, where t1 , t2 , ... , t . . .+1 3 3 q: represent the qj + 1 time points at which the observations * t are made. Let Y: ij( tlj ), Yij(t2j)' ... , Yij(tqj)' * ei (t ) represent the true scores and errors of measure- j qj+1 ment, respectively, for the i§§_individual in the jth treatment group at the q3. + 1 time points. The most gen- eral form for a growth model that will be considered here is that where classical measurement theory assumptions are made and where the functional relationships * Y..(t 1] qj+l) ij(Yiju:l ) Y: jctz ). j 3 (1-1) * ooo'Yo (t )'t It 'ooo't ) i . l. 2. .+1 j q: 3 3 q: hold, where the fij's are continuous functions, which may be different for each individual in each of the J groups. The classical measurement theory assumptions of particular interest are, for each time t and for each j; and o - * . (l) '11th) "" Yij (t) + eij (.t) 1 (ii) E(eij(t)) = 0 ; (iii) Cov(Y;j(t), eij(t)) = o ; (iv) Cov(eij(t), eij(t')) = 0 for any time t' f t . The models considered by Bryk, Strenio, and weisberg (1980), Strenio, Weisberg, and Bryk (in press), Olejnik (1977), and the models that will serve as the basis of the remainder of this dissertation can all be seen as special cases of the above general model. The most general model considered by Bryk, Strenio, and Weisberg (1980) is when equation (l-l) reduces to Y* * ij‘tzj) 3 Yij(tl )+b. (t -t )+a.(_t ) (1-2) j lj 23' 1:1 3 23' where bij is some constant for the ith individual in and the jth group and aj(t) is the treatment effect for group j at time t ; * Yij(tl.) + bi.(t2 — t1.) represents natural 3 3 j 3 growth (i.e., growth which occurs in the absence of any treatment effects or errors of measurement). For this model and for all other continuous growth.models to be discussed, the assumption is made that treatment effects are additive. That is, the treatment causes an increase or decrease of exactly the same amount for all individuals in the same group over the amount of growth accounted for by natural growth. The assumption of addi- tive treatment effects is standard in the experimental design literature (e.g., Cox, 1958). Notice that this model states that each person's true growth over time is linear, albeit possibly a different line for each person. Strenio, Weisberg, and Bryk (in press) describe a more general model of natural growth than that considered in equation (1-2). In this more recent paper they extend their model of natural growth.to L. 3 2 j(t )= 2 1r.. -(t -t ) j(1:. ). (1-3) Zj i=1 131 23 13. 1j where the Lj's are predetermined integers and the "iji's are undetermined constants. The reason for the absence of any treatment effects in the model given by equation (1-3) is that in the paper they were only concerned with the estimation of natural growth curves. Notice that this model allows for natural growth over time which is a polynomial of any degree. Additionally, the coefficients of the various terms in the polynomial need not be the same for each person. Olejnik (1977) discusses two different models of natural growth. One of his models assumes that each in- dividual's natural growth.over time is linear, that there is a correlation of +1 between true scores at any two points in time and that the fan spread hypothesis holds. Although he never explicitly expresses this model in terms' of individual growth curves, the model can be expressed as a: a: Yij(t) = [b(t —-tlj) + ll‘Yij(tlj) + C(t -tlj) + aj (t) r where b and c are positive constants and t is any point in time. Also, this model requires that the within groups standard deviation is the same fOr all J groups at any time point. Even though the expression given here is in terms of any number of groups, it should be pointed out that Olejnik only deals with two-group designs in his disser- tation. Olejnik's other model will be discussed briefly in Chapter 3. The growth models to be considered here are an exten- sion of the model of Olejnik (1977) just described. As will be discussed later, this extension is both more and less general than the model considered by Strenio, Weis- berg, and Bryk (in press). These models assume only three things: (1) Classical measurement theory holds. (2) The correlation within each group between true scores at any two points in time is +1. and (3) Treatment effects are additive. These growth models can be expressed symbolically as * * Yij(t) = gj(t)'Yij(t1j) + hj(t) + (ll-(t) and (1-4) * Yij (t) a Yij (t) + eij(t) for all t, where gj(t) and hj(t) are continuous functions; * gj(t)'Yij(t1 ) + hj(t) represents natural growth: 3 and aj(t) represents the population treatment effect. Let T be the time at which the treatments are initiated. Notice that aj(t) E O for all t £.T- Further, notice that gj(t) > 0 for all t, since a correlation of +1 is being assumed. There are several reasons for considering the class of growth models represented by equation (1-4). First, as will be shown in Chapter 3 these models are more general than the fan spread hypothesis. As mentioned earlier, Campbell and Boruch (1975) and others have stated their belief that in many actual situations, data conform to the fan spread hypothesis. Hence, they and others (e.g., Kenny, 1975; Olejnik, 1977) have mainly restricted their attention to situations where the fan spread hypothesis holds when discussing the problem of measuring change. Since the growth models being assumed for this disserta- tion are more general than the ones considered by others who have worked on the problem of measuring change, they are an important class of models to study. Second, as will be shown in Chapter 2, these growth models allow for a variety of different functional forms for individual growth and hence for group growth. They are not restricted to linear growth or even to polynomial growth as has been done in the past literature on the problem of measuring change (e.g., Kenny, 1975; Olejnik, 1977; Strenio, Weis- berg & Bryk, in press). Finally, these models of natural growth have led to the development of very different ap- proaches to data analysis than heretofore considered. Nevertheless, it is recognized that the models under study in this dissertation are only a small subset of all pos- sible growth models. Overview This dissertation has three purposes. The first pur- pose is to explicitly show what types of individual growth are possible under the growth model given by equation (1-4). This will comprise Chapter 2. The second purpose is to 10 explore the relationships between the growth models of equation (1-4). the fan spread hypothesis, and the various methods of data analysis that have been suggested in the past with respect to the nonequivalent control group design. This will comprise Chapter 3. Consistent with the models described by equation (1-4), the discussion of methods of data analysis will be restricted to those methods which do not require the use of additional in- formation such as background variables, covariates other than the measure itself, nor replicates of the measure of interest within a point in time. As will be seen, cur- rently proposed methods of data analysis are not suffi- ciently general as to apply to the full set of growth models represented by equation (1-4). Hence, new methods of data analysis must be found. The third and main purpose of this dissertation is to describe the data analysis pro- cedures which have been developed. These data analysis procedures will be described in Chapters 4 through 6. The basic idea behind these new methods is to collect data at enough pretest time points so that one is able to estimate the natural growth patterns. Next, the estimated natural growth patterns are projected into the future. Point esti- mation and hypothesis testing methods are then developed which are based on the projected natural growth and the observed posttest data. Chapter 7 will provide a summary 11 of the results from Chapters 3 through 6 and will also con- tain a discussion of some directions for further research. CHAPTER 2 EXAMPLES OF NATURAL GROWTH In this dissertation the term natural growth will be used to denote the relationship between true scores at time tlj and at any other time, t, that would have occur- red if no treatments were applied. There are an infinity of different continuous natural growth curves represented by the models given in equation (1-4). The purpose of this chapter is to illustrate the flexibility of the model by discussing several specific examples. These examples were chosen because they model growth curves found in the educational and behavioral science research literature. For simplicity of presentation, initial discussion is limited to a single group and hence, the j subscript will be dropped temporarily. Further, without loss of generality, it can be assumed that t1 = 0. Hence, 3' natural growth under equation (1-4) can be expressed for a one-group design as 'k * Yi(t) = g(t)-Yi(0) + h(t) . (2-1) Some specific examples of natural growth are: (1) Parallel growth. Parallel growth is defined by ‘k t 7 g(t) E 1, so that, Yi(t) = Yi(0) + h(t), where h(t) is any 12 13 continuous function. Figure 1 provides a pictorial repre- sentation of an example of parallel growth. For ease of illustration, Figure 1 and all remaining figures will only show the growth curves for three individuals in the group. ? * Yi (t) Person 3 Person 2 Person 1 ._1 .—D t Figure 1. An example of parallel growth. (2) Differential linear growth. Differential linear * * growth is defined by Yi(t) = (b-t + l)-Yi(0) + C't, where b and c are real-valued constants with b # 0. Figures 2 through 4 provide pictorial representations of examples of differential linear growth. * Q Yi(t) Person 2 Person 3 Person 1 (u 1? rr Figure 2. Differential linear growth when b < 0 and c 3 0. 15 4v * Yi(t1 a t Person Person Person '6 Figure 3. Differential linear growth when b < 0 and c = 0. l6 Y* i(t) Person 1 Person 2 V t— s—ot Person 3 v Figure 4. Differential linear growth when b > 0 and c < 0. (3) Exponential growths Exponential growth is de- t * fined by Yi(t) = [b°c + (l — b)]°Y;(O), where b and c are real-valued constants with c > 0. (Since measures of growth take on only real values, values of c < O, which 17 * yield complex values for Yi(t) are not allowed.) Figures 5 through 8 provide pictorial representations of examples of exponential growth. . * T TPerson 3 Yi(t) Person 2 Person 1 i Figure 5. Exponential growth when b = 1 and c > 1. 18 * vim 49 Person 3 Person 2 Person 1 1't Figure 6. Exponential growth when 0 < b < 1 and c > 1. 19 Person 3 Person 2 . * Yi(t) Person 1 J. Figure 7. Exponential growth when 0 < b < l and 0 < c < l. 20 9 * Yi(t) \ 3"---" -"'""' "" """""“" \ F I c at Person 1 Person 3i Y Person 2 Figure 8. Exponential growth when b < 0 and c > 1. The inclusion of monotone decreasing functions (see e.g., Figures 3, 7, and 8) as representatives of natural growth was motivated by learning theorists interest in forgetting curves. Other forms of natural growth included under equation (2-1) are: 21 (4) Logarithmic growth. Logarithmic growth is de- fined by t * Yi(t) - [logc(b-t + C)]°Yi(0). where b and c are real-valued constants with c > 0 and c i l . (5) Cumulative normal (Normal Ogive) growth. Cumula- tive normal (Normal Ogive) growth is defined by t 2;”) = 2.4.3.... 2 * 'eXP(-%v ) dv}°Y.(0) 1r 1 (6) Logistic growth. Logistic growth (Lord & Novick, 1968) is defined by (1 + cm:t * ° Y-(O), 1 + c-dt 1 * — Yi(t) where c and d are real-valued constants with c > 0 and d > 1 . (7) Polynomial growth. Polynomial growth is defined by a» P n * q k cn't )-Yi(0) + Z dk't , n k=1 1 where the cn's; n = 1,2, ... , p and the dk's; k = 1,2, ... q are real-valued constants with cp # 0 and dq # 0 I 22 Figures 9 through 12 provide pictorial representations of logarithmic growth. Figure 13 provides a pictorial representation of Cumulative normal and Logistic growth. Figure 14 provides a pictorial representation of polynomial growth when p = 2, q 8 1, c2 > 0, c1 > O, and d1 < 0. As stated previously the examples given here represent only a small fraction of the types of natural growth allowed under equation (2—1). Continuous functions of any form are allowed for g(t) and h(t), with the only restrictions being that g(t) > 0 (because, it is being assumed that the correlation of true scores at any two points in.time is +1) and that g(O) = l and h(O) a O. The reason for the restric- tions g(O) = l and h(O) = 0 is consistency. For equation (2-1) to hold at time t = 0, it is necessary to have Y:(0) * g(O)°Yi(O) + h(O) for each individual. Consequently, g(O) l and h(O) = 0. All of the types of growth possible for single-group designs are also possible for multi-group designs. For multi-group designs, the natural growth curves may be 1) exactly the same for all the groups 2) of the same form for all the groups, but with different constants specifying the functions. For example, a three-group design where all three groups follow exponen— tial growth patterns would be expressed as and where and cl, or 3) 23 * - t 1 b * o t . t y* * b - t y* 0 b1' b2, and b3 take on possibly distinct values c2, and c3 take on possibly distinct values. of different forms for each group (e.g., group 1 follows logarithmic growth, group 2 follows exponential growth, group 3 follows polynomial growth, etc.). W 24 9 Y;(t) Person 1 Person 2 Person 3 ‘ .i --————-——---——————_————- Figure 9. J. Logarithmic growth.when c > 1 and b > 0. 25 4 * vim I I I I I Person 1 I I Perso 2 ' I p I erson 3 ' I I I I GLe . I I I I I I I I I I I 11' I. it Figure 10. Logarithmic growth when c > 1 and b < 0. ’It 26 I I 3‘11””) Figure 11. Logarithmic growth when 0 < C < 1 and b > 0. Person 1 Person 2 Person 3 27 * T Yi(t) t k Person 1 Person 2 Person 3 \ Figure 12. Logarithmic growth when 0 < c < l and b < 0. 28 4 * Yim ----------_ Person 3 Perso Person 1 __V . ’t V Figure 13. Cumulative Normal or Logistic growth. 9 \ 29 T. Yi(t) Person 1 Person 2 Person 3 Figure 14. \ An example of polynomial growth when Y:(t) = (c °t2 + c * 2 l't + l)°Yi(0) + d with c2 > 0, c1 > 0, and d1 < 0. l °t CHAPTER 3 REVIEW OF THE LITERATURE In this chapter the various methods that have been suggested for the analysis of data arising from the application of nonequivalent control group designs are discussed. Following the lead of Campbell and Stanley (1966) most of the literature on the problem of measuring change has restricted its attention to designs having only two groups--a treatment group and a control group. This sole attention to two group designs is unfortunate. Meth- ods of data analysis should be discussed in the context of designs with any number of treatment groups and with or without the presence of a control group. For the remainder of this dissertation the discussion will usually be in the context of both one-group and multi-group designs where random assignment has not taken place. For multi-group designs, the presence of a control group will not be assumed. The discussion here will be restricted to two- group designs only when absolutely necessary. The literature on the problem of measuring change is very confusing, in that each author(s) makes different assumptions and oftentimes the assumptions are implicit, rather than explicit. The attempt here is to clarify the literature by separately discussing many of the methods of data analysis that have been suggested in the literature. 30 31 The discussion for each analysis method will include an explicit statement of the assumptions being made and a short description of the method. Since the fan spread hypothesis is the most widely made assumption in the past literature on the problem of measuring change (e.g., Bryk & Weisberg, 1977; Campbell, 1971; Kenny, 1975; Kenny & Cohen, 1980; Olejnik, 1977), a discussion of the relationship between the fan spread hypothesis and the growth models defined by equation (1-4) is offered first. This discussion will help to clarify the motivation for and the appropriateness of some of the analysis strategies to be discussed later in this chapter. Relationship Between the Fan Spread Hypothesis and Natural Growth Models In the previous literature the fan spread hypothesis has only been discussed in the context of two-group designs. For those designs the fan spread hypothesis states that at the population level the ratio of the difference of the group means to the standard deviation common to the popula- tions is constant over time when there are no treatment effects (Kenny, 1975). In the past literature it is not made clear whether the standard deviation is that for the true scores or that for the observed scores. The discussion 32 here will be in terms of both true scores and observed scores. Symbolically, the fan spread hypothesis on the true scores can be expressed «g ‘ * * _; * u Y1 2 0;“:1) 0*(t2) where t 1 and t2 are any two points in time; i ”Y (tk) is the population mean for group j on i the measure of interest at time tk; j=l,2; k=1,2; * and 0Y(tk) is the standard deviation common to both populations for the true scores on the measure of interest at time tk; k = 1,2 . An extension of the fan spread hypothesis to multi- group designs is straight-forward by assuming that at the population level the fan spread hypothesis holds for every pair of two groups chosen without replacement from the J groups. Hence, the extended fan spread hypothesis for true scores will be defined as * * * * J 3 3 3 (3-1) * o;(t1) oY(t2) where j, j' represent any two of the J groups. 33 Recall that natural growth on latent variables was defined in the system of equations (1-4) by Y* (t) (t) 1* (0) + h (t) (2 1) ij ’ 93 ij j ' By deriving the means and variances for these variables, it can be seen under what conditions natural growth satisfying equation (2-1) conforms to the fan spread hypothesis. First, taking the variance of both sides of equation (2-1) yields * 2 2 * 2 [CY (t)] = [9.(t)] '[°y (0)] . (3-3) . 3 . J 3 But as stated, the fan spread hypothesis requires that the variances for any time, t, be the same for all groups. That is, it is required to have * * CY (t) = oY(t) for all j. 1 Hence, by equation (3-3) it is necessary to have gj(t) 5 g(t), for all j at each time t, and to have a; (0) E 0;(0) J for all j in order for the fan spread hypothesis on true scores to hold. Consequently, equation (3-3) can be rewritten as [o; + hjuyj'(0) + hj.(t2>1 / 92(t2) to;<0)12 + 0:“:2) If, in addition o:(t) = [g(t)]z-O:(0) for each t, then equation (3-11) simplifies to equation (3-8). Hence, hj(t) E hj.(t). Therefore, the fan spread hypothesis for observed scores holds when natural growth under equation (1-4) reduces to * Yith) * g(t)-Yij(0) + h(t) and when o;j(0) = o;(0) and oej(e = g(t)'oe(0). The statement that oe (t) J g(t)oe(0) is equivalent to requiring that the reliability of Y be constant over time and across groups. Olejnik (1977) has asserted that the fan spread hypothesis for either true scores or observed scores can hold even when there is not a correlation of +1 within each group between true scores at any two points in time. 38 Although he gives no mathematical formulation of individual growth in this case, Figure 15 shows the example he gave in his dissertation. For this example, he has assumed that at the population mean level both groups (he deals only with two-group designs) follow the same differential linear growth pattern, and hence, he has only drawn growth curves for individuals from one group. It should be noted that Olejnik's restriction to differential linear growth is not necessary. Figure 16 gives a sketch of an example where the fan spread hypothesis holds for an exponential type model for group mean growth and where a correlation of +1 between true scores within each group is not assumed. For simplicity, only two groups have been drawn. In Figures 15 and 16 the solid lines represent population mean growth and the dotted lines indicate the growth curves for some selected individuals. Figure 15. Olejnik's example. 39 * Yij (t) 4 >1: Figure 16. Exponential group growth under the fan spread hypothesis. 40 In summary, the fan spread hypothesis for true scores holds when equation (1-4) reduces to * * Yij‘t’ g(tIYij(0) + h(t) + aj(t) * and (3-12) * o (t) = 0Y(t) . The fan spread hypothesis for observed scores holds for those growth models where the system of equations (3-12) are fulfilled and, in addition, a (t) = g(t)-ae(0). e 3 Further, both the fan spread hypothesis for true scores and for observed scores can hold for some models of individual growth where there is not a correlation of +1 within each group. No mathematical formulation of these models has yet to be developed. Relationship Between the Fan Spread Hypothesis and Differential Linear Growth A discussion of the relationship between the fan spread hypothesis and differential linear growth is needed since in the past literature the distinction between these two con- cepts has been blurred. The concepts of differential linear 41 growth and of the fan spread hypothesis are, however, distinct concepts. In the previous two sections it was shown that the fan spread hypothesis holds for many forms of natural growth other than differential linear growth. Hence, linear growth that conforms to the fan spread hypothesis is a subset of all natural growth that conforms to the fan spread hypothesis. Further, differential linear growth conforms to the fan spread hypothesis only in rare cases. Differential linear growth is represented in g Y Y O C C t I 1 Y . . O I c I t ’ That is, when gj(t) = bj-t + 1 and.hj(t) = cj-t . But, for the fan spread hypothesis (either for true scores or ob- served scores) to hold it is necessary to have gj(t) 5 g(t) and hj(t) E h(t) . Hence, the fan spread hypothesis does not hold under differential linear growth unless all of the bj's are equal to some common value and all of the cj's are equal to some common value. Analysis Strategies For the remainder of this dissertation the only con- tinuous growth models that will be considered are those represented by the system of equations (1-4) 42 * * Yij(t) 9j(t)'Yij(tlj) + hj(t) + aj(t) and. (1-4) (t) * Yij(t) + e..(t) - Y.. 13 13 Recall that aj(t) represents the amount of growth for group 3: over that which would have occurred under natural growth. Let “a(t) represent the mean of the aj(t)'s. Then, Yj(t), as defined by Yj(t) = aj(t) ~ ua(t), represents what has traditionally been called a treatment effect under comparative experiments. The past literature on the problem of measuring change has been divided into two groups with respect to the dis- cussion of treatment effects. The majority of the litera— ture has been concerned with treatment effects as defined by the Yj(t)'s. A small subset of the literature has been concerned with treatment effects as defined by the ej(t)'s. Those analysis strategies which deal with treatment effects as defined by the Yj(t)'s will be discussed first and with respect to three criteria: (1) Under what additional conditions over those of equation (1-4) does the analysis strategy provide unbiased point estimates of the differences between treatment effects? (2) If under certain conditions the analysis strategy provides unbiased point estimates of the differences between 43 treatment effects, than how are interval estimates of the differences constructed? and (3) How is the hypothesis testing of J 2 Ho: 2 (Y.(t)) = 0 i=1 3 (i.e., Ho; All of the Yj(t)'s are equal) versus J 2 H1: 2: (Y.(t)) #0 i=1 3 (i.e., H1: All of the Yj(t)'s are not equal) accomplished? Before describing the analysis strategies it should be pointed out that, by definition, Yj (t) "" Yjv(t) = [OI-j (t) " Ua(t)] "' [ajl(t) " ua(t)] Hence, Yj(t) — Yj.(t) = aj(t) — aj,(t). Since the growth models described by the system of equations (1-4) are in terms of the aj(t)'s rather than the Yj(t)'s, the discussion of the three criteria will be in terms of the aj(t) — aj.(t) 5. ANOVA of Index of Response Analysis of Variance (ANOVA) of Index of Response re- fers to a group of analysis strategies, rather than a single 44 method. Much of the literature on the problem of measuring change has considered various forms of ANOVA of Index of Response as approaches to data analysis for nonequivalent control group designs. A general discussion of ANOVA of Index of Response will be given first. Then, specific forms of ANOVA of Index of Response will be discussed. A description of ANOVA of Index of Response is also being included here since the methods of data analysis to be developed in Chapters 4 through 6 can be thought of as generalizations of this group of approaches. To employ ANOVA of Index of Response it is only neces- sary to have scores on the measure of interest at two points in time for two or more groups. Taking data at the first point in time as a pretest, without loss of generality, it can be assumed that the pretest observations are taken at time t1 8 0 for all j. The second time point is considered 3' to be a posttest,_given at some time, t, past when the treatments have been initiated. It is assumed here and throughout the remainder of the dissertation that all indi- viduals in all of the treatment groups start receiving the treatment at the same time. A new score is formed, defined by zij(t) = Yij(t) - K'Yij(0), where K is some known con- stant (Cox, 1958; Porter, 1973). The method of data analysis then used is an ANOVA with the Zij(t)'s as the 45 dependent variable. Notice that ANOVA of Index of Response with K = 0 reduces to an ANOVA of the Yij(t)'s. (The linear model for ANOVA of Index of Response is as for ANOVA, Zij (t) = “2“” + (1111):] + (fIR)j-jr where uz(t) = uY(t) — K-uY(0) (3-13) is the population grand mean for Z(t); uY(t) is the population grand mean for Y(t); (TIR)j = uzj(t) —'uz(t): (3-14) IR denotes Index of Response; and (fIR)ij is the error term for an individual. First, notice that “z.(t) = “Y.(t) —- K°uY0(0) . (3-15) 3 J 3 Second, by substituting equations (3-13) and (3-15) into equation (3-14) (TIR)j = “Y.(t) —-uY(t) —-K(uY.(0) —-uY(0)) . (3-16) 3 3 Consequently, for any two groups j and j': (3-17) RWY. (0) - uY .(0)) J j 46 Next, taking the means on both sides of the system of equations (1-4) yields uyj(t) = ngt)uyj(0) + hj(t> + ath) . Solving for aj(t) gives aj(t) = “Yj (t) — [gj(t)qu(0) + hj(t)] . (3-18) So, aj(t) - aj.(t) {my (t) — [gj(t)uY.(0) + hj(t)1} — J 3 ({HY. ' (t) "" [gj I (t) HY (0) + hj.(t)]} . J 1' Therefore, ath) — aj.(t) uY (t) — uY.'(t) - {[gj(t)uY.(0) + 3 3 3 (3-19) hj(t)] ‘ [gjdtII-lqu) 7" hjt(t)]} . 3 Hence, by comparing equations (3-17) and (3-19), ANOVA of Index of Response theoretically provides correctly defined differences in treatment effects between groups j and j' if and only if [gj (t)uyj(0) 4" hj (t)] "' IngItmyj'm) + hjv(t)] = K(“Y.(0) - ”Y. (0)). J J' 47 That is, if and only if [gj (t)qu (0) + hj (t)] - [SJ-.(tII-lyj'W) + hju(t)] (3-20) I: (0) - u (0) Yj Yj, Hence for two—group designs ANOVA of Index of Response with [91(t)uyl(0) + hi‘t’l“ [92(t)uY2(0) + h22<0)'°Y‘°’ °Y(0)Y(0) 8* * Y (t)-Y (0) . Further, by equation (3-27), BY*(t)-Y*(0) is equal to g(t). There- fore, the Dij(t)'s are the same as the zij(t)'s as defined in the ANOVA of Index of Response with K = g(t). Thus, True Difference Scores provides correctly defined differ- ences in treatment effects and correctly tests for nonzero differences in treatment effects when * Yij(t) * g(t)-Yij(0) + hj (t) + “j (t) and Yij(t) * Yij (t) + eij (t) . 58 ANOVA of Estimated Standardized Change Scores ANOVA of Estimated Standardized Change Scores (Kenny, 1975; Olejnik, 1977) is the analysis strategy in which an ANOVA is performed on the zij(t)'s as defined by SY(t) Zij(t) = Yij(t) - ~Yij(0) . SY(0) ANOVA of Estimated Standardized Change Scores is an attempt to develop a method of data analysis when oY(0) and oY(t) are not known, which is analogous to ANOVA of Observed Standardized Scores. There are, however, several problems which arise when trying to go from observed standardized change scores to estimated standardized change scores. The first problem is that the distribution of the zij(t)'s may not be a normal distribution, even if the vector (Y(0),Y(t)) has a bivariate normal distribution. The second problem is that in order to decide whether ANOVA of Estimated Standardized Change Scores yields correctly de- fined differences in treatment effects at the population level, it is first necessary to derive an expression for E(zij(t)). Unfortunately, the problem of finding E(Zij(t)) is an unsolved problem, even for the case when it can be assumed that the vector (Y(0),Y(t)) has a bivariate normal distribution. So, one can not determine theoretically 59 under what conditions the E(Zij(t))'s lead to correctly defined differences in treatment effects. For large enough sample sizes (i.e., N 3 30) it has been shown that ANOVA is robust with respect to the viola- tion of the assumption of a normal distribution for the zij(t)'s (Glass, Peckham, & Sanders, 1972). Further, asymptotically aY(t) u (1:) =11 (t) - -u (0) . zj Yj oY(0) Yj Hence, asymptotically, ANOVA of Estimated Standardized Change Scores provides correctly defined differences in treatment effects and correctly tests the hypothesis of nonzero differences in treatment effects under the same conditions as does ANOVA of Observed Standardized Change Scores. Until the problem of finding the expected value of the Zij(t)'s has been solved and a test statistic has been defined with known distribution, it is recommended that ANOVA of Estimated Standardized Change Scores not be used as a method of data analysis, even for true experi- ments, when small sample sizes are present. 60 ANOVA of Estimated Residual Gain Scores ANOVA of Estimated Residual Gain Scores (Manning & Dubois, 1962; Olejnik, 1977; Porter & Chibucos, 1974) is an analysis strategy in which an ANOVA is performed on either the Vij(t)'s or Wij(t)'s (depending on the particular reference) as defined by A Vij(t) = Yij(t) — BY(t)-Y(0)°Yij(°) A . 8Y(.t.) «(0) wij(t) = Yij(t)'_ A .Yij(0)' panama) where BY(t)-Y(0) is the least squares estimate of the slope of the Y(t) on Y(0) regression line and where °Y(0)Y(0) is some estimator of the reliability of Y(0). Two problems arise when one performs the ANOVA pro- cedures on the Vij(t)'s or Wij(t)'s. These problems parallel the problems discussed in the section on ANOVA of Estimated Standardized Change Scores. The first problem is that neither E(Vij(t)) nor E(Wij(t)) is known (Draper & Smith, 1981). The second problem is that the distributions of the Vij(t)'s and Wij(t)'s may not be normal, even if the 61 vector (Y(0),Y(t)) has a bivariate normal distribution. Asymptotically, however, and IBY(t)'Y(0) "w (t) = my (t) — -uY(0) . j 3 “Y(omm Hence, asymptotically, ANOVA of Estimated Residual Gain Scores using the Vij(t)'s provides correctly defined treat- ment effects and correctly tests the hypothesis of nonzero differences in treatment effects under the same conditions as does ANOVA of Raw Residual Gain Scores. Recalling that BY(t) «(0) BY*(t)-Y*(O) = , then asymptotically, ANOVA of panama) Estimated Residual Gain Scores using the Wij(t)'s provides correctly defined treatment effects and correctly tests the hypothesis of nonzero differences under the same conditions as does ANOVA of True Residual Gain Scores. As with ANOVA of Estimated Standardized Change Scores, it is recommended that ANOVA of Estimated Residual Gain Scores not be used when small sample sizes are present. 62 Analysis of Covariance Analysis of Covariance is a method of data analysis where the slope of the Y(t) on Y(0) regression line is estimated, as in ANOVA of Estimated Residual Gain Scores, but where the statistical difficulties involved in doing an ANOVA of Estimated ReSidual Gain Scores are eliminated. The methodology involved in performing an Analysis of Co- variance (ANCOVA) is well-known (see e.g., Glass & Stanley, 1970; Seber, 1977; or Winer, 1971) and will not be repeated here. As with ANOVA of Index of Response, ANCOVA requires that the observations be taken at the same time points for all J groups. Without loss of generality, assume that there are a pretest and a posttest, given at times 0 and t, respectively. For the models under consideration here Y(0) is the covariate and Y(t) is the dependent variable for the ANCOVA. In general, when deriving the Sum of Squares to be used for an ANCOVA, the assumption is made that the covariate is fixed. In many settings such as those assumed in this dissertation it is unreasonable to assume that the covariate is fixed. In these settings, the covariate must be considered as a random variable. DeGracie and Fuller (1972) have shown, however, that ANCOVA still works when the covariate is taken as a random variable. A second assumption of ANCOVA is that the covariate is measured 63 without error. Since the covariate and the dependent var- iable here are the same measure taken at two different time points, it will be assumed here that both Y(0) and Y(t) are measured without error. A further assumption made for * ANCOVA is that there is a linear relationship between Y (t) * and Y (0) when all of the subjects are considered as coming from one population. That is, ANCOVA can only be used for the models under consideration in this dissertation when * t The linear model for ANCOVprith a random covariate is Yij(t) = uY(t) + (rm)j + By(t,.Y(o)(Yij(0) - “Y(0), + (fAC) ij where AC denotes Analysis of Covariance; (TAC)ij = [uyj'(t) - uY(t)] and (f ) AC is the error term for an individual. 13' In the section on ANOVA of Raw Residual Gain Scores it was shown that BY(t)-Y(0) is well-defined only when, for each time t, CY (t) a oY(t) for all j. Hence, ANCOVA provides 2) correctly defined differences in treatment effects and 64 correctly tests for nonzero differences in treatment effects if and only if, for each time t, (i) no errors of measurement are present in the data; 0 o * * , I 0 (ii) Yij(t) = g(t) Yij(0) + h(t) + aj(t) .. . . * t and (iii) OYj(t) = oY(t) . The estimates of the treatment effects under ANCOVA are given by A 4 A (Seber, 1977). a- ' It has been shown that E(TAC)j = (TACIj ./\ Further, by simple algebra it can be shown that E[(1:AC)j - A (tAC)j'] = aj(t) - aj,(t) . Consequently, (TAC)j -(TAC)j. provides an unbiased estimator of aj(t) — aj,(t). The procedures for interval estimation and for the testing for nonzero differences in treatment effects are given in Glass 8 Stanley (1970) and Winer (1971). Estimated True Scores Analysis of Covariance Estimated True Scores Analysis of Covariance was developed by Porter (Porter, 1967; Porter & Chibucos, 1974) as an extension of Analysis of Covariance techniques to 65 situations where a random covariate is used which contains A errors of measurement. Estimated true scores, Tij's, are defined by Tij = jlaj I ”Y(0)Y(0)'(Yij(°) ‘ jIGS), where pY(0)Y(0) is the reliability of the measure, Y(0), which is assumed the same for all J groups. An Analysis of Covariance is then performed using the Tij's as the covar- iate and the Yij(t)'s as the dependent variable. In addi- tion to assuming that the reliability of Y(0) is the same for all J groups, Estimated True Scores ANCOVA makes all of the usual ANCOVA assumptions. The assumption of equal reliability, combined with the usual ANCOVA assumption that t O Y (0) = o;(0) for all j, implies that o (0) = oY(0) for j i all j. Hence, Estimated True Scores ANCOVA can be used Y for the growth models under consideration in this disser- tation only when * Y. O l] * (t) g(tI'YijW) + h(t) + aj(t) oyj(0) = oycoz * Yij(t) + eij(t) 66 and * * .(0) = OY(0) . OY J The linear model for Estimated True Scores ANCOVA is given by A Yij (t) = “Y(t) + (TEAC)j + BY(t) “g(tij — 1,1,3) + (fEAC)ij ; where EAC denotes Estimated True Scores ANCOVA; (TEACH =- qu (t) - uY(t) -— BY(t) ,.’1‘.(u.’1‘.j — LIE.) ; A “T represents the population mean for the j A l ' o Tij s for group J I u; represents the population grand mean for the f. '0 ijs' BY(t)-T is the slope of the Y(t) on T regression line; and (fEAC)ij is the error terms for an indiv1dual . Treatment effects are estimated using where BY(t)-T = BY(t)~Y(O)/°Y(0)Y(0)' USing Monte Carlo 67 A Simulations Porter (1967) showed that (IEAC)j — (TEAC)j' ..(t), for any two groups provides an estimator of oj(t) — 013 j and j', which contains no identifiable bias. He further showed that the resulting test statistic from an ANCOVA on the Yij(t)'s using the Tij's as a covariate is approximately distributed as an F statistic with J — l and N —’J — 1 degrees of freedom when there is an equal number of indi- viduals in each group. The properties of the test statistic have not been studied for those situations where there is an unequal number of individuals in the groups. The remaining analysis strategies are concerned with the direct assessment of treatment effects (i.e., the aj(t)'s) as well as differences in treatment effects. These remaining analysis strategies will be discussed with respect to the estimation of treatment effects and the testing of the hypothesis of nonzero treatment effects, as well as testing differences in treatment effects. Rogosa's Method Rogosa (1980) has developed a method for estimating treatment effects for two-group designs in those situations where no errors of measurement are present, hj(t) E 0, and the observations are taken at the same time points for both groups. Rogosa's method is included here because the 68 methods to be developed in Chapter 4 can be thought of as generalizations of this method. Under Rogosa's assump- tions the system of equations (1-4) reduces to Rogosa rewrites the set of equations (3-28) as Yij(t) = Y1 + yz-T.. + Y3-Yij(0) + Y4-Yij(0)-Ti. , 13 3 where 1 if j = 1 T.. 8 3 13 o ifj=2 Y1 = 32(t) 3 Y2 = 0-1 (t) "' “2 (t) 3 Y3 = 92(t) ; and Y4 = gl(t) - gz(t) . Next, he computes the least squares estimates of Y1: Y2: A A Y3' and 74- Call these least squares estimates Y1: Y2: Y3! and Y4. Rogosa assumes, as is done in traditional regression analysis, that the only values of Y(0) which are of interest are the observed values from the sample used. Hence, the A Yi's are unbiased estimators of the Yi's. But, the deri- vations given in DeGracie and Fuller (1972) can be applied 69 here to show that the 71's are unbiased estimators of the 71's even when Y(0) is taken to be a random variable. Hence, 71 + 72 provides an unbiased estimate of a1(t), Y1 provides an unbiased estimate of a2(t), and 72 provides an unbiased estimate of al(t) — a2(t). Rogosa does not dis- cuss how to test whether the difference in treatment effects is nonzero or how to test whether the treatment effects themselves are nonzero. Notice that Rogosa's method can provide unbiased estimates of treatment effects and differences in treatment effects for designs with any number of groups, as long as the assumptions detailed in the first paragraph of this section hold, by repeating his procedure for each combination of two groups chosen from the J groups. Adjusted Gain Scores Olejnik (1977) developed a method of data analysis for those two-group designs where it is assumed that mean group growth at the population level is linear over time when no treatment effects are present. The reason for including Adjusted Gain Scores here is that the methods to be de- veloped in Chapters 4 and 5 can be thought of as general- izations of this method. Olejnik requires that two pretest 70 observations and one posttest observation are taken on the measure of interest. The pretest observations are taken at times 0 and t2 and the posttest observations are taken at time t3. Figure 17 provides a pictorial representation of Olejnik's model. M (ter-IY (t2)) I ‘ 1 22 (0.1: (on I I .. Y (t :11 (t ) / 1 I ' 3 Y2 3 I (t :u (t )) I (o (0)) ' 2 Y2 2 ' (12(5)) / 'uYZ I I ‘ I I l l at ‘i_ I Figure 17. Olejnik's model. 71 Olejnik defines an Adjusted Gain Score by t He shows that the WthI's provide unbiased estimates of the treatment effects (i.e., the aj(t)'s). It should be noted here that the development of the estimates for treatment effects using Adjusted Gain Scores does not depend upon having exactly two groups in the design. Hence, unbiased estimates of treatment effects can be found using Adjusted Gain Scores for any growth situa- tion for which * * Y..t -"= b.'t+l °Y.. 0 + .°t+ ..t l:’( ) ( J I 13( ) cJ °j( ) and * Yij(t) = Yij(t) + eij(t) , where the bj's and cj's are real-valued constants with bj # 0. The hypothesis of nonzero treatment effects is tested by performing a one-sample t-test, separately, on the Wij(t)'s for each of the J groups. The hypothesis of nonzero differences is tested by performing an ANOVA on the W1j (t) ' s. 72 Empirical Bayes Estimation Strenio, Bryk, and Weisberg (Bryk, Strenio, & weisberg, 1980; Strenio, weisberg, and Bryk, in press) have applied the ideas of empirical Bayes estimation (Fearn, 1975) in order to estimate treatment effects when certain types of continuous growth models are assumed. The most general continuous growth model assumed by Strenio, weisberg, and Bryk (in press) is Li * 2 k ) i=1 3 lj and (3-29) * Yij (t) a Yij (t) 4" eij (t) I where the kiji's are unknown real-valued constants, the Lj's are predetermined positive integers, and t is the 1. 3 time of the first observations for the jth group. The reason for the absence of a treatment effect in equation (3-29) is that Strenio, Weisberg, and Bryk consider only natural growth in their paper. The model given by equation (3-29) and the models of growth given by the system of equations (1-4) overlap only in the case where, for each j, the k. I ijl s are equal to some common value (say ka) across individuals. That is, (3-29) and (1-4) overlap only when 73 L. ( ) 1 i k (t t )£ g.t = + -' - 3 i=1 3“ 1j and hj(t) 0 . (3-30) The idea of empirical Bayes estimation, in general, is to obtain estimates for the kijz's for each person by using a weighted sum of the information available for that person and the information available about the remainder of the individuals in the group. One of the requirements of the empirical Bayes method is that the variance-covariance * t * * matrix of the vector Y. a Y. t , Y. t , ... Y. t __1 (3(2j) 3( 3j) I J(ij be nonsingular, where t2 , t3 , ... , tp are the times of j j j the additional observations on the measure of interest. * * . Th ' - ' t ' f Y., V Y. , under (3-30) is e variance covariance ma rix o .1. (.1) " '1 r T g.(t )g.(t ) ... g.(t ) g.(t ) J 2j 3 3j J Pj 3 2j g.(t )g.(t ) g.(t ) g-(t ) J 2j J 3j J pj J 3j 2 - - - -[o (t )1 Y3. lj g.(t _ I J Pj l L 3 JJ L- J .JI 74 'k * Notice that, Rank (V(Yi)) = 1. Hence, V(Yj) is nonsin- gular. Consequently, the method of empirical Bayes estimation cannot be used for the models being considered in this dissertation. CHAPTER 4 RESULTS FOR CASES WHEN NO ERRORS OF MEASUREMENT ARE PRESENT In Chapter 3 it was shown that presently available methods of data analysis can be used.to estimate and test for differences in treatment effects for the growth models under consideration (see equation 1-4) when one of the following conditions hold: . * * (1) Yij(t) = g(t)-Yij(0) + h(t) + aj(t); (ii) There are only two groups in the design and the values of gl(t), 92(t), hl(t), h2(t), ”Y (0), and uY2(0) 1 are known; _.. t . . t . or (iii) Yij(t) - (bj t + 1) Yij(o) + cj t + oj(t) . For designs where condition (i) holds, Classical ANCOVA can be used if there are no errors of measurement present and CY (t) = oY(t) for all j. Estimated True Scores ANCOVA 3 (Porter, 1967) can be used when errors of measurement are 0 a * _ * 0 present if pY(0)Y(0) is known, OYj(t) - oY(t) for all j, and CY (t) = 0Y(t) for all j. If the values of g(t) are 3' known, then ANOVA of Index of Response with K set equal to 75 76 g(t) can be used. For designs where condition (ii) holds an ANOVA of Index of Response with [91(t)qu(0) + hl(t)] — [92(tIuY2(0) + h2(t)] qum) - uY2(0) can be used. For designs where condition (iii) holds, Olejnik's (1977) Adjusted Gain Scores can be used. Further, in Chapter 3 it was shown that point esti- mates of treatment effects as defined in equation (1-4) can be found by presently available methods of data analysis only when no. * - . . * . . (iii) Yij(t) - (bj t + l) Yij(0) + cj t + aj(t) , or (iv) There are no errors of measurement present in the data and hj(t) : 0 for all t and for all j. For designs where condition (iv) holds, Rogosa's (1980) method is appropriate. For designs where condition (iii) holds, Adjusted Gain Scores is appropriate. Finally, it was shown in Chapter 3 that an appropriate test of nonzero treatment effects (i.e., Ho: aj(t) = 0 versus H1: aj(t) # 0) exists only when condition (iii) holds, by using Adjusted Gain Scores. Hence, new methods of data analysis need to be devel- oped which provide estimates and hypothesis tests for 77 treatment effects and differences in treatment effects for data sets conforming to equation (1-4) and where conditions other than (i), (ii), (iii), and (iv) hold. The remainder of this dissertation is devoted to the discussion of new methods of data analysis developed by the author. As has already been illustrated, the development of methods of data analysis can be facilitated through placing various constraints on the parameters in equation (1-4). First, since the hj(t) and aj(t) terms are confounded, some information about the hj(t)'s is necessary. Three possible types of available information are: (a) the exact natures of the hj(t)'s are known (e.g., .. -.2 .8 - hl(t) — 3 t, h2(t) — 4 t + 3 t , ... , hJ(t) — loglo(5°t3 + l): (b) the functional forms of the hj(t)'s are known °t, h2(t) = k °t2 + k -t*, ... , hJ(t) = (e.g., h1(t) = k 2 3 l loglo(k4°t3 + 1), where k k2, k3, and k4 are unknown 1! real-valued constants); and (c) for each time t, the hj(t)'s are equal to some common value, say h(t) (i.e., for each t, hl(t) = h2(t) = --- = hJ(t) = h(t)). 78 In this chapter and in Chapters 5 and 6 methods of data analysis are developed under each of these types of infor- mation. The discussion of methods of data analysis under each type of information is further broken down into six cases according to whether the exact natures of the gj(t)'s are known, the functional forms of the gj(t)'s are known, or nothing is known about the gj(t)'s and according to whether or not errors of measurement are present (see Figure 18). Cases where errors of measurement are present in the data are more important for educational research and so will receive greater attention in this dissertation. Never- theless, a discussion of those cases where errors of mea- surement are not present will also be included, since there are educational and behavioral research settings in which it can be assumed that there are no, or perhaps negligible, errors of measurement present (e.g., settings where elapsed time, weight, or height is the measure of interest). The results for those cases when no errors of measurement are present are discussed in the remainder of this chapter. The results for those cases when errors of measurement are present are discussed in Chapters 5 and 6. Case 1 For Case 1 there are no errors of measurement present in the data and the exact natures of the gj(t)'s and hj(t)'s 79 .HoQOE nusoum Hmumsom we» no mmmmoosm .mH musmwm 0H EH 0H ma vd Md oasu ooau oaoo onuu onou mono .u.:m.uvfln a” He en a e a e30:x~u..a.fie mono Geno undo ouou coco Undo oBMOu assauuocsm u n v n N a :30:3 one one one on 0 U U 0 MO moo onso. ...».fis usoaoum usoaousnsoz usoooum usoeuusuaox asoooum acoeuusoooz unoaousnsoz «o ououuu acoflousmso: uo ououun acoeoususo: «0 whose” uo uncuuu 02 no ououuu 02 no ououuu oz u..u.nu usono tions statues ...... :21. ac usu0u announces» esosx o.A».fim 80 * are known. Notice that Yij(t) = Yij(t) in this case and for the remaining cases where no errors of measurement are present. Hence, the system of equations (1-4) reduces to .) + tht) + aj(t) (4-1) Yij(t) = gj(t)oYij(tlj ; i = 1,2,..., Nj j = 1,2,..., J , where gj(t) and hj(t) are known continuous functions and Nj represents the number of individuals in the sample from group j. Notice that the pretests for the different groups need not occur at the same time. Solving equation (4-1) f . t , or aj( ) Otj(t) = Yij(t) '- [gj (t)‘Yij(tlj) + tht)]. (4-2) Hence, only a sample of size equal to one from each group is necessary in order to determine the aj(t)'s. No esti- mation or hypothesis testing procedures are necessary, since equation (4-2) determines the aj(t)'s exactly. Cases 3 and 5 For Cases 3 and 5 there are no errors of measurement present in the data and the exact natures of the hj(t)'s are known. In Case 3 the functional forms of the gj(t)'s 81 are known while in Case 5 nothing is known about the gj(t)'s. Under equation (4-1) any two individuals from the jth group are represented by Ylj(t) a gj(t)-Ylj(tlj) + hj(t) + aj(t) (4-4) and yszt) = gj(t)°Y2j(tlj) + hj(t) + aj(t) . (4-5) Subtracting equation (4-4) from equation (4-5) gives Y2j(t) - Ylj(t) = gj(t)°[Y2j(tlj) -‘Ylj(t1j)] . Hence, gj(t) = . (4-6) So, for Cases 3 and 5 the values of the gj(t)'s are easily derived solely from knowledge of the pretest and posttest scores of two individuals. Once the values of the gj(t)'s have been determined using equation (4-6), Cases 3 and 5 reduce to Case 1. Nothing is gained by knowing the func- tional forms of the gj(t)'s. Further, if one assumes incorrect exact expressions for the gj(t)'s, then the values computed for the oj(t)'s will, consequently, be incorrect. Hence, it is recommended that when no errors of measurement 82 are present, no assumptions be made about the gj(t)'s, but instead equation (4-6) should be used to determine the values of the gj(t)'s at the time points at which obser- vations have been collected. For those cases where errors of measurement are present, however, the distinction between those cases where the exact natures are known, the functional forms are known, or nothing is known about the gj(t)'s will be important. Cases 7, 9, and 11 For Cases 7, 9, and 11 there are no errors of measure- ment in the data and the functional forms of the hj(t)'s are known. The general functional forms of the hj(t)'s are given in terms of undetermined constants, the values of which must be calculated. The method used to calculate the estimates for these constants is a generalization of Olejnik's (1977) Adjusted Gain Scores to nonlinear growth. Following Olejnik's lead, the presence of multiple pretests is introduced. For group j, let tl , t2 , ... , t J 3' p jrj represent the times at which the pj pretests are adminis- tered, where pj denotes the number of pretests. Let tp +1 j represent the time of the posttest for the jth j I group. Again, the times of the pretests and posttest need 83 not be the same for the different groups. For simplicity, the j subscript will be dropped because the aj(t)'s are computed separately for each group. Consider any two individuals from the jth group, say individuals 1 and 2. By equation (4-1) [and droPping the j subscript] Y1(tk) g(tk)'Yl(tl) + h(tkI and. (4-7) Y2(tk) g(tk)'Y2(t1) + h(tk) ; k = 2,3,...,p , since a(t) 0 for all t : tp. Solving the system of equa- tions (4-7) for g(tk) and h(tk) in terms of Y1(tl), Y2(t1), Y1(tk)’ and Y2(tk) yields g(tk) = (4-8) Y1(t1)'- Y2(t1) and = 0 (4‘9) h(tk) Hence, the values of h(tk) for k = 2,3, ... , p can be determined from knowledge of the pretest scores for only two individuals. 84 Once the values of the function h are derived at the time points t2, t3, ... , and, tp, the values of the un- known constants in the general expressions for h(t) can be determined. The method for determining the values of the constants is, however, dependent on the functional form of h(t). To illustrate the method for determining the values of the constants, the following two examples will be used: 3 d a polynomial form: h(t) = 1 + I cd-(t — t1) (4-10) d=l _ and an exponential form: h(t) = b-c(t't1’ + (1 —-b) . {4-11) Polynomial Form The object of the method to be described is to deter- mine the values of cl, c2, and c in the polynomial from 3 knowledge of the h(tk)'s; k = 2,3, ... , p. For simplicity, assume that p = 5, t1 = 0, t2 = 1, t3 = 4, t4 = 5, and t5 = 7. The method to be described will work for any p Z 4 and for any set of values for the tk's. Substituting t1 = 0, t2 = 1, t3 = 4, t4 = 5, and t5 = 7 into equation (4-10) yields 85 h(l) = 1 + cl + c2 + c3, h(4) = l + 4cl + 16c2 + 64c3, h(5) = l + 5cl + 25c2 + 125c3, (4—12) and h(7) = 1 + 7c1 + 49c2 + 343c3. The system of equations (4-12) is then solved for cl, c2, and c3 in terms of h(l), h(4), h(5), and h(7). This can be done since the values of h(l), h(4), h(5), and h(7) have already been determined using equation (4-9). Exponential Form The object of the method to be described in this sub- section is to determine the values of b and c in (4-11) from knowledge of the h(tk)'s. The method to be described will work for any p‘: 3 and for any set of values for the t Recall that the values of h(t2), h(t3), h(t4), ... , k's. h(tp) were already determined by using equation (4-9). Next, the values of the t 's and h(tk)'s; k = 2,3, ... , p k are substituted into equation (4-11) to yield a new system of equations analogous to (4-12). This new system is then solved for b and c. For the sake of illustration, assume that p = 4, t = 0, t = l, t = 4, and t = 5. 1 2 3 4 86 Substituting these values into equation (4-11) yields the system h(l) = 1m: + (1 - b), (4-13 i) h(4) = 1m:4 + (1 — b), (4-13 ii) and h(S) -- b°c5 + (1 - b) . (4-13 iii) Since these equations are not linear in the parameters, b and c, they cannot be solved algebraically. One must take the different possible subsystems of two equations each and see if the same solutions for b and c are found. If the solutions agree, then b and c have been found. If the solutions do not agree, then no general solution exists for the system (4-13). Contradictory solutions could have arisen for one of several reasons. A discussion of those situations where contradictory solutions occur will be postponed until the end of this chapter. The method for obtaining the actual solutions for b and c will be described using equations (4-13 ii) and (4-13 iii). This same method can be applied if instead the pair of equations (4-13 i) and (4-13 ii) or the pair of equations (4-13 i) and (4-13 iii) is used. Solving equations (4-13 ii) and (4-13 iii) for c gives 87 h(4) - (,l.-b) 1“ c a (4-14) and h(S) - (l-b) 1’5 c = , respectively. (4-15) b Equating (4-14) and (4-15) gives h(4) - (1 -»b) 1/4 h(5) —.(1 -)b) 1/5 b b which must now be solved for b. There is, however, no algebraic method for directly solving equation (4-16) for b. Hence, numerical analysis techniques must be used. (e.g., Newton-Raphson method [Froberg, 1965; Thomas & Finney, 1979]). From the preceding discussion the general approach should be apparent. First, the values of the h(tk)'s; k = 2,3, ... , p are determined using equation (4-9). A system of equations analogous to (4-12) or (4-13) is then generated which relates the values of the h(tk)'s to the parameters involved in the general expression for h(t). This system of equations is then solved for the parameters using algebraic techniques (when possible) or numerical analysis methods. The minimum number of pretest time points required in order to obtain the values of the unknown 88 parameters is mh + l, where mh is the number of unknown parameters in the functional form expression for h(t). By substituting the computed values of the parameters into the functional form expressions for each of the hj(t)'s, the exact natures of the hj(t)'s are determined. For Case 9, where the functional form expressions of the gj(t)'s are known, and for Case 11, where nothing is known about the gj(t)'s, the values of the gj(t)'s are determined using the method described for Cases 3 and 5. For Case 7, the gj(t)'s are already known. Knowing both the gj(t)'s and hj(t)'s, the aj(t)'s are determined by oj(t) = Yij(t) - Igj(t)-Yij(0) + hj(t)]. where i is any individual from the jth_group. As with Cases 1, 3, and 5, since assumptions about the gj(t)'s are unnec- esary and may even lead to incorrect values for the aj(t)'s, it is recommended that for Cases 7 and 9 the assumptions on the gj(t)'s be ignored and that these cases be treated as if they were part of Case 11. Since the determination of the values of the treatment effects is done separately for each group, the groups in a design need not all belong to the same case. All that is 89 necessary is sufficient information to place each of the J groups into one or the other of Cases 1, 3, 5, 7, 9, or 11. Cases 13, 15, and 17 For Cases 13, 15, and 17, hj(t) E h(t) with h(t) un- known and no errors of measurement are present in the data. For these three cases only differences in treatment effects can be calculated since the h(t) and aj(t) terms are con- founded. For Case 13 the exact natures of the gj(t)'s are known, while for Case 15 only the functional form expres- sions for the gj(t)'s are known and for Case 17 nothing is known about the gj(t)'s. For all three cases the system of equations (1-4) can be rewritten as Yij(t) = gj(t)°Yij(tlj) + h(t) + aj(t) . (4'17) Case 13 Pick one person from group j, say person i, and one person from group j', say person i', where j and j' are two distinct treatment groups. Then by equation (4-17), for the i'Eh person from group j', Yi'j'(t) = gjs(t)°Yitjl(tlj') + h(t) + aju(t) . (4‘18) 90 Subtracting equation (4—18) from equation (4—17) yields 3 gj(t).Yij(tlj) + (1)-(t) -' gjl(t) 'Yiojl(tlj') - aja(t) . Hence, “j (t) - aj,(t) 3 Yij(t) - Yiljl(t) ’ [gj(t)°Yij(tlj) - gjl(t) 'Yiujl(tlj')]o So, the exact value of aj(t) — aj,(t) can be determined by taking a subsample of size equal to one from each of the groups. Notice that no interval estimation or hypothesis testing procedures are needed since the exact values of the aj(t) - aj,(t)'s have been determined. Cases 15 and 17 Define new functions, Hj(t)'s, by Hj(t) = h(t) + aj(t). Then, for these two cases the system of equations (4-17) can be rewritten as Yij(t) = gj(t)°Yij(tlj) + Hj(t) . (4-19) Cc Si f1 ti 91 Consider two individuals from the jth group, say individuals 1 and 2. Then, by equation (4-19) Y1j(t) = gj(t)-Ylj(tlj) + Hj(t) and (4-20) ‘Y2j(t) = 9j(t)'Y2j(tlj) + Hj(t) . Solving the system of equations (4-20) for gj(t) and Hj(t) in terms of Y1j(tl ), Y2j(tl.)' Ylj(t), and Y2j(t) yields 3 J Ylj(t) -'Y2j(t) gj(t) = and Y2j(t)°Ylj(tlj) — Y2j(tlj)-Ylj(t) H(t) = . (4-21) Similarly, for group j', consider a subsample of size two from that group, say individuals 1' and 2'. By a deriva- tion analogous to that for group j, 92 Yltjn (t) "’ thjt (t) gj'm ( ) ( ) Y . . .t - Y ... t and (4-22) Yz'j"t"Y1'3'(tlj'i " Yz'j"tlj-’°Y1'j"t’ Hj.(t) Y1.j,(tlj') - Yz'j'(t1j.) Subtracting equation (4-22) from equation (4-21) yields 1j(t1j) - Y2j(t1j) -Y1j(t) .._. (4-23) Y2,j,(t)°Y1,j,(tl Y2j(t)°Y )"Yc 0(t )°YI°I(t) j. 2 j lj. 1 J chjt(tlj') "' qujv(tlj') But, by definition Hj(t) - Hju(t) [h(t) + oj(t)] - [h(t) + aj.(t)] SO, Hj(t) — H.,(t) J 03. (t) - aj , (t) . (4-24) 93 Consequently, by equating equations (4-23) and (4-24) Y2j(t)'Ylj(tlj) - Y2j(tlj)°Ylj(t) Y .(t ) -‘Y .(t ) l] lj 23 lj YZ'j' (t) 'Y10j1(tlj') - Y20j1(tlj')'ylcju(t) Yl'j'(t1ju) _ Yz'j'(tljc) Since this is the exact value of aj(t) — aj.(t), no interval estimation or hypothesis testing procedures need be dis- cussed. Notice that here, as with Cases 3 and 5, there is no advantage in having knowledge of the functional forms of the gj(t)'s. Contradictory Solutions Each of the methods for computing the aj(t)'s con- sidered thus far has required data for only two subjects per group, except for Cases 1 and 13, where only one sub- ject per group was needed. If the conditions describing the cases hold exactly, then the solutions for the aj(t)'s will be invariant across pairs of subjects. Thus, if using different pairs of subjects yields contradictory solutions 94 for aj(t), then one or.more of the conditions describing the cases must be false. That is, either (i) Errors of measurement are present in the data ; (ii) There is not a correlation of +1 between true scores at two points in time within group j ; and/or (iii) For Cases 1, 3, 5, 7, 9, and 11, depending on the case, either the exact nature or the general func- tional form of hj(t) has been misspecified. Also, when discussing the methods for Cases 7, 9, and 11 it was mentioned that different, and hence contradictory, solutions may arise when different combinations of time points are used for the pretest observations. The possible reasons for these different solutions are the same as those just listed. The problem, however, is that one does not know which of the reasons caused the contradiction. If only reason (i) is the cause, then the results to be derived in Chap- ters 5 and 6 should be used. If only reason (iii) is the cause, then the correct exact nature or functional form expression for hj(t) must be determined. If both reasons (i) and (iii) are the case, then the results of Chapters 5 and 6 should be used, but first the correct expression for hj(t) must be determined. If reason (ii) is the case, then the methods developed in this dissertation are not 95 applicable. It should be noted here that the methods of data analysis developed by Strenio, Bryk, and Weisberg (Bryk, Strenio, and Weisberg, 1981; Strenio, Weisberg, and Bryk, in press) seem very promising for some of the situa- tions where there is not a correlation of +1 between true scores . CHAPTER 5 POINT ESTIMATION WHEN ERRORS OF MEASUREMENT ARE PRESENT In this chapter point estimation procedures for those cases where errors of measurement are present in the data are discussed. The interval estimation and hypothesis testing procedures for these cases will be given in Chapter 6. Case 2 Estimators For Case 2 the exact natures of both the gj(t)'s and hj(t)'s are known and errors of measurement are present. Recall that the general growth model is * * Yij(t) = gj(t)°Yij(tlj) + hj(t) + oj(t) and (1—4) at Yij(t) = Yij(t) + eij(t) . Taking the population mean on both sides of equation (1-4), uyj(t) = gj(t)°uyj(tlj) + hj(t) + aj (t) . (5-1) 96 97 Hence ' aj(t) = uyj(t) — Igth)-uyj(tlj) + hj(t)1 . (5-2) If ”Y.(tl.) and qu(t) are known then equation (5-2) gives 3 J 3 aj(t) exactly. If, as is usually the case, the population means are not known, then for a wide variety of statistical distributions, a point estimator of aj(t) is given by a"~ aj(t) = Yth) — [gj(t)°Yj(t1j) + hj(t)] . (5-3) Overview of the Procedures for the Remaining Cases Each of the procedures to be discussed is a variation of a several stage method. At the first stage, for Cases 4, 8, 10, 12, 14, and 16, estimates are obtained for the unknown constants (i.e., parameters) in the functional form expressions for the gj(t)'s and hj(t)'s. The method used to obtain the estimates of the parameters in the func- tional form expressions is the same whether the parameters for the gj(t)'s, the hj(t)'s, or for both are being esti- mated. Hence, the discussion of the first stage will be done in general. The second stage concerns the estimation of treatment effects (i.e., the oj(t)'s in equation (1-4)) 98 given the estimates obtained at the first stage and the particulars of the case of interest. Hence for the second stage, each of the cases must be discussed separately. For Cases 6 and 18, the first and second stages are replaced by a process which directly estimates the ej(t)'s. The third stage concerns methods for interval estimation and hypothesis testing of both treatment effects and differ- ences in treatment effects. These third stage methods are the same for all the cases, once estimates of the a.(t)'s have been obtained, and are described in Chapter 6. 3 Stage 1: Estimation of the gj(t)'s and hj(t)'s Stage 1 is divided into two substages. At the first substage, estimates of the gj(tk )'s and hj(tk )'s are 3' 3' obtained, where the tk 's; k = 1,2,3,...,pj are the times 3' of the pj pretests for the jth group. At the second sub- stage, these estimates of the gj(tk )'s and hj(tk )'s are 3' 3' used to obtain estimates of the unknown constants in the functional form expressions of the gj(t)'s and/or hj(t)'s. For simplicity, the j subscript will be dropped since the estimation of the unknown constants in the functional form expressions for the gj(t)'s and hj(t)'s is done separately for each of the J groups. Dropping the j 99 subscript and considering only the pretest time points, equation (1-4) becomes * * Yi(tk) = g(tk)°Yi(tl) + h(tk) (5-4) and * Some of the decisions made at substage l are dependent upon the context of what is to be done at substage 2. Since the second substage provides the motivation for the methods chosen to complete the first substage, it will be discussed first. Substage 2: Estimation of the unknown parameters in g(t) and h(t) Once the substage 1 estimates of the g(tk)'s and h(tk)'s are calculated, estimates of the unknown parameters in the general functional form expressions for g(t) and h(t) can be computed. For simplicity, the discussion here will be in terms of g(t). The procedures to be used for h(t) are analogous. The second substage of Stage 1 begins by setting up a system of equations similar to the systems (4-12) and (4-13) which relates the estimates of the g(tk)'s to the 100 unknown parameters. For example, if g(t) = logc[b°(t — t1) + c] then the system of equations would be ) g(tz) = logc[b'(t2-tl) + c] , z"* g(t3) = logclb'(t3-tl) + C] r . (5-6) A g(t ) = logCIb'(tp-tl) + c] . In general, define mg to be the number of unknown parameters in the general expression for g(t). If p :_mg then estimates of the unknown parameters can not be found. If p = mg + 1 then the system of equations is solved for the unknown parameters using the appropriate algebraic and numerical analysis techniques, as was done in Chapter 4. It should be remembered, however, that the resulting answers for the unknown parameters in the general expres- sion of g(t) are now estimates of the parameters rather than their exact values. When p > mg+1 the method of least squares can be used to provide estimates of the parameters. The methods de- veloped in Chapter 4 for the situations where p was greater than mg can not be used here because errors of estimation 101 are present. Once errors of estimation are present, sys— tems of equations such as (5-6) usually become contra- dictory simply because of the presence of these errors and not for the reasons discussed at the end of Chapter 4. The method of least squares resolves this contradiction by finding the estimates for the unknown parameters which minimize the inconsistency. For example, consider g(t) = logc[b°(t-tl) + c] and let p = 5. Let 57:3), 57:37: 572:): ”*\ and g(ts) denote the estimates of the g(tk)'s which were computed using one of the substage 1 methods to be described later in this section. The quantity, 5 ‘,a\ 2 L =k:2[g(tk) - logc(b‘(tk-tl) + c)] represents the inconsistency, in the least squares sense, inherent in the system of equations (5-6). The method of least squares finds the values of b and c which minimize L. These values are considered to be the estimators of b and c. In general, for any differentiable function g(t), the estimates of the unknown parameters are found by find- ing the vector of values for the parameters for which the minimum value of the expression a"~ L =k§2[g(tk) - g(tkn2 occurs . 102 Notice that the method of least squares can be used to find estimates of the unknown parameters whenever p is at least two greater than the number of unknown parameters. Further, in general, the precision of least squares esti- mates improves as the number of pieces of available infor- mation used increases. Here, the pieces of information are the g(tk)'s; k a 2,3,..., p. Hence, the greater the number of pretest time points, the better the precision of the estimators of the unknown parameters. Substage 1: Estimation of the g(tk)'s and h(tk)'s Equation (5-4) represents a linear structural (Madane sky, 1959; Moran, 1971) or functional relation (DeGracie & Fuller, 1972; Lindley, 1947), where g(tk) and h(tk) are * the slope and intercept of the Y*(tk) on Y (t1) regression line. The general problem of how to estimate the slope and intercept of a linear structural relation is known as the errors-in-variables problem and has been widely dis- cussed in the literature, especially in the area of econo- metrics (e.g., Johnston, 1972; Madansky, 1959; Moran, 1971; Sprent, 1966). The econometrics literature deals mostly with those situations where there is only one independent variable and one dependent variable. 103 The maximum likelihood structural equations approach developed most notably by J6reskog (Goldberger & Duncan, 1973; J6reskog, 1969; J5reskog, 1977; Magidson, 1979; W. Schmidt, 1975), however, allows for any number of inde- pendent and dependent variables. Also, as has just been shown, in order to complete the second substage of the process for determining the unknown parameters, it is necessary to have at least one more pretest time point than the number of unknown parameters. So, if the func- tional form under consideration has two or more parameters, then the necessary number of pretests is three or greater. For these situations, only the maximum likelihood approach can be used. Further, since the more pretests used, the better the precision of the substage 2 estimators, it is recommended that pretest observations be collected at as many time points as possible. It is realized that the number of pretest time points is, however, constrained in ' educational settings by the amount of money and investi- gator time available. Also, in most settings, the number of tests is constrained because the subjects can only take a certain number of tests without either reactivity, fatigue, or attrition occurring. Finally, as will be shown later in this section, when only two pretest obser- vations are available, additional assumptions must be made before estimates of the g(tk)'s and h(tk)'s can be deter- mined. Hence, it is recommended that observations always 104 be collected at three or more pretest time points, for those cases where Stage 1 is implemented, and that maximum likelihood estimation be used to estimate the g(tk)'s and h(tk)'s. Since the maximum likelihood approach is appli— cable in a wider variety of situations than the other approaches, it will be discussed first. Maximum Likelihood Approach For a design with p pretests, the system of equations represented by equations (5-4) and (5-5) can be described pictorially as in Figure 19. * Y (t2) 0 e(t2) a 2 (t3) { e(t3) /*“4’ 3* “ “‘4’ i 0 . o efla)-———-%’Y (t1) ‘ \ * \ 2 (tp_1)q e(tp_1) _ * _ . Y (t )‘(E e(tP) Figure 19. Pictorial representation of the structural relation. 105 The system can be written in vector form as * * Yi(tk) = g(tk)-Yi(tl) + h(tk) (5-7) and * Yi(tk) = Yi(tk) + ei(tk) . (5-8) Maximum likelihood requires expressions for the means, variances, and covariances of the observed variables under consideration. Taking the mean on both sides of equation (5-7) uYItk) = g(tk)°uY(tl) + h(tk) (5-9) I where u (t ) is the vector of means. The variance of Y(t ) .Z_;E_ 1 is given by 03(t1) = [0;(t1)12 + 02(t1) . (5-10) where 02(t) represents the variance of the errors of mea- surement at time t. The variance of Y(tk) is given by 2 _ 2, * 2 2 _ 0Y(tk) - [g(tk)] [0Y(tl)] + 06(tk) . (5 11) for k=l,2, ... , p. The covariance of Y(tk) and Y(tl) is given by °Y 0, as is being assumed for the growth models under consideration here, then maximum likelihood estimates of g(tz) and h(tz) exist and are given by z"\ g(tz) = (t) —Asz(t) + ‘52 (1:) Wu) +4HS )2 2 y 1 2 N K 1 Y(tl)Y(t2) ZS Y(tl)Y(t2) and ’f‘\ ‘ 2 oe(t2) where l = ————————-. Appendix B provides a derivation of o2 e(tl) /\ A . . g(tz) and h(tz) under Assumption 4. Appendix B also pro- vides derivations of the maximum likelihood estimates under Assumptions 5 and 6, since a search of the literature did not reveal any place where the estimates under these assumptions are discussed. 113 The maximum likelihood estimate of g(tz) under Assump- tion 5 is SY(t2) SY(tl) Under Assumption 6, 2 SY(t2) g(t ) = -p . 2 s Y(t2)Y(t2) Y(tl)Y(t2) 1"~ a"~ Under both assumptions, h(tz) = Y(tz) -g(t2)'Y(El) . Other Approaches When p a 2, there are also approaches other than maxi- mum.likelihood available to estimate g(tz) and h(tz). When an independent estimates of 02(t1) is available DeGracie and Fuller (1972) have suggested using 114 ’/"~ SY(tl)Y(tk) g(tk) = 255,77 .zf””a~‘:‘\e 1 //’\‘ Mtl) SY (tl )Y (t1) + N- 2°2 e‘ti) + _,—+-'*-~1 S a a Y (t1)Y (t1) 4\ 2oe(t1) + A q.S * * Y (t1)Y (t1) as the estimator for g(tk), where N SY(1:1)Y(tk)"fi=l-'I £1”; <1: I - Y‘T‘MY “‘k’ “‘YT‘k“ ; //’~‘~ o:(t1) represents the independent estimate of 0: (t1) ; N SY(t1)Y(tl)' N=I 2 (Y 1(t1) ‘iYZt1)’2’ i=1 ’,,a/’”‘-.\~ S * * .- ~ /2\ {\ SY(tl)Y(tl) ‘ °2e(t1) if SY(t1)Y(tl) ‘ °e(t1) 1 2 > $171- Ge(tl) .x”‘~ N31 02 e(t1) otherwise: and q is the number of degrees of freedom for the x2 115 /\ distribution of which the distribution of o:(t1) is a multiple. The estimator for h(tk) can then be defined as /\ h(tk) = Y(t? — @«T‘tfi . Notice that DeGracie and Fuller's method is only con- cerned with two time points. Any number of pretests can be used, however, but the determinations of the estimates for g(tk) and h(tk) are done separately for each k; k=2,3, ... , p. Another approach for estimating the g(tk)'s is given by Spiegelman (1979). Spiegelman's approach requires a knowledge of real analysis on the part of the data analyst. Even though Spiegelman's technique is a possible method for estimating the g(tk)'s, it will not be discussed here because of its complexity. Several other approaches discussed in the literature are variations of the techniques of instrumental variables or method of grouping (Johnston, 1972; Madansky, 1959; Wald, 1940). The methodology of the instrumental variables and method of grouping techniques requires that the errors of measurement in the independent variable be statistically independent of the instrumental variable or grouping 116 variable. As has been pointed out by P. Schmidt (1978) and Madansky (1959), unless there is information available besides the pretest and posttest scores, the errors of measurement will be correlated with any instrumental or grouping variable used. Hence, these methods should not be used in the data collection situations considered here. The discussion so far in this chapter has been focused on single-group designs. For multi-group designs the methods developed here can be applied to each group separately to find the estimates of the unknown parameters in the expressions for the gj(t)'s and hj(t)'s. Treating each group separately will suffice except in those instances where it is assumed that gj(t) E gj,(t) and/or hj(t) E hj,(t) for two distinct groups, j and j'. If it is assumed that 92)“) E gj'(t)l 113-(t)E hjc(t)r Pj a Pjur and tkj = tk,' for k 3 1:2: --- , Pj(=Pj.), then the two groups j and J j' should be combined and considered as one group for the purposes of this first stage. If gj(t) gj.(t), but hj(t) i hj.(t), and still pj = pj, and tkj = tkj' for k = 1,2,...,pj , then the two groups should be combined when A A . the gj(tk)'s (and hence, the gj,(tk)'s) are determined and when the estimates of the unknown parameters in the expres- sion for gj(t) (and hence, gj.(t) ) are calculated. Call 117 . A the common estimate of gj(tk) and gj.(tk) by gj,j'(tk) , .r"\~ for k a 2,3,..., Pj' Then determine the values of hj(tk) ,1’"‘~ and hj'(tk) separately using a/"\ a”"“‘~ hj(tk) = Y57tk) - gj'j.(tk)-?;Tti) (5-22) and A A The values of the unknown parameters in the expressions for hj(t) and hj.(t) are computed separately using the values of the estimates given by equations (5-22) and (5-23), respectively. If gj(t) E gj.(t) and if for some k, tk. # tk.. then 3 J the methods just described can not be used. Methods need to be developed which will insure that the estimates of the unknown parameters in the expressions for gj(t) and gj,(t) are the same in these situations. The development of these methods is left as an open question. Further, if gj(t) Z gj.(t) and hj(t) E hj,(t), something should be done to insure that the estimates of the unknown parameters in hj(t) and hj,(t) are the same. How this should be accom- plished is also left as an open question. 118 Stage 2: Point Estimation of Treatment Effects The discussion of the determination of the point estimators of treatment effects will be discussed separately for each case, since the process is slightly different in each case. Case 4 In Case 4 the exact natures of the hj(t)'s are known but only the functional forms of the gj(t)'s are known. For this case, there are two methods possible for determining z/“\ . the aj(t)'s. For the first method, recall that at Stage 1, estimates were obtained for the unknown constants in the functional form expressions for the gj(t)'s. For the jth group, let gj(t) denote the function formed by substituting these estimates of the constants into the general functional form expression for gj(t). For example, if gj(t) = 0‘) logCIb-(t - t1) + c], b = 1.2, and c = 4.17, where and c are the Stage 1 estimates of b and c, then gj(t) log4 l7[l.2(t — t1) + 4.17]. Point estimators of treatment effects are then given by fi 33' (t) = Yj(t) - [gj(t)‘Yj(tlj) + hj(t)] . 119 Define mg to be the number of unknown parameters in the 3' functional form expression for gj(t). The method just described will work only if p3. > mg . J' The choice of Yj(tl ) as the pretest to be used as the J' exogenous (i.e., independent) variable for estimating aj(t) was arbitrary. The general growth model could have been stated alternatively as t * h 2 Yij(t) - gjk(t) Yij(tkj) + jk(t) + aj(t) , (5- 4) where the gjk(t)'s and hjk(t)'s are some continuous func- tions and where k is set equal to either 1, 2, ... , pj-l’ or pj. Notice that equation (1-4) is the special case of (5-24) when k = 1. For a fixed k, then, Y;(tk ) can be . i thought of as the exogenous variable and the remaining * * t * pretGStS Yj(tlj)l Yj(t2j)' on. ' Yj(tk-1j)' Yj(tk+lj)l * ... , Yj(tp ) can be thought of as the endogenous variables 2') in a structural equations causal model. It is now possible to generate pj different point estimates of aj(t), one from each of the pj different versions of equation (5-24), where each time a different pretest becomes the exogenous variable. 120 It is left as an open question as to how point estimators of the aj(t)'s can be developed, which are improvements in the sense of increased precision over the pj separate estimates, by using some function, such as the mean or median, of the pj separate estimates of aj(t). For the second method, define a new variable, Wij(t) = Yij(t) - hj(t). The system of equations (1-4) can then be rewritten as Wij (t) * gj(t) wij(tlj) + aj(t) and (5-25) * Notice that the system of equations (5-25) is, for each j, a linear structural relation with a slope of gj(t) and an intercept of aj(t). Hence, for any particular time t, estimates of gj(t) and aj(t) can be obtained directly by using the techniques described for substage l of Stage 1, with Wj(t1 ) as the independent variable and Wj(t) as the 3' dependent variable. It should be recalled here that in order to compute the estimates in situations where there is only one independent and one dependent variable, it is necessary to make one of the assumptions listed in substage l 121 (or some other assumption that will make the system iden- tified). These direct estimates of the aj(t)'s are considered to be the Stage 2 estimates and are labelled as the‘5;(t)'s. Notice that Stages 1 and 2 are combined for this method. This is possible because the exact natures of the hj(t)'s are known. As with the first method, the use of Wj(tl ) as the 3' independent variable is arbitrary. Rewriting equation * (5-25) in terms of W.(tk ) yields 3 j 'k * Wij(t) = gjk(t)-Wij(tkj) + aj(t) . (5-26) * Equation (5-26) is a linear structural relation with Wj(tk ) 3' as the independent variable and w;(t) as the dependent variable. Hence, pj different estimates of aj(t) can be calculated, one for each of the pj pretests, using the techniques described for substage l of Stage 1, once one of the additional assumptions is made. As with the first method, it is left as an open question as to how point estimators of the aj(t)'s can be developed, which are improvements, in the sense of increased precision, over the pj estimates of aj(t) generated separately using only one 122 pretest at a time, by using some function, such as the mean or median, of these pj different estimates. Since two methods have been proposed when Case 4 holds, the question of when each method is appropriate needs to be discussed. The first method has the advantage over the second method in that no additional assumptions need be made in order to implement it. The disadvantages of the first method are that it is computationally more complex than the second method and that a minimum of mg + l J pretest time points is needed. Hence, when observations are available for at least mg + 1 time points and the data j analyst is not willing to make any additional assumptions, the first method must be used. Further, when mg or fewer j pretest time points are available then the second method must be used and one of the additional assumptions made in order to insure the identifiability of the system. When there are at least mg + l pretest time points available 1 and one of the additional assumptions seems reasonable, then a choice must be made between the two methods. The method of choice should be that method which leads to the smallest standard error of estimate (i.e., the better precision). It is left as an open question as to which method possesses better precision. 123 Case 6 In Case 6 the exact natures of the hj(t)'s are known and nothing is known about the gj(t)'s. For this case, the 'aj(t)'s are determined using the second method described for Case 4 situations. Case 8 In Case 8 the exact natures of the gj(t)'s are known but only the functional forms of the hj(t)'s are known. For this case estimators of the hj(tk )'s are given by J ,r"‘~ ___—___ ‘,/~\_ hj(tkj)_= Yj(tkj) — gj(tkj)-Yj(tlj). The hj(tkj)'s are then used, exactly as was described for the second substage of Stage 1, to provide estimates of the unknown parameters in the functional form expressions of the hj(t)'s. A new function, called hj(t), is formed by substituting the estimates of the unknown parameters that were obtained in Stage 1 into the functional form expression for hj(t). Recall that in order for this method to be implemented it is necessary to have at least mh + 1 pretest time points, 3' where mh is the number of unknown parameters in hj(t). j 124 Point estimators of treatment effects are then given by z/‘\- A aj(t) 8 Yj(E) — [gj(t)-Yj(tlj) + hj(t)] . Case 10 In Case 10 the functional forms of both the gj(t)'s and hj(t)'s are known. The point estimation method for this case begin by using substage l of Stage 1 to generate /\ . the hj(tk)'s. Substage 2 is then used to find the estimates of the unknown parameters in the functional form expres- sions for the hj(t)'s. The hj(t)'s are then formed by substituting the estimates of the unknown parameters into the functional form expressions for the hj(t)'s. The method then proceeds as in Case 4, except that the hj(t)'s of Case 4 are replaced by the hj(t)'s. Case 12 Case 12 occurs when only the functional forms of the hj(t)'s are known and when nothing is known about the gj(t)'s. In this case a new variable is formed by defining Uij(t) = Yij(t) — hj(t), where hj(t) is defined as in Case 125 10. The method described for Case 6 is then used to obtain the 5;?t)'s by replacing the Wij(t)'s of Case 6 with the Uij(t)'s. In developing the point estimators of the treatment effects it was assumed that for any particular nonequiv- alent control group design, the known information about the growth curves for each of the J groups belonged to the same case. Since the determination of the point estimators is done separately for each group, insisting that all of the groups belong to the same case is overly restrictive. Hence, it should be assumed that for each of the J groups, the known information about the growth curves allows the data analyst to place each of the groups into one of the o O A cases 2,4,6,8,10, or 12. Once this is done, the aj(t)'s are derived separately for each group by using the methods given in this chapter for the case to which the group's growth curves belong. Case 14 For Case 14 the exact natures of the gj(t)'s are known and hj(t) E h(t), with h(t) unknown. In this case, the system of equations (1-4) can be rewritten as 126 i * Yij(t) = gj(t)'Yij(t1j) + h(t) + aj(t) and (5-27) * For this case, and for Cases 16 and 18, only differences in treatment effects can be estimated since the h(t) and aj(t) terms are confounded. Taking the population mean on both sides of equation (5-27) yields uyj(t) = gj(t)'uyj(tlj) + h(t) + Gj(t) . (5-28) Similarly, for any other group, j', uyj (t) = gj.(t)'uyj'(tlj') + h(t) + aj.(t) . (5'29) Subtracting equation (5-29) from equation (5-28) yields “Y.(t) ' “Y. (t) = J 3' Hence ' aj(t) " ajl(t) = HY (t) -’uY '(t) - [gj(t)uy_(tl ) J :3 3 3 gj.(t)qu'(tlj')] . (5-30) 127 If the values of “Y.(tl ), “Y. (t1. ), “Y.(t)’ and “Y. (t) J j 3' 3' J J' are known, then equation (5-30) determines the value of .,(t) exactly, and no interval estimation or aj(t) —'a3 hypothesis testing procedures need be discussed. It is, however, rarely the case that uY (tl ). "Y (tl ). “Y (t), j j j' j' j and "Y '(t) are known. When these populations means are 3 unknown, a point estimator of aj(t) — aj,(t) is given by ,z””'~““~ aj(t) _ aj|(t) a 3' Case 16 For Case 16 the functional forms of the gj(t)'s are known and hj(t) a h(t), with h(t) unknown. Define Hj(t) to be h(t) + aj(t). The system of equations (5-27) can then be rewritten as 'k * Yij(t) = gj(t)°Yij(tlj) + Hj(t) and (5-31) * Yij (t) = Yij (t) + eij(t) 128 The two methods described under Case 4 for estimating aj(t) can now be used here to estimate Hj(t), simply by replac- ing aj(t) by Hj(t) in the Case 4 discussion. Once the A o o D Hj(t)'s have been determined, point estimates of the aj(t) — aj,(t)'s, for any two groups j and j', are formed by using /\ /\ Case 18 For Case 18 nothing is known about the gj(t)'s and hj(t) E h(t). The second method for estimating aj(t) under Case 4 can be used here to estimate each of the Hj(t)'s in the system of equations (5-31), simply by replacing the aj(t)'s by the Hj(t)'s in the Case 4 discussion. Once the A Hj(t)'s have been determined the point estimates of the differences in treatment effects are determined as in Case 16. Bias of the Point Estimators The point estimation techniques introduced in this chapter lead almost always to biased estimates of treatment effects. The discussion of this bias will be broken down into three categories according to the information available 129 about the gj(t)'s. The cases where the exact natures of the gj(t)'s are known will be discussed first, followed by those cases where the functional forms of the gj(t)'s are known and then by those cases where nothing is known about the gj(t)'s. Cases 2, 8, and 14 comprise those cases where the exact natures of the gj(t)'s are known. For Cases 2 and 14, the formulas given in the previous section provide unbiased estimates of treatment effects and differences in treatment effects, respectively, since for all t, E(Yj(t)) = “Y (t) 3' under a wide variety of statistical distributions. For Case 8, however, the aj(t)'s almost always provide biased estimates of the aj(t)'s. The Case 8 estimation method begins by finding estimates for the hj(tk )'s, where the 3' tk 's are the pretest time points. These estimates are :i . /\ -—___ — . given by hj(tk.) = Yj(tk.) - gj(tk.)'Yj(t1.)' Notice that J J J J A these hj(tk )'s are unbiased estimates of the hj(tk )'s j 3' under a wide variety of statistical distributions. In order to find estimates for the parameters in the functional form expressions of the hj(t)'s, a system of equations 130 analogous to (5-6) is set up which relates the hj(tk )'s j to the unknown parameters. This system is then solved for the unknown parameters. Hence, expressions are obtained which are estimates of the unknown parameters in terms of the hj(tk )'s. That is, estimates of the unknown parameters 3' are given as functions of the letk.)'s and gj(tk.)'s; J J k = 1,2,...,pj. Even though the sztk )'s are unbiased 3 estimators of the "Y (tk )'s, it is rarely the case that j 1 functions of the sztk )'s are unbiased estimators of the 3' corresponding functions of the “Y (tk )'s (Bickel & Doksum, J J W) uyj(t1j) 1977). For example, E ——————— # -————-—— , when a bivar- Y.(t ) u (t ) 2. Y. 2. 3 3 3 3 iate normal distribution is assumed for Yj(t1 ) and Yj(t2 ) j 3' (Cochran, 1977). Hence, the estimates of the unknown parameters are usually biased. Therefore, the hj(t)'s, which were formed by substituting the estimates of the un- known parameters back into the functional form expressions of the hj(t)'s, are biased estimates of the hj(t)'s. Con- sequently, the aj(t)'s, as defined by, 131 A A aj(t) = Yj(t) — [gj(t)-yj(tlj) + hj(t)]. are almost always biased estimators of the aj(t)'s. Since the aj(t)'s are functions of the letk )'s, by 3 taking limits as the sample sizes approach infinity, one can see that the aj(t)'s are, however, consistent esti- mators, provided that the limits exist. It should be pointed out here that it is an accepted technique in the applied statistics literature when a point estimator is desired of a quantity which is a function of several parameters, to first find unbiased, or even just consistent, point estimators of the parameters. These estimates of the parameters are then substituted into the original function to yield a useable point estimate of the desired quantity. 2 °Y SY For example, a point estimate of is given by ———— . OX S2 X Since in many instances in the past literature the tech- niques used for Case 8 lead to point estimators with an acceptable amount at bias, it is conjectured that the Case 8 techniques will also lead to point estimates with an accept- able amount of bias. Further study of this bias is needed. For Cases 4, 10, and 16, where only the functional form expressions for the gj(t)'s are known, maximum likelihood 132 techniques are first used to generate estimates of the gj(tk )'s. It is well known that maximum likelihood tech- 3' niques often lead to biased estimation (Bickel & Doksum, 1977: Mbod, Graybill, and Boes, 1974). Maximum likelihood estimators are, however, usually consistent (Patel, Kapadia, & Owen, 1976). Further, maximum likelihood esti- mation is probably the most widely used and accepted esti- mation technique, since it has been found in a wide variety of situations to produce estimators which have a negligible amount of bias and which are asymptotically efficient when compared to a large class of possible estimators (Zacks, 1971). The maximum likelihood estimates of the gj(tk )'s are 3' then used to generate estimates for the unknown parameters in the functional form expressions of the gj(t)'s, in a manner analogous to that just described for finding esti- mates of the unknown parameters in the expressions for the hj(t)'s in Case 8. Further, for Case 10, maximum likeli- hood estimates of the hj(tk )‘s are generated using 3' /\ /\ = —- o ' hj(tkj) Yj(tkj) gj(tkj) Yj(t1j), where the gj(tkj) s are the maximum likelihood estimates of the gj(tk )'s. 3' These maximum likelihood estimates of the hj(tk )'s are 3' 133 then used to replace the unbiased estimates of the hj(tk )'s 3 used in Case 8, and the hj(t)'s are then generated as in Case 8. Hence, for Cases 4, 10, and 16 maximum likelihood estimation along with the techniques from Case 8 are used to generate estimates of treatment effects and differences in treatment effects. Consequently, these estimators are usually consistent estimators. Since both the maximum likelihood estimation and the Case 8 techniques have been found to lead to acceptable levels of bias, it is conjec- tured that a combination of these two methods will still lead to estimators with an acceptable amount of bias. The nature of the bias needs to be studied further. For Cases 6 and 18 maximum likelihood techniques are used to directly arrive at estimates of the treatment effects and/or differences in treatment effects. Case 12 combines the maximum likelihood techniques used in Case 6 with the techniques used in Case 10 to find the hj(t)'s. The hj(t)'s are then used as in Case 8, to find estimates of the treatment effects and differences in treatment effects. Hence in these three cases the estimators used are also almost always biased, although usually consistant, estimators. In these cases, also, the nature of the bias of the estimators still needs to be investigated. For these cases, as well as for Cases 4, 8, 10, and 16 computer 134 simulation techniques appear to be the only feasible method for further study of the bias. CHAPTER 6 INTERVAL ESTIMATION AND HYPOTHESIS TESTING PROCEDURES In Chapter 5 methods were developed for the point esti- mation of treatment effects for those cases where errors of measurement were present and where either the exact natures or the functional forms of the hj(t)'s were known. The beginning of this chapter is concerned with the interval estimation of treatment effects and with the testing of the hypotheses of nonzero effects (i.e., Ho: aj(t) = 0 versus H1: aj(t) # 0). Next, procedures for the interval estimation and hypothesis testing of differences in treat- ment effects are developed. In order to develop interval estimates and hypothesis testing procedures it is necessary to have estimates of the variance of the Egjtx's. Since the probability distribu- . z”‘\ tions of the aj(t)'s are unknown, except in Case 2, tradi- tional methods can not be used to estimate the variances a’”‘~ of the aj(t)'s. Two techniques which have been suggested in the literature for estimating variances when nothing is known about the probability distributions are the 6-method (Bishop, Feinberg, and Holland, 1975) and jackknifing (Tukey, 1958). The 6-method quickly becomes computationally 135 136 intractable in many cases. Moran (1971) has shown that this intractableness occurs for the growth models and de- signs being considered here when even only two pretest observations are available. Because of its computational difficulties, it was decided not to use the 6-method to a"\ find estimates of the variances of the aj(t)'s. The method of jackknifing, however, avoids the computational diffi- culties inherent in the 6-method. Further, jackknifing was originally designed as a bias reduction method (Quenouille, 1956). Hence, besides providing estimates 0 A o I o o o of the variances of the aj(t)'s, jackknifing will also in most cases (Gray & Schucany, 1972) provide a reduction in the bias of the point estimators of the treatment effects and differences in effects. The technique of jackknifing begins by drawing a random sample from a specified population. Let N denote the number of subjects in the sample. The N subjects are then divided N into m disjoint subsets, each of size ——— . Let Y be the m parameter of interest and Y be an estimator of Y. Further, AT A let Y be the value of Y when all N subjects are used, and .(i) A let Y be the value of Y when the subsample of size N N - ———, where the £th_subset has been deleted, is used. m 137 Next define J£(Y) by AT A(1) J2”) = m-Y - (In-1W ; k=l,2,..u m and define J(Y) by A l . m A J(Y) = -—— 1: J (Y) . m i=1 2 An estimate of the variance of J(Y) is given by (Tukey, 1958) "Na Ugh?) — J(Y)]2 m — 1 Think of m as being fixed. Gray and Schucany (1972) have 5- [J(Y) - Y] shown that is asymptotically distributed JSJ N (as ———-—+ co) as a t-random variable with m —'1 degrees of m freedom. An open problem is to determine how m should be man?) -— v] chosen so as to allow to be distributed 2 SJ approximately as a Student's t random variable and at the same time allow for enough degrees of freedom so that the power of the test is not too low (Miller, 1974). 138 A The estimators of interest here are the aj(t)'s from Cases 2, 4, 6, 8, 10, and 12. It should be kept in mind here that once the aj(t)'s have been determined it is no longer important to consider which of the cases the growth curves belong to. To apply the jackknifing technique, first divide the Nj subjects from the jth_group into mj disjoint subsets. Next define a§£)(t) to be the value of al"~ the estimator, aj(t), when the ith subset is deleted from the sample for the jthgroup. Notice that the values of the estimates of the gj(tk)'s and hj(tk)'s change when the 2§h_subset is deleted, and hence also the values of the estimates of the unknown parameters in the expressions for gj(t) and hj(t) will change once the 25g subset is deleted. /\ Hence, in order to compute all of the a§£)(t)'s, the entire two-stage process described in Chapter 5 must be /\ repeated mj times for the jth group. Once the a§£)(t)'s have been computed, new estimators of the aj(t)'s are given by 139 J(@)) a-Tt—mj 3,671») . j 221 3 where (6-1) A /\ fl? J£(aj(t)) = mj-aj(t) - (mj - 1)'aj (t) . ,/’\~ These new estimators, the J(aj(t))'s, have two advan- A tages over the original estimators, the aj(t)'s. The first advantage is that in most situations the bias of the A J(aj(t))'s as estimates of the aj(t)'s is less than the a"\ bias of the aj(t)'s. There are, however, situations in a"~ which the bias of J(aj(t)) is greater than the bias of A ' aj(t) (Gray & Schucany, 1972). Gray and Schucany (1972) discuss various conditions under which jackknifing increases, decreases, or does not affect the bias of the estimator of interest. The conditions, however, demand knowledge of some of the properties of the distribution of the estimator. But, for the estimators of interest here (i.e., the aj(t)'s) nothing is known about their distributions, so one can not know for sure whether jackknifing increases, decreases, or does not affect their biases. 140 /\ /\ The second advantage of J(aj(t)) over aj(t) is that A /\ once the J£(aj(t))'s and J(cj(t))'s are computed, interval. estimation and hypothesis testing procedures are available by observing that af“\ 511-3” [J(aj(t)) - aj(t)] /)j is asymptotically distributed as a random variable with a Student's t distribution with m.‘— 1 degrees of freedom 3 z [J,> — J(a.(t))]2 2.21 3 3 where (53(t)). = . 3 m. — l 3 An approximate (l - a)% confidence interval for aj(t) (and hence, an a-level test for nonzero aj(t)) is then given by 2 . A a /12 k 2 9 k 9 2 Y 1 °Y(t2)Y(tl) g(t2)-[o;(tl)12 g(tk) for k=3,4,...,p . That is, 152 153 U Y(t3)Y(t2) g(t3) 3 I 0' Y(tz) Y(tl) O Y(t4)Y(t2) g(t4) = , I (A-3) OY(t2)Y(t 1) o Y(tP)Y(t2) g(tp) = O' Y(t2)Y(t1) By interchanging the roles of t2 and t3, it can be seen that c Y(t3)Y(t2) a Y(t3)Y(t1) Substituting equation (A-4) into equation (A-l) gives 0 Y(t3)Y(t2) * 2 o = . [o (t )1 . (A-S) Y(t2)Y(t1) 0Y(t3)Y(tl) Y 1 154 Solving equation (A-S) for [0;(tl)]2 yields 0 '0' * 2 Y(t2)Y(tl) Y(t3)Y(t1) O’ Next, solving the set of equations (5-9) for the h(tk) yields F W) - ‘ r W) h(tz) uY(t2) g(tz) h(t3) uY(t3) g(t3) = o — o 0UY(tl) . (A’7) h(t ) “Y(t ) g(t ) Substituting the system of equations (A—3) and equation (A-4) into the set of equations (A-7) yields 155 1r "'[ P ‘7 ‘ r"¥(t-.3)x(t2)mH h(tz) uY(t2) °Y(t3)Y(tl) °Y, ... , g(tp), h(t2)' h(t3). ... . h(tp), 2 2 2 2 . oe(tl), oe(t2), ce(t3), ... , and ce(tp) can be written in terms of the parameters on the left hand side, namely P(P + 1) 2 parameters in the variance-covariance matrix of Y(tl), Y(tz), Y(t3), ... , and Y(tp). Consequently, the system of equations (5-9) through (5-13) is identifiable whenever p Z 3- When p = 2 there are 6 parameters on the right hand side of equations (5-9) through (5-13) [namely, uY(tl), [0*(t )]2 (t ) h(t ) 02(t ) and 02(t )1 and 5 para- Y 1 ' g 2 ’ 2 ' e l ' e 2 meters on the left hand side [namely, uY(t1), uY(t2). 03(t1), 03(t2), and 0 Since the number of 1. Y(tl)Y(t2) parameters on the right hand side is greater than the num— ber on the left hand side the system is automatically under— identified. APPENDIX B APPENDIX B DERIVATION 0F MAXIMUM LIKELIHOOD ESTIMATES UNDER ASSUMPTIONS 4, 5, AND 6 Assumption 4: The ratio of 02(t2) to c:(t1) is known Let A represent the ratio of o:(t2) to 02(t1). Hence equation (5-20) can be rewritten as o§(t2> = [g(t2)]2'[0;(t1)]2 + A-c:(t (B-1) 1’ ' Solving the system of equations (5-17), (5-18), (5-19), (B-l), and (5-21) for g(tz) in terms of “Y(tl)’ uY(t2). OY(tl), GY(t2), and OY(t1)Y(t2) gives '2 ((1.12( (t2) — Aoy(tl) i ‘ 2 2 2 4/?% (t2) - Aay(t1)) + 4AIOY‘t1’Y‘t2’],) 20 Y(tl)Y(t2) 9(t2) = * Since g(tz) > o and [oY(t1)]2 > 0, then by equation (5-21) oY(tl)Y(t2) > 0. Consequently, 158 159 2 (02 Y(t2)- AaY(t1) + 2 2 2 K = > O 20 Y(t1)Y(t2) and (02(t)—Aa§(t)- Y 2 1 ‘ 2 2 2 2 J(cY(t2) — AoY(tl)) + 4)‘[°Y(tl)Y(t2)] ) K 2 < 0 o 20 Y(t1)Y(t2) Hence, since g(tz) > 0, K2 can never be a solution for g(tz). Consequently, Kl provides the unique solution for g(tz) in terms of uY(t1), uY(t2), 03(tl ), 02 Y(t2) and °Y(t1)Y(t ). Therefore, a one-to-one transformation exists between the set of parameters uY(t1), uY(t2), 02(t1), 2 0Y(t2), and °Y(tl)Y(t ) and the set of parameters ”Y(tl)’ [0*(t >12 (t ) h(t ) and 02(t ) as defined b Y 1 ' g 2 ' 2 e 1 Y “Y(tl) = uY(t1) : 160 2[°Y(t1)Y(t2)] * 2 . [oy(t1)] 2 . }( Y(t2)- AoY(t1) + *2 2 **2‘ 2 d/(OY(t2) - AoY(t1) + 4A[cy(tl)y(t2)] J) 2 2 (oy(t2)i— 10Y(t1) + 2 (OY(t2)W(t1) + 4A[0Y(tl)Y(t2)] .) g(tz) = ,1 ; 20Y(t )Y(t ) 1 2 (B-2) h(tz) = uY(t2) - g(t2)uY(tl) ; and 2[a ] 2 2 Y(tl)Y(t2) 2 (02 (t2)— AcY(tl) + 2 2 2 2 \/Q°Y(t2)" A°Y(t1)) + 4A[OY(t1)Y(t2)] L) One of the properties of maximum likelihood estimation is that if one set of parameters is related to another set of parameters by a one-to-one transformation, say f, so that f(2) = X where §_and 1 represent the first and second A sets of parameters respectively, then X = f(g) where g and A ‘2 represent the maximum likelihood estimates of g and g '161 (Mood, Graybill, & Boes, 1974). Under assumption 1, g = 2 2 _ * 2 2 (uY(t1), [0Y(t1)] , g(tz), h(tz), Oe(t2)). Assuming that the distribution of the vector (Y(ti), Y(t2)) is multi- variate normal, the maximum likelihood estimator of g is A 2 2 2 given by g = (Yltl), Yltz), SY(tl), SY(t2), SY(tl)Y(t2))' 2 2 . where SY(t1), SY(t2), and SY(t1)Y(t2) are given by equa- tions (5-14) and (5-15). Substituting 2 into the system of equations (8-2) gives the maximum likelihood estimates of 3. That is, //’\~ uY(t1) = YZtl) ; 2 ,/:"‘- 2[SY(t1)Y(t2)] [oyulnz = ; 2 2 (Sy(t2) - st(t1) + 2 2 2 2) J(SY(t2) - ASY(tl)) + 4l[Sy(tl)Y(t2)] 2 2 (SY(t2) - ASY(t1) + 2 2 2 (SY(t2) — XSY(tl)) + 4A[SY(t1)Y(t2)] > o I g(tz) [ 1 2 S Y(tl)Y(t2) 162 /\ /\ h(tz) = thZ) - g(t2)th1) 3 and ’2/\\ 2 ’/:’~\\‘2 oe(t1) = 52(t1)" [aY(t1)l . 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