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"1. 4%; '3‘ m“ -‘. l. : .-' .f' 1' ~ 1‘ Eu: "1-. _f:‘ i?" ‘ "$1.1“ "LII 1:}. f ‘ .T-‘J';. v', _-_3§,.t - _:.r" 4“” c- 2 Sr 46???" 92‘1“ 4. '. - ~x. ,, . *3“ 4 on E I- ‘ ' ‘Jcflym . .1 . 4 1'— ‘ _flL.."‘ :- "t: .1 ‘9‘5332 i’.’ v E: 4‘ .F- -'-. * “fr/.1.” .4-‘. A ‘IPC- .. 5" . ;?-1.¢,4: o. “1‘34"; ‘ ’~.~.\ ' $3 '° a- )—" ‘ *. .‘ ; 1:1. ‘; - —- 1 . I 6:21:93: --‘_c _- 1"} 1' IIII'J £1311“ #11 1'11 THE/ fl Ul This is to certify that the thesis entitled Robustness and Power of Multivariate Tests for Trends in Repeated Measures Data Under Variance-Covariance Heterogeneity presented‘by Gabriella Belli has been accepted'towards fulfillment of the requirements for Ph. D. degree in Counseling , Educational Psychology, & Special Education (Statistics & Research Design) 0&8le V Major professor 0-7639 MSU is an Affirmative Action/Equal Opportunity Institution MSU RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from —;-—IL your record. FINES will be charged if book is returned after the date stamped below. .-; f s“ j” :5, 2;- a ~ * . l f— 7" :T'N‘Ai‘l 3" :13» .i crawls. 3* " — W- __...‘_~.__ _,~ . ROBUSTNESS AND POWER OF MULTIVARIATE TESTS FOR TRENDS IN REPEATED MEASURES DATA UNDER VARIANCE-COVARIANCE HETEROGENEITY by Gabriella M. Belli 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 1983 ©Copyright by GABRIELLA BELLI 1984 Mu. 0f homo groups). ordered group d and (3) effects Stoop h tests 0 A“ aEgo: f°r int. but tha' Thi Was to ‘ differe; 0f main evaluate tw° wit! and (2) ABSTRACT ROBUSTNESS AND POWER OF MULTIVARIATE TESTS FOR TRENDS IN REPEATED MEASURES DATA UNDER VARIANCE-COVARIANCE HETEROGENEITY BY Gabriella M. Belli Multivariate statistics are subject to the assumption of homoscedasticity (iAL, equal covariace matrices across groups). In a repeated measures (RM) design with time ordered data, three hypotheses are tested: (1) between- group differences, (2) within-group trends over occasions, and (3) group by occasion interactions. Although the effects of assumption violation on tests of the between- group hypothesis have been investigated, the effects on tests of within-group and interaction hypotheses have not. An argument is presented indicating that multivariate tests for interactions should behave like between-group tests, but that tests for within-group trends should not. The primary purpose of this Monte Carlo investigation was to determine whether heteroscedasticity has a differential effect on the robustness of multivariate tests of main effects in a RM case. A secondary purpose was to evaluate the robustness and power of multivariate tests of two within-group hypotheses: (1) overall tests of trends, and (2) subsequent tests of trends higher than linear, undei sampl Hotel Wilks inves trend than (2) w sligh' (3) D. for w. Size, betwei Stomp: r0bus1 heter‘ fatter inctea heterc factor a dec; lathe: Gabriella M. Belli under various combinations of number of groups and equal sample sizes. The test statistics were: Roy's largest root, R, Hotelling-Lawley trace, T, Pillai-Bartlett trace, V, and Wilks' likelihood ratio, W. The following are the major conclusions drawn from the investigation. (1) Multivariate tests of within-group trends are considerably more robust to heteroscedasticity than are multivariate tests of between-group differences. (2) Within-group tests of trends higher than linear are slightly more robust than overall tests of trends. (3) Departures of empirical Type I error from nominal alpha for within-group tests increase as heterogeneity, sample size, or alpha increase, but not as dramatically as for between-group tests. (4) Increasing the number of equal groups does not have a consistent detrimental effect on robustness of within-group tests. (5) For low and moderate heterogeneity (i.eu, covariance matrices differing by factors of two or four), power of within-group tests increases as total sample size, N, increases. (6) For high heterogeneity (i.e., covariance matrices differing by a factor of nine), power of within-group tests increases with a decrease in the number of discrepant score vectors, rather than with an increase in N. In a; and comma: wish to t] friend, ccl strong notI teacher ar Programs. I worl the many :I COnStant 5. of this rel Floden, an providing I Von D" J0e L. t0 Hr. Jef were eSSER MOSt husband, D. and EnCou r5 ACKNOWLEDGEMENTS In appreciation for his support, insightful questions and comments, and willingness to let me pursue my ideas, I wish to thank Dr. Andrew C. Porter. He has been a valuable friend, colleague, and chairperson and has provided a strong model for professionalism and excellence as both teacher and researcher throughout my masters' and doctoral programs. I would also like to thank Dr. Richard F. Houang for the many thought provoking discussions, as well as for his constant availability, during the conceptualization stage of this research. I wish to express my appreciation to the rest of my committee, Drs. James H. Stapleton, Robert E. Floden, and William H. Schmidt, for reviewing my work and providing suggestions for improvement. I would further like to express my appreciation to Dr. Joe L. Byers, who helped provide the computer time, and to Mr. Jeff Glass, who coded the FORTRAN program. Both were essential components in making this research possible. Most importantly, an expression of gratitute to my husband, Dr. Robert E. Krapfel, without whose moral support and encouragement this work would not have been completed. His patience and understanding have been immeasureable. ii LIST OF T LIST OF P Chapter I. STATI H.Hmt III. Revre TABLE OF CONTENTS Page LIST OF TABLES OCCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO v LIST OF FIGURES 00.0.0...OOOOOOOOOOOOOOOOOOOOOO0.00... Vi Chapter I. STATEMENT OF THE PROBLEM IOOOOOOOOOOOOOOOOOOOOOOO 1 II. MULTIVARIATE ANALYSIS OF VARIANCE ............... 8 General Multivariate Linear Model .......... 8 Multivariate Generalization for Repeated Measures .............................. 10 Hypothesis Testing ......................... 14 Tests of Significance ...................... 18 Theoretical Comparison of Tests ............ 21 III. REVIEW OF THE LITERATURE OOOOOOOOOOOOOOOCOOIOO... 24 Strategies for Investigating Robustness to Heterogeneity ...................... 25 Consequences of Non-independence and NOD‘nOrmality 0000000000...00.00.00.000 28 Consequences of Heterogeneity .............. 30 Fixed-model ANOVA ..................... 30 Mixed-model Repeated Measures ......... 32 Two-sample MANOVA - Hotelling's T2 .... 35 General MANOVA Test Statistics ........ 38 Iv. METHOD COIOOCOOCOOCOOOOOOOOOCOOOOOOCOOO0.00.00... 46 Reduction to Canonical Form................. 47 Parameters Of the Study OOOOOOOOOOOOOOOOOOOO 48 iii VI. 13; APPEND] DOW? ("1 o Procedures ................................. 51 Determination of Critical Values ...... 53 Design for the Study .................. 53 Empirical Comparisons of Tests ........ 57 Analysis of Within-group Tests ........ 58 Interpretation of Obtained Probability Values ............... 58 "ante carlo TeChniques ......O.....OO....O.. 59 Random Number Generator ............... 60 Creation of Normal Deviates ........... 61 Transformation to Desired Structure ... 62 v. RESULTS ......................................... 64 Comparison of Tests on Robustness .......... 65 Power of Within-group Tests of Trends ...... 73 Robustness Under Various Conditions ........ 79 Sample Size and Robustness ............ 80 Number of Groups and Robustness ....... 83 Power Under Various Conditions ............. 87 Sample Size and Power ................. 88 Number of Groups and Power ............ 92 Total Sample Size and Power ........... 92 VI. DISCUSSION ................O..................... 95 conC1USion8 .............O.......O.....OOOO. 95 Guidelines for the Researcher .............. 98 Suggestions for Future Research ............100 APPENDICES ...........................................103 A. B. C. D. Computational Details and Computer Program ...103 Monte Carlo Critical Values ..................136 Significance Levels for Between-group Tests ..143 Significance Levels for Within-group Tests of Non-linearity ......................146 Power Rates for Within-group Tests ...........l49 BIBLIOGMPHY .........................................154 iv LIST OF TABLES Table Page 2-1 Multivariate Analysis of Variance (k-sample) .... 15 2-2 Multivariate Analysis of Variance for Repeated Measures O...................O..00...... 15 2-3 SSCP Matrices for RM Tests ...................... 17 2-4 Multivariate Test Statistics .................... 20 4-1 DeSign for the StUdy ............................ 54 4—2 Standard Errors for Nominal Alpha Levels and Number of Replications Used in the Study ........ 59 5-1 Percentage Exceedance Rates of Monte Carlo Critical Values for Multivariate Tests under True NUll Hypotheses O O O . O O O O O O O O O O . O O O O O O O 68 5-2 Differences in Percentage Exceedance Rates Under True Null Hypotheses ...................... 72 5-3 Percentage Exceedance Rates for Within-group Tests under True Alternatives .....OOOOIOOOOOOOOO 75 5-4 Percentage Exceedance Rates for Within-group Tests with Modified True Alternatives ........... 78 5-5 Percentage Exceedance Rates Under a True Null for Tests Of Trends Withk=3 ......O........... 81 5-6 Percentage Exceedance Rates Under a True Null for Tests of Trends with n = 20 ................. 84 5-7 Average Percentage Exceedance Rates Under True Alternatives for Tests of Trends with k = 3 ..... 89 5-8 Average Percentage Exceedance Rates Under True Alternatives for Tests of Trends with n = 20 .... 91 Figure 5'1 Tre: Tab of 5'3 Pow for LIST OF FIGURES Figure Page 5-1 Trend transformation for power results of Table 5-3 with mean vectors: (0 .4 .8 .5 .1) for p a 5 (0 .4 .8 .5) for p = 4 ..................... 77 5-2 Second trend transformation for power results of Table 5-4 with mean vectors: (0 .6 .7 .2 .05) for p =- 5 (0 .6 .7 .2) for p = 4 ..................... 77 5-3 Power curves averaged over four test statistics for different total sample sizes N, where: N = 30 40 60 120 150 k 8 3 2 3 6 3 nalo 20 20 20 so ......OOOOOOOOOOOOO 92 vi CHAPTER I STATEMENT OF THE PROBLEM Classical experimental research involves investigating the effect of manipulating one or more independent variables on a single dependent variable. This involves either testing the null hypothesis of equal group means against a general alternative or testing for specific planned comparisons among the group means. The test statistic used is the F-test (or t-test for two groups). Given parametric assumptions, this is the uniformly most powerful test that is invariant with respect to linear transformations (Scheffé. 1959). Generalizing to the multivariate case, where there are two or more dependent variables (say, p). the corresponding null hypothesis is that of no differences among the I: group vectors. where each vector consists of the group means on the p dependent measures. The F-test is a univariate test statistic, and several generalizations of it have been proposed for significance testing in the multivariate case. Among those tests that are invariant under linear transformation of the dependent variables, Hotelling's T2 statistic is the uniformly most powerful for one-sample tests of means and two-sample tests of mean differences (Anderson. 1958). Four other commonly used test statistics are Roy's largest root. R. Hotelling-Lawley trace. T. Pillai-Bartlett trace. V. and Wilks' likelihood ratio. W. However. for situations where there are multiple dependent variables or more than two groups. no test has emerged that is both invariant with respect to linear transformations and uniformly most powerful. A specialized case of the multivariate analysis of variance (MANOVA) deals with situations where the same measure is repeatedly taken over the same individuals. The design on the measures. or occasions of testing. may reflect the passage of time. with the same measure taken at equally spaced intervals. or it may represent a factorial structure. with the same measure taken after various treatment interventions. In addition to the usual multivariate hypothesis of group differences. hypotheses about the occasions and. if there are multiple groups. about group by occasion interaction may be tested. The null hypothesis for occasions is that of no differences among the p occasion vectors. where each vector consists of the occasion means for the k groups. ‘When there is only one group or when no group by occasion interaction exists. of even greater interest is the testing of hypotheses about the trend the data follow. assuming equally spaced time points. or about contrasts among the various measures. assuming a factorial design. Tests for these hypotheses are E tests conc. toxr null simi then Viol can the 0P6: thre the [Obu bala Sand 519: “her are all within-group tests as opposed to between-group tests in the usual MANOVA sense. In both the univariate and the multivariate cases. the test statistics used are based on certain distributional assumptions. These are that the random errors or error vectors for the p measures are: (1) independently and (2) normally or multivariate normally distributed (3) with a common variance or variance-covariance matrix. Violations of these assumtions may lead to erroneous conclusions. However. if a particular test is insensitive to violation of one or more of the assumptions when the null hypothesis is true (i.eu. if it leads to conclusions similar to what would be expected given the assumptions). then the test is said to be "robust" with respect to the violation. The assumption of independence is critical and no test can favorably withstand its violation. Non-independence of the observartions or of observational vectors due to faulty operationalization of experimental design is a serious threat to nominal alpha levels. In univariate situations. the F-test for fixed effects has been shown to be fairly robust with respect to violation of normality and. for balanced designs. of homogeneity (see Glass. Peckham. and Sanders. 1972). However. severe departure from nominal significance level may occur under heterogeneity conditions when samples are small and unequal (Scheffé. 1959). Re situatj violati 1980). result: are sim robust heteros varianc‘ but are two 9:0 number 7 With mo are equé moderat« level a1 T: Violati‘ within~. SitUati‘ (1975) S to high. latQQr . likely t treat“! between Regarding between-group differences in multivariate situations. the several tests respond differently to violations of the assumptions (for a review. see Ito. 1980). Generalizing. it may be said that robustness results for fixed effects of at least some of these tests are similar to those in the univariate case. They are robust to non-normality and also fairly robust to heteroscedasticity (i.e.. violation of homogeneity of variance and covariance) in balanced. two-group designs. but are not so for unbalance designs. However. even with two groups. the tests become liberal with increases in number of dependent variables or amount of heterogeneity. With more than two groups. tests are robust only if samples are equal and extremely large. If they are unequal. even moderate heterogeneity has large effects on significance level and power (Ito and Schull. 1964). To date. no studies have considered the robustness to violation of multivariate test assumptions for tests of within-group differences in a repeated measures (RM) situation. Due to the nature of RM studies. Morrison (1976) states that I'many experimental conditions which lead to higher mean values may also produce responses with larger variances" (p. 141). Different populations are likely to respond differently to successive measurements or treatment conditions. thereby also causing correlations between measures to differ from group to group. This is particularly true in studies of naturally occurring groups (e.g.. a comparison of learning disabled and normal children on learning retention rates over time). Subjects within a classification group may be expected to respond in a similar fashion. but it is unrealistic to expect that scores for the two groups come from the same multivariate normal population. Hence. it is important to determine the validity of the multivariate tests of RM in the presence of heterogeneity conditions. Just as findings from robustness studies for tests of between group differences have parallels in the univariate and multivariate cases. it may be presumed that similar parallels would hold for tests of within-group differences when homogeneity is violated. However. results from mixed- model RM studies would not apply to multivariate tests since the univariate tests are based on the assumptions of equal variances and equal pairwise correlations across the measures. which are unnecessary for multivariate tests to be valid. 'The effect on within-group tests when using a covariance matrix that is pooled from heterogeneous population covariance matrices is not known. The robustness of a parametric test is idiosyncratic rather than general with respect to any violation and changes in one parameter may produce different levels of departures from nominal significance level. Tests of within-group differences are based on transformations of the dependent variables and the assumptions are made on the transformed scores. It will be shown in Chapter II that multivariate tests of between-group and within-group differences are based on sums of squares and cross products (SSCP) matrices that are different in both form and size. and that the relationship between the eigenvalues needed for calculating the test statistics for the two tests is not obvious. Hence. it is not possible to predict the behavior of one type of test from that of the other. Since the current robustness results from studies of multivariate between-group tests may not apply directly to within-group tests. separate investigations need to be made. Furthermore. subtests of particular trends for RM data make use of subcomponents of the appropriate SSCP matrices for hypothesis and error. Since it is known that between- group tests become more robust with lower dimensionality of variables. it is expected that tests of successively higher order trends should show greater robustness than tests of lower order trends. The present research was an investigation of the robustness and power of multivariate within-group tests for a repeated measures design with the same measure taken over a series of equally spaced time points. Non-normality does not seem to cause serious problems under any circumstances thus far investigated. whereas heterogeneity may be a serious problem in certain cases. Therefore. given that he tk vi ha St to gn cox exa dii of con tes lev amc Als tru cor: Pre mu} lit end dis heterogeneity is typically a violation of greater concern. the focus of this study was limited to the effect that violation of the assumption of a common covariance matrix has on the sampling distributions of four multivariate test statistics. The purpose of the first part of the investigation was to determine whether tests of between-group and within- group hypotheses differ in their reactions to heterogeneous covariance matrices across groups. The second part was to examine whether covariance matrix heterogeneity produces differential effects on within-group tests when the number of groups or of subjects within groups are varied. Third. comparisons were made between overall tests of trends and tests of non-linearity. In all cases. actual significance levels obtained under a true null hypothesis and a given amount of heterogeneity were compared to nominal values. Also. actual powers for within-group tests obtained under a true alternative and a given amount of heterogeneity were compared to expected nominal powers if no violation was present. The following chapters will present the general multivariate and repeated measures models. along with their hypotheses and test statistics. a review of the robustness literature. the method used for investigating robustness and power of multivariate within-group tests. results and discussion of results. CHAPTER II MULTIVARIATE ANALYSIS OF VARIANCE In this chapter. the mathematical models for the general multivariate analysis of variance (MANOVA) and for the multivariate generalization to repeated measures (RM) are described. These are followed by a description of the hypothesis testing procedures through the separation of the total source of variation into component parts. the tests of significance used in multivariate analyses. and the assumptions on which they are based. The final section deals with a comparison of the sums of squares and cross products (SSCP) matrices used to test between-group and within-group differences. General_nultixariare_Linear_nodel Assuming there are nj (3' = l.....k) independent observations in each of k groups. the ith observation in the jth group is a pxl vector consisting of a constant term A. a group effect 21. and a random error component iij Yij ' E|+ Ej + Eij' The Y13 and the sij are distributed in the population of subjects as No.1,. 2) and N(D_. 2). respectively. where 2 is any pxp symmetric positive definite matrix. The null hypothesis tested in MANOVA is that the pxl mean vectors of all groups are equal. Ho: £1 I 2,2 I ... 3 11-k- By letting Ej - g,+ gj. this hypothesis is equivalent to testing that all the 221 - 0 (i.e.. that all the treatment or group effects are equal) (Bock. 1975). The general MANOVA model for k group means may be expressed in matrix terms as Y. 8 A3 + E. where: Y. - a kxp data matrix of k group means on p measures A I a kxm known design matrix . an mxp matrix of unknown paramenters E. - a kxp matrix of random errors The error matrix B. is distributed N(Q5D‘10£) where D a diag(n1.n2.....nk)- Since A typically is not of full rank. A'1 does not exist. and therefore solving for the unknown parameters is not possible. One solution is to reparameterize the model. which may be done by factoring A into the product of two matrices. K and L. A 8 KL where L is an lxm contrast matrix that describes a set of 1 linear combinations of the paramenters in : and K is the ! corresponding kxl column basis for the design matrix A. Then. E(Y.) =- A5 = KLE = K9 where e is an lxp matrix of new paramenters describing the resulting linear combinations that reflect the research interest regarding differences among the groups (Bock. 1975. pp. 239-240). WWW Multivariate analysis of variance of repeated measures (MANOVA of RM) is a variation of MANOVA that includes a test for the occasions or repeated measures. What distinguishes these data from general multivariate data is that in RM the multiple dependent scores are assumed to be in the same metric (i.e.. having the same origin and unit). whereas in general the scores are qualitatively distinct (i.eu. having different origin and unit). The underlying model for the ith observation in the jth group is a pxl vector that contains a component for occasions 1. for groups gj. and for random subject error Eij' lij - l + Ej + 513. As before. the gij are distributed N(.0,. 2 ). But. unlike the general MANOVA model. where the common term a does not provide any additional information. the common term in this model. 1. represents a pxl vector of constants and general means for the p occasions. The second term. fljr is a 931 vector of effects for the jth group that incorporates both group and group by occasion interaction effects. The model allows for a design on the occasions and a design on the subjects (Bock. 1975). 10 In the one-sample case or. assuming no interactions. in the k-sample case. the objective is to characterize the occasion vector 1. The appropriate characterization depends on the structure of the repeated measures dimension. If the measures correspond to points along a continuum. a polynomial representation is used. .1; - X f where x is a regression model matrix and 8, is a vector of unknown regression coefficients. If the measures correspond to a factorial classification. then a treatment contrasts and interaction representation is used. 1 = A.§ where A is a design matrix for the occasions and g is a vector of unknown occasion effects. In the former case. x is of full rank while. in the latter case. A is not and the model may be reparamenterized a second time. While this reparamenterization follows the same pattern as before. with A .. KL. A is now the design matrix for the occasions and not for the groups. Under the usual MANOVA model. the general occasion effect 1 is not estimable and hypotheses on it are not testable in the presence of group effects. Bock (1963) and Potthoff and Roy (1964) have suggested a variation of MANOVA that involves transforming the dependent variables to within-subject differences. A new set of measured variables is formed as linear combinations of the original 11 measures. Yij" 3 P'yij where P is a matrix representation of the design over the measures. In terms of the previous discussion of the characterization of 1. P is either: (1) the regression model matrix x, if the measures are taken at ordered time points or (2) the orthonormalization of K. where K is the basis for the reparameterization of A. the design matrix for the occasions. Assuming a full rank model for group means. the transformation in matrix terms consists of postmultiplying the components of the MANOVA model by a known matrix P. which may be any pxq matrix. Preferably. P should be an orthogonal matrix and this is now assumed. Then. Y.P 8 K9 P + E.P or equivalently. Y.* = KO P + E.* where: Y.*= a kxp matrix of transformed scores K = a kxl basis matrix for transformations on groups 9 a an lxp matrix of parameters P = a pxp basis matrix for transformations on occasions E.*- a kxp matrix of transformed errors Analysis now proceeds as usual with the transformed scores in Y.* replacing the original dependent measures. The fact that the standard procedures apply can be seen 12 since the transformation 2. - Y.*P'1 reduces the RM model to a standard MANOVA.mode1 (Timm. 1980. p. 76). Furthermore. if P is orthogonal (i.e.. P'P - I). so that P‘1 s P”. each vector of scores may be transformed using P'. as was shown previously. When P is either non-singular or has rank p. the transformation has nice properties with respect to the distributional assumptions. Given that the yij are independent and distributed N(£.2). then the Yij* are also independent and are distributed N(P'u. P'XP) (Bock. 1975. p. 140). Three basic hypotheses are of interest with k-sample RM data. These deal with comparisons among the mean curves or profiles of the groups. and may be phrased in terms of the following questions: (1) Are the curves or profiles of the k groups parallel? (2) If parallel. are they also coincident? and (3) If coincident. are they also constant? (Bock. 1975). The first question is asking about the presence of any group by occasion interactions. The second relates to group differences and the third to occasion differences. Subhypotheses to assess the effect of the treatment structure or the trend over the occasions may also be tested. Assuming a polynomial representation for the RM dimension. this involves partitioning the sources of variation for occasion and for group by occasion into 13 constant. linear. quadratic. etc. terms. Then a hypothesized trend may be tested by a multivariate test that all higher order trends are zero. The interpretation for these tests on occasions is straightforward and relates information about the type of trend the RM follow over time. However. a q-degree trend among the interactions implies that “any contrast among the groups can presumably be described as a polynomial of this degree. For example. a degree-2 interaction would imply that differences between groups. in addition to a possible linear trend. are accelerating or decelerating with respect to occasions" (Bock. 1975. p. 474). H I] . T l' The multivariate hypothesis testing stage involves partitioning the sums of squares and cross products (SSCP) matrix for total variation into a constant. a between- groups. and a within-groups part. The MANOVA table for the general multivariate analysis is given in Table 2-1 (adapted from Book. 1975). The SSCP matrices for RM may be calculated directly by substituting Y* for Y in Table 2-1. The same results may be obtained by transforming the MANOVA SSCP matrices as shown in Table 2-2 (Bock. 1975). 14 Table 2-1 Multivariate Analysis of Variance (k-sample case) Source of df SSCP (pxp)* Variation Equal n's General Constant 1 0c - (n/k)Y.'ll'Y. (1/N)Y.'Dll'DY. (occasion effect) Between groups k-l Qb a ny.'y. - Qc Y.'DY. - Qc (group effect) Within groups N-k Qw = Y'Y - nY.'Y Y'Y - Y.'DY. error Total N Qt a Y'Y Y'Y * where D = diag(n1.....nk) and l a a unit vector. Table 2-2 Multivariate Analysis of Variance for Repeated Measures Source of Variation df SSCP (pxp) Constant 1 QC* 8 P'QCP Between groups k-l Qb* = P'QbP Within groups error N-k Qw" = P'QwP Total N Qt* = P'QtP 15 The multivariate test statistics are functions of the appropriate SSCP for hypothesis and error (say. H and B. respectively). The MANOVA hypothesis of equal group means may be tested by setting H - Qb and B - Qw' For RM. the matrices in Table 2-2 may be partitioned in the following manner: Qc* ‘ PC 1 1 Qb* ’ Pb : 1 Qw* 8 'w : 1 --T---- --T ...... F"“ I I I :c :3 :w L.‘ J b. . L'J Assuming a polynomial decomposition. the scalars c. b. and w represent the sums of squares for constant. group effect. and error terms that would be used in a univariate analysis. The (p-1)x(p-l) matrices C. B. and W are the SSCP for occasion effects. group by occasion effects. and subject within group by occasion error. The diagonal elements of these submatrices are the univariate sums of squares for the respective linear. quadratic. etc. trends. Table 2-3 shows how these matrices are used for the three omnibus tests in a RM situation. With no group by occasion interaction. the full matrices 0b* and Qw* are the H and E matrices for group effect and corresponding error for a multivariate test of group differences. When P is orthogonal. a test using these transformed matrices gives the identical results as with Oh and Qwr because test statistics based on either determinants or trace functions remain invariant under 16 orthogonal transformation (Anderson. 1958. p. 277). Table 2-3 SSCP Matrices for RM Tests Hypothesis H B Dimension parallelism (interaction) 3 w (p-1)x(p-1) Coincidence (group effect) Qb* Qw* pxp Constancy (occasion effect) C W (pP1)x(p-l) The submatrices C. B. and W may be partitioned further to provide tests for particular trends. To test for any q

0 for all x =- 0) (Anderson. 1958. p. 337). This will usually be the case if the number of dependent variables (p) is less than the degrees of freedom for error (dfe). Let 11 z 12 2 ...z 13 > 0 where s - min(dfh. u) with dfh 8 degrees of freedom for hypothesis and u - the number of variables after any transformation. Then. four commonly used multivariate test criteria are defined in Table 2-4 (Timm. 1975). These are exact tests. with known central and noncentral distributions. When 3 - l (i.e.. if p a l or k - 2). they are equivalent and may be represented as an exact P distribution. There also are P approximations for the multivariate tests (see e.g.. Tatsuoka. 1971). The only parameters necessary to define the distribution of the statistics under valid assumptions and true null hypothesis are number of variates. degrees of freedom for hypothesis. and degrees of freedom for error (Ito. 1962). .Additionally. noncentrality parameters are needed under true alternatives. Based on these parameters. Timm (1975) provides tables for the upper percentile points of R. T. and V and for the lower percentile points of W. The null hypothesis is rejected at significance level a if the obtained value of W is less than the 100a-centile of the null distribution. For the other tests. the null is rejected if the obtained value of a statistic exceeds the 100(1-o)-centile of the corresponding distribution. 19 Table 2-4 Multivariate Test Statistics Roy's largest root R 3 _il_ 1+1l s Hotelling-Lawley trace T = 2 xi = tr(HE"1) 181 s Pillai-Bartlett trace v - 2 _’)j_ . tr(H(H+E)'1) 1+Ai i=1 8 Wilks' likelihood ratio w . H_1_ = lsl-(|H+E|)'1 i=1ini 20 .Thenretica1_£nmnaxison_of_Tests Although the multivariate test statistics for tests of between-group and within-group differences are identical. they operate on different.SSCP matrices. The question of interest. then. is whether these matrices. whose expected values are functions of the common covariance matrix. are equally subject to violations of homoscedasticity. The following discussion outlines the relationship between the matrices used for the two tests. Multivariate test criteria are functions of the eigenvalues of HE'l. where H and E are the SSCP matrices for hypothesis and error. respectively. For a within-group test HE'1 is the lower (p-1)x(p-l) submatrix of the appropriately transformed QcQw'l (1) and for a between-group test. it is the pxp matrix chow-1 (2) where QC. 0b! and Qw are defined in Table 2-1. From the robustness literature. which is reviewed in Chapter III. we have general conclusions about the effects of particular types of homogeneity violations in the population covariance matrices when (2) is used to test for group differences. These results are based on distributions of the p eigenvalues of (2). Tests for group by occasion interactions are based on the eigenvalues of the order-(p—l) submatrices of (2). 21 Since the same SSCP matrices are used for both interaction and between-group tests. the lower dimensionality in the portion of those matrices used for interaction tests should tend to make them slightly more robust than the between- group tests. Tests of occasion differences with RM data are based on the p-l eigenvalues of the order-(p—l) submatrix of (1). By substitution. QcQw-l = (Y.'DY. - Qb)Qw'l . (Y.'DY.)Qw'1 - obow'l. Even though a relationship exists between matrices (1) and (2). knowledge about the distributions of eigenvalues of (2) does not provide direct information about the distribution of eigenvalues of (1). Since within-group tests are actually based on a submatrix of (1). it would further be necessary to establish the relationship between the p eigenvalues of the full matrix (1) and the (p-l) eigenvalues of the submatrix used for these tests in order to fully specify the relationship between the matrices for the two types of tests. Bach subsequent within-group test of successively higher order trends is based on submatrices of (l). which decrease in dimension. Therefore. each test of a higher order trend should result in slight increases in robustness over the previous within-group test. 22 It is not obvious whether heterogeneity in the population covariance matrices would differentially effect the robustness of between-group and within-group tests and the mathematics needed to demonstrate the necessary relationships are intractable. Therefore. an empirical study was conducted to determine if the distributions for any of the four multivariate test statistics presented earlier are comparable for testing the two types of hypotheses. In this way. it could be determined if the tests respond similarly to the same violation to homogeneity; .A further comparison of the robustness between within-group tests for any trend across time and the subsequent tests for trends higher than linear was also conducted. The study involved the simulation of a large number of experiments so that the actual significance levels could be compared to nominal levels with minimal standard error. The second part of the study was an investigation of the effects on robustness and power of within-group tests when sample size and number of groups are varied. 23 CHAPTER III REVIEW OF THE LITERATURE Consequences of assumption violations have been thoroughly investigated for univariate test statistics from both the large sample and small sample points of view. Only recently have similar studies been undertaken for multivariate test statistics. While some of this work has been theoretical. involving large sample theory and asymptotic approximations. most of it has been empirical. Since the mathematics involved in a theoretical study of multivariate statistics are quite complex. Ito and Schull (1964) remarked that ”the small sample treatment of the problem ... is very difficult if not impossible” (p. 72). Researchers in the multivariate area have focused on a one-way fixed effects classification for the independent factor and have considered tests of between-group differences on multiple dependent measures. Robustness studies of within-group tests have dealt only with violations of the univariate mixed-model assumptions of equal variances and covariances across the repeated measures (RM). Typically. comparisons have been made between the usual F-test and the I? adjusted by a correction factor (e.g.. Collier. Baker. Mandeville. and Hayes. 1967) or between univariate and multivariate analyses (e.g.. 24 Scheifley. 1974). However. in all cases with more than one group. groups were assumed to have a common covariance matrix. The following review will briefly summarize the fixed and mixed model univariate results and then present the multivariate results in greater detail. As a preliminary. an overview of a common strategy used to model heterogeneity will be presented. . . -; .- .. :.. : .-:; . :- - ..-.- Variance heterogeneity in univariate studies is easily portrayed by a ratio of population variances. For multivariate problems. modeling is more complicated since there are many ways of introducing heterogeneity in population covariance matrices. Two single-valued multivariate analogs to a variance are the trace and determinant of the covariance matrix. The trace represents the total variation and the determinant represents a generalized variance (Tatsuoka. 1971). Ratios of covariance matrix determinants parallel the univariate case. forming a convenient index of multivariate heterogeneity. A typical tactic used in empirical studies of robustness of multivariate test statistics against violation of the assumption of homoscedasticity is to reduce the problem to canonical form. This procedure. which was used in all but two of the multivariate studies 25 reviewed. produces diagonal covariance matrices. thereby reducing the number of parameters that need to be considered by p(p-1)/2. The procedure is based on theorems for matrix transformations (see Tatsuoka. 1971. pp. 125-129). It consists of applying a linear transformation. say C (where C is orthogonal. i.e.. C'C = I and ICI - l). to the matrix of observations x. thus producing a new set of uncorrelated variables Y - XC. The matrix C represents a rigid (or angle-preserving) rotation from the original variates to the principal axes and consists of columns of eigenvectors of the original covariance matrix 2. Using the same transformation matrix. 2 is transformed into a diagonal matrix C'XC " diag(11,12,,,,,ls), with the variances of the canonical or transformed variates (eigenvalues) as diagonal elements. This is called 'diagonalizing the matrix“ (Tatsuoka. 1971. r» 128). The trace and determinant of the original covariance matrix are equal to the trace and determinant of the transformed matrix. A multivariate analysis on the canonical variates produces the same results as those obtained with the original ones since the MANOVA test criteria are invariant under any linear transformation (Anderson. 1958. r» 277). The operationalization of this procedure in MANOVA robustness studies of heterogeneity relies on the fact that two population covariance matrices. say V1 and v2, may be 26 ' linearly transformed to the identity matrix. I. and to a diagonal matrix. D. whose diagonal elements are the eigenvalues of V2V1'1 (Holloway and Dunn. 1967). o is called the diagonal matrix of latent roots. MANOVA test criteria for a given test based on any mixture Of N(D..V1) and N(.Q.V2) are equivalent to a mixture of N(Q.I) and N(Q.D) (Olson. 1973). To model situations with non-zero mean vectors. a mixture of N(u1,1) and N(g2.D) may be used to represent the canonical forms of N(C"'1.21.V1) and N(C"'1P.2.V2). This applies to both the central case. with equal population means. and the noncentral case. with unequal population means. Heterogeneity is typically introduced either equally in all of the canonical dimensions. with D - 61. or in only one dimension. with D - diag(d.1.....l). Variations on this theme allow for heterogeneity to vary across canonical dimensions. with some di a 1 while other d1 = d. In this way. a researcher need only vary values of d to simulate a variety of heterogeneous conditions. For more than two groups. either one or more groups are sampled from a population with covariance matrix D and the rest from a pOpulation with covariance matrix I. An alternative is to sample groups from populations with covariance matrices I. D. and multiples of D. 27 WW3! Violation of the independence assumption is quite serious. For analysis of variance (ANOVA). positive correlations among the errors yield a liberal test (i.e.. too many significant results) and negative correlations yield a conservative test (i.e.. too few significant results). This is true for both equal and unequal sample sizes and the discrepancy between nominal and actual levels of significance increases as the absolute value of the correlations increase (see Scheffé. 1959). For a two-group matched-subjects design in univariate situations. use of the correlated or dependent t-test is an appropriate technique to handle the problem. For correlated observations that arise from a RM situation. the problem is identical to that of a mixed-model analysis and two avenues are open. One is to use the correction factors of Box (1954) or Greenhouse and Geisser (1959). These adjust the degrees of freedom for the F-test and the latter produces conservative results. The other method is to use exact multivariate tests. which do not make the ANOVA assumption of independence of errors across measures taken on the same subject. However. independence of errors between subjects must still be maintained. Glass. Peckham. and Sanders (1972) provided a thorough review of the univariate literature for fixed-effects designs. General conclusions were that violation of the 28 normality assumption does not present a problem for either the t-test or the F-test in an analysis of mean differences. For both equal and unequal sample sizes. discrepancies between actual and nominal significance levels are slight and. with equal n's. the F-test proves to be robust even in the extreme case of dichotomous data. However. non-normality does effect inferences about variances. such as in tests of random-effects or equality of variances (Scheffe. 1959). Considering six multivariate tests and using equal rfls. Olson (1973. 1974) found that departures from normality in the direction of positive kurtosis (occasional extreme observations) had only minor conservative effects on Type I error rates. From the asymptotic expressions for central and non-central distributions of Hotelling‘s T2 and T02 (a generalized T2 for more than two groups). which were obtained by Ito (1969). approximate values for actual significance and power may be found. In a recent review. Ito (1980) mathematically demonstrated that. for sufficiently large sample sizes. non-normality did not appreciably effect either the significance level or the power of these test statistics. The question of what sample size is to be considered "sufficiently large” was left open. since this is difficult to demonstrate theoretically. Ito (1980) further stated that. from Monte Carlo studies. the T2 test in the two-sample case has been 29 found to be particularly robust against non-normality for tests about means. However. as in the univariate case. non-normality has serious consequences for tests of equality of covariance matrices. WW Studies of both univariate and multivariate cases indicate that violation of the homogeneity assumption may cause serious discrepancy between actual and nominal significance levels and that this is typically a more serious problem than non-normality. Since this violation is more serious. as well as being the focus of the present research. greater attention will be given to studies of robustness in the face of heterogeneity. Consequences in both the univariate and multivariate cases will be reviewed. £1xed:mndel.ANQ!A Extensive work has been done to examine the consequences of departures from homogeneity of variance for univariate test procedures (for reviews. see Scheffé. 1959. Chapter 10 and Glass. Peckham. and Sanders. 1972). In the univariate two-sample case. inequality of population variances has little effect on either significance level or power of the t-test if sample sizes are about equal. However. if sample sizes are markedly disparate. large deviations from the nominal error rate occur for both large and small sample cases. The test is conservative if the 30 larger group has the larger variance and is liberal if the larger group has the smaller variance (Scheffé. 1959). For more than two groups. heterogeneity does have a slight effect on the Type I error rate of the F-test even when groups are of about equal size. in which case the test is liberal (Scheffé. 1959). However. general conclusions from both theoretical and empirical work have been that the ANOVA F-test is robust to heterogeneity of variance. A major exception is in the case of small and unequal sample sizes. where the effects are serious. Results for unequal rfls follow the same pattern as for the t-test. with either conservative or liberal results. It should be noted. however. that these general conclusions have boundary conditions. which depend on sample size or ratio of sample sizes. on the amount of heterogeneity. and on the value of nominal alpha. Ramsey (1980) found that even for the equal sample t-test. robustness depends on certain conditions. For example. with n's greater than 15. the t-test will not exceed a significance level of .06 at a nominal level of .05 regardless of the amount of heterogeneity. but robustness may be achieved with n's as small as five if the ratio of variances in the two populations is 1:4 or less. Also. there is an inverse relationship between nominal alpha level used and sample size needed for robustness. 31 Mixed:model.8eneated_Measures I In univariate mixed-model analysis. the RM dimensions are treated as additional design factors. Two assumptions are made for a valid univariate test: (1) equality of covariance matrices across levels of the between-group factor. and (2) uniformity of the common covariance matrix (i.e.. equality of the variances of the RM and of the pairwise correlations between these measures). In a RM situation. variances might change between observations. possibly due to treatment effects on each occasion. Also. there is potential for lack of independence between error components of the observations. particularly if the RM factor reflects time. Huynh and Feldt (1970) demonstrated that uniformity is merely a sufficient and not a necessary condition for validity of within-group F-tests. What is required is that the assumptions stated above be met by the covariance matrices of orthonormalized variates rather than of the original variates. Nevertheless. the majority of the robustness literature in the mixed-model case has focused on violation of the uniformity assumption with the original variates. Some of these studies are reviewed below. While the studies in this section have a different focus from the rest of this paper. since variances and covariances are equal across groups. they are included as backgound to a study of consequences of assumption 32 violations in a RM study. Also. they provide another indication of the idiosyncratic nature of the behavior of test statistics under different forms of violations. In a theoretical study. Box (1954) assessed the approximate effects of unequal variances and serial correlations in one factor of a two-way design with one observation per cell. He showed that these conditions reduced the apparent number of degrees of freedom in both the numerator and denominator of the F-ratio and that the effect was to produce a slightly liberal test. Empirical results for a k-sample RM study (with k = 3 and p a 4) were obtained by Collier. Baker. Mandeville. and Hayes (1967). They compared Type I error rates for three ANOVA F-tests: unadjusted. adjusted by Box's correction factor. and by Greenhouse and Geisser's conservative lower bound for the correction factor. They considered 15 different patterns of covariance matrices. where both variances and pairwise correlations were varied. although covariance matrices were common across groups. As expected. their results showed that the P-test for group differences had a close agreement between empirical and nominal alpha. but that the F-test for occasions and group by occasions effects did not. In both cases the unadjusted P was liberal and the adjusted R was fairly robust with Box's correction factor but conservative with the lower bound test. .An unexpected finding was that 33 departures from nominal alpha did not significantly decrease. and in some cases actually increased. when sample sizes increased from five to 15. A similar. but smaller study conducted by Mendoza. Toothaker. and Nicewander (1974) upheld the above conclusions. In an empirical study comparing the mixed-model ANOVA. MANOVA of RM. and analysis of covariance structures (ANCOVST). Scheifley (1974; Scheifley and Schmidt. 1978) considered a one-group RM case with a 2x2 design on the measures. Three covariance matrices were used. where one matrix conformed to the assumptions of each analysis. When the ANOVA assumption of uniformity was not met. all three tests were generally conservative. ANCOVST had the greatest power when a significant difference in means was present in only one of the RM factors and MANOVA of RM had the greatest power when the null for both RM factors and the interaction were false. Significance level results for the univariate test under violation of uniformity in the above study were not consistent with the previous two empirical studies in this section. where results tended to be liberal. This may be partly due to the fact that the two covariance matrices used to model univariate assumption violation in Scheifley"s (1974) study had variances that were fairly close to being equal. while the other two studies had larger discrepancies between variances. Another 34 possibility is that the opposite results were due to the different patterns for the covariance matrices. The first two studies considered successive trials on one RM factor and the covariance matrices had simplex patterns (i.e.. successive diagonals had lower values). Due to the two-way factorial structure on the RM in the third study. those covariance matrices had circumplex patterns (i.e.. values in successive diagonals first increased and then decreased). W2 Unlike the mixed-model case. in multivariate analysis the separate repeated measurements are considered as multiple criterion variables. They may have unequal variances and a general pattern of correlations. The assumption is that this general covariance matrix is common across groups. To test for differences among the dependent variables. the original variables are transformed into contrasts of interest. Hotelling's T2 statistic is the multivariate analog to the t-test. and is the uniformly most powerful test for comparing two groups on p variables (Anderson. 1958. pp. 115-118). Several researchers have found it to behave in a fashion similar to the t-test under heterogeneity conditions. In an empirical study using Monte Carlo methods with relatively small samples (N = n1+n2 ranging from 10 t0 40). Hopkins and Clay (1963) examined the Type I error rates of 35 Hotelling's T2 statistic for testing the equality of two independent mean vectors in the p - 2 case. The two populations studied were N(Q.0121) and N(Q.0221). where heterogeneity between covariance matrices was present equally in both canonical dimensions and of the form 022/012 - 1.6 and 3.2. Under these circumstances. they found that with n1 . n2 > 10. heterogeneity had little effect on test results. but that. as in the univariate case. this robustness does not extend to unequal sample sizes. Everything else being equal. the greater the heterogeneity. the greater the departure of the observed significance level from the nominal alpha level. Furthermore. regardless of the amount of heterogeneity. the T2 test was conservative if the larger group had more variability and liberal if it had less variability. Another empirical study of the effect of inequality of covariance matrices and of sample size on the distribution of Hotelling's T2 statistic was conducted by Holloway and. Dunn (1967). They considered both level of significance and power with number of variables ranging from one to 10 and total sample sizes from five to 100. In canonical form. the covariance matrix for one population was equal to the identity. I. and for the other it was either dI or diag(d.1.....1). with d - l. 1.5. 3. 10. and 100. They confirmed the robustness of T:2 for p :- 2 as found by Hopkins and Clay and concluded that the actual level of 36 significance increases when any of the following occur: (1) number of variables increases. (2) total sample size with equal groups decreases. or (3) number of heterogeneous dimensions increases (i.e.. all di . d). They also stated that.'equal sample sizes help in keeping the level of significance close to the supposed level. but have little effect in maintaining the power of the test“ (p. 125). In general. power was often considerably reduced by departures that left the significance level satisfactory. In a third empirical study of the robustness of T2. with p - 2. 6. or 10. Hakstian. Roed. and Lind (1979) did not use covariance matrices in canonical form. However. all variances in one population were equal to one and covariances had an irregular pattern. Two distinct matrices were used for a second population. where all elements were greater than in the first by a factor of 1.44 or 2.25. For two variates. robustness was evident with equal sample sizes as small as six. For unequal sample sizes. their results paralleled the previous studies. Additionally. they found that increasing the total sample size while keeping the ratio of sample sizes constant does not help. and may actually hurt. the situation. In summary. while the T2 test is robust to covariance matrix heterogeneity with equal n's. it is not robust with unequal n's. The latter is true even for relatively mild departures from equality of the covariance matrices and of 37 sample sizes. 5eneLa1_HANQ¥A_TEEL_SLAIIELIGE The MANOVA tests discussed in this section are all functions of the eigenvalues of HE'l. where H and E are the hypothesis and error SSCP matrices. For two groups. the tests are equivalent. Hotelling's T02 is a generalization of the T2 test. which may be used with more than two groups. The T statistic is often used in place of T02. since they are directly related (i.e.. T02 - dfe'r). Robustness studies of multivariate test statistics for more than two groups have shown that. in general. these test statistics behave in a manner comparable to the univariate F-test. One of the earliest and most cited theoretical studies of multivariate robustness to heterogeneity of covariance matrices was conducted by Ito and Schull (1964). They investigated the asymptotic distribution of Hotelling's T02 statistic. with one to four variables and two to five groups. For the case of two large samples of equal size. they showed analytically that the test is fairly well behaved. with respect to both significance level and power. in the presence of heterogeneity. Also. for samples of nearly equal size. robustness holds as long as the characteristic roots of 2221'1 fall in the range (.5.2). For two large samples of unequal size. the departure from a nominal alpha level of .05 increased as: (l) the ratio of 38 sampl of he depa1 incr« the: effe Howe even effe« test. signj the a had t sample sizes (I 3 n1/n2) departed from one. (2) the degree of heterogeneity (d - the characteristic roots of 2221'1) departed from one. or (3) the number of dependent variables increased. For more than two groups and equal samples. there was a tendency to overestimate significance. but the effect was not serious with moderate heterogeneity. However. if one or some of the groups were of unequal size. even moderate heterogeneity conditions produced large effects on the significance level and the power of the test. In all cases with unequal sample sizes. actual significance was greater than .05 if the larger group had the smaller variance and less than .05 if the larger group had the larger variance. In an empirical study of the robustness properties of Hotelling's T. Wilks' likelihood ratio criterion W. and Roy's largest root R with small equal samples (n =- 5 or 10). Korin (1972) specified departure from equality of covariance matrices in two ways. symbolized by A(d) and Bid) with d a 1.5 or 10. A(d) represents cases where only one population covariance matrix differed (i.e.. (I.I.dI) for k 8 3 and (I.I.I.I.I.dI) for k = 6). while B(d) represents cases where two differed (i.e.. (I.dI.2dI) for k - 3 and (I.I.I.I.dI.2dI) for k - 6). Results showed that the three tests were somewhat comparable and that. although they were all liberal. R tended to be more so than did the other two. The discrepancy between nominal and actual 39 values was slight with small violations of covariance homogeneity (d all:1.5). but was pronounced with larger violations (d = 10). This indicates that. unlike the large sample case. with small n even equal samples do not guarantee robustness. A very extensive Monte Carlo investigation of the performance of six multivariate test criteria under heterogeneity conditions was conducted by Olson (1973. 1974). He considered groups of equal sizes (n - 5. 10. and 50) with both number of variables and of groups equal to 2. 3. 6. and 10. With populations having distributions N(Q.I) or N(Q.D). he used two types of contaminating covariance matrices (i.e.. where all canonical dimensions varied equally. D18 61. or where only one dimension varied. D =- diag(pd-p+1.l.....1)). with d . 4. 9. or 36. For a given value of d. total variability in both matrices. as measured by the trace of D. were equal. Therefore. only the manner in which variability is allocated was varied for a given d. and not the total variability. The latter being varied by different choices of d. Under various combinations of these factors. Olson examined Type I error rates and power of Roy‘s largest root. R. two trace-type tests (Hotelling-Lawley‘s T and Pillai-BartlettIs V). and three determinental tests (Wilks' likelihood ratio. W. Gnanadesikan's criterion. U. and Olson's alternative criterion. S). The U and S tests 40 tend favc furt test libe susp asym that error hypot were andw thoug robus equal overe exten sUbst four Chapt GXCee. dimem inCreE A180, tended to be quite conservative and did not respond favorably to violations and so will not be discussed further. Olson concluded that. although the remaining four tests all tended to be liberal. the R test was far too liberal and should be rejected if any heterogeneity is suspected. For large samples. the V. W. and T tests are asymtotically equivalent and he suggests as a rule of thumb that they may be so considered whenever degrees of freedom error are at least 10p times larger than degrees of freedom hypothesis. For smaller samples. the T. W. and V tests were robust against mild heterogeneity. but in general. T and W did not fare as well. Findings showed that even though it tended to be liberal. the V test was the most robust under the conditions examined. These results with equal samples uphold Korin's (1972) conclusions of overestimation of significance level for small samples and extend them to even moderately large samples (n - 50). Although departures from assumptions have substantially different effects on the distributions of the four test statistics to be considered in this study (see Chapter IV). general conclusions for equal samples are that exceedance of nominal alpha may be decreased by reducing dimensionality. p. or number of groups. k. However. increasing sample size with equal n's does not always help. Also. even though the percentage exceedance tended to be 41 greater at larger nominal alpha. Olson (1974) found that ”different proportions of contamination showed their effects in much the same way at all three significance levels" (i.e.. for .01. .05. and .10) (p. 898). In general. exceedance rates increased with greater heterogeneity. but they "tended to increase more as d increased from 1 to 4 and from 4 to 9 than as it increased from 9 to 36" (p. 898). Furthermore. regardless of p and k. effects were relatively minor when only one canonical dimension varied (D - Ch”) but severe when they all did (D - d1). For situations where D = CH. larger n's corresponded to lower exceedance rates for R. T. and W whereas for V. rates either decreased or increased as necessary to converge to T and W for large n. This is due to the fact that. for small n. V was significantly better than the other tests in many of the cases. It should be noted that. for equal samples. when D a dI ”effects of kurtosis and heterogeneity tend to be in opposite directions. the former yielding conservative rates and the latter producing too many significant results“ (Olson. 1974. p» 901). With respect to power. differences among the R. T. V. and W statistics were typically small. However. the R statistic tended to have slightly higher power if differences in the population mean vectors were confined to one of the 8 dimensions. while the V statistic had a slight 42 advantage if the differences were equally pronounced in all the 3 dimensions. Furthermore. holding the noncentrality parameter constant. increasing the number of groups tended to decrease power. while increasing group size had no consistent effect on power. Another Monte Carlo study on the significance levels of -R. T. W. and V test criteria with equal n's. but where heterogenetity was modeled on the original covariance matrices and not on the canonical dimensions. was conducted by Ceurvorst (1980). He considered a variety of situations that included varying the number of dependent variables (2 and 3). number of groups (2. 3. and 6). degrees of freedom error (18. 60. and 180). and both type and degree of heterogeneity. For differences of type. he considered inequality of variance alone. of correlations alone. and of both together. with combinations of three variances (l. 4. and 9) and three correlations (.2. .5. and .8). For heterogeneity of correlation he found only mild liberal exceedance rates for the four test statistics using a .05 nominal alpha. The observed significance levels were always less than..09 and proved to be fairly robust in most cases. Results for heterogeneity of variance confirmed previous results for canonical forms. plus indicated that the effects did not depend on the magnitude of the common within-group correlation(s) for any of the cases considered. Comparisons among the test criteria showed 43 that they were consistently ordered R-T-W-V from highest to lowest exceedance rate of the nominal alpha. The most serious discrepancies occured when k = 6 and five groups had variances equal to unity. while the sixth had variances equal to nine. When both heterogeneity of variance and of correlation were present. results differed depending on the relative size of variances and correlations. If groups with the largest variances had the largest correlations (LVLC). violations became increasingly more serious than for heterogeneity of variance (HV) alone. If groups with the largest variances had the smallest correlations (LVSC). the reverse was true. with violations being less serious. Comparisons of the criteria under LVSC conditions were similar to the HV situations with the V test being uniformly most robust. followed in order by W. T. and R. Under the LVLC conditions. no criteria was uniformly best. When only one variance differed. R was often the best choice. but it was the worst when all variances differed. Also. when R was best. the other tests generally had exceedance rates that were .07 or less. Pillai and Sudjana (1975) studied the effects of unequal covariance matrices on the R. T. V. and W statistics in the exact case by deriving central and noncentral distributions and applying them in a numerical study with n - 5. 15. and 40. Considering p = 2. they 44 stated that low heterogeneity produces modest changes in the powers of the test statistics. but that changes become pronounced as heterogeneity increases. None of the four statistics showed an advantage over the rest. In summary. the discrepancy between actual and nominal alpha tends to decrease with lower degrees of heterogeneity. and with smaller number of variables and of groups. It appears that. for two equal samples. neither the significance level nor the power of Hotelling's T2 is seriously affected by heterogeneity. but that this is not necessarily true for unequal n's. For more than two groups of large equal samples. robustness may be achieved with moderate departures from homogeneity. but even moderate heterogeneity produces large effects on both significance level and power when samples are unequal. For several small or moderately large groups. even equal samples do not protect against departure from nominal significance levels. with test criteria tending to be liberal. Consequences of violation of the homogeneity assumption through a contaminating covariance matrix is generally worse if all canonical variances differ by an equal amount than when only one differs. The case of only some equally discrepant variances falls between the two extremes. In general. Roy's largest root. R. appears to be the worst of the invariant tests and Pillai-Bartlettls trace. V. the best. with respect to both robustness and power. 45 CHAPTER IV METHOD Previous work exploring the robustness of MANOVA test criteria to violation of homogeneous covariance matrices across groups has dealt only with fixed-effects between- group tests in a one-way classification. In the present research. the effect of violating the assumption of homoscedasticity was considered in a repeated measures (RM) situation with data from ordered time points and a fixed- effects. one-way design over the subjects. The purpose of the study was twofold: (l) to compare the robustness of multivariate test statistics for between-group and within- group tests. and (2) to analyze the behavior of within- groups tests under various conditions with respect to both robustness and power. When data in the p-variate response measures reflect the passage of time. and assuming no group by measures interaction. overall within-group tests encompass all p-l degrees of freedom (df) for trend. thereby testing the null hypothesis of no trend in the data or. equivalently. of equality of occasion means. Subsequent tests may be confined to any p-q-l degrees of freedom (df) remaining after a q s p degree trend is hypothesized. 46 In this chapter. details are presented about the use of covariance matrices in canonical form for RM analyses. the parameters and procedures for the study. and the Monte Carlo techniques that were used. W The assumption of homoscedastisity for tests of between-group differences relates to population covariance matrices for the original score vectors. For simplicity. canonical forms of the covariance matrices are typically used in MANOVA robustness studies (see Chapter III). For MANOVA of RM. the score vectors are linearly transformed to reflect the design on the measures. The transformed vectors remain multivariate normal if the original vectors are multivariate normal (Finn. 1974. p. 62) and the assumption now relates to the transformed covariance matrices. With time ordered data. the transformation consists of a matrix of normalized orthogonal polynomial coefficients. When such a matrix is applied to populations with covariance matrices reduced to the canonical forms I . dI. or C(d) - diag(d.1.....1). the transformation does not alter I or dI. Although C(d) becomes a general matrix. it is reduced to C(d) when diagonalized. Therefore. the same underlying violation is modeled for both between-group and within-group tests when the original covariance matrices are in canonical form. 47 mu nu he so be n01 pal to vic COR set 9E0 wer trel Ovex big} nUll ofa Stat largt trace W A major problem in any study of robustness of multivariate test statistics comes from the seemingly vast number of ways in which the assumption of covariance homogeneity can be violated and the many factors that have some bearing on levels of robustness. Therefore. it becomes necessary to specify these factors and nonconforming populations in terms of some relevant parameters and to choose particular levels of each in order to have a systematic coverage of different forms of violations under various conditions. The parameters considered in this study are described in this section. Tests_nf_unltiyariatg_Hypgtheses. For each simulated data set. tests of a between-group hypothesis and two within- group hypotheses were conducted. The hypotheses tested were: (1) the null of no between-group differences on p-variate mean vectors. using k-l df. (2) the null of no trends across the p-variate data. using p-l df. for an overall test on occasions. and (3) the null of no trend higher than linear. using p-2 df. Rejection of the second null hypothesis. but not of the third implies the existence of a linear trend across time. Test_£riteria. For each hypothesis. four multivariate test statistics. defined in Table 2-4. were calculated: Roy's largest root. R. Hotelling-Lawley trace. T. Pillai-Bartlett trace. V. and Wilks' likelihood ratio. W. 48 W. Experiments were simulated with p - 4 or 5 response measures. This enabled both within- group tests to be multivariate. Since a multivariate test for linearity uses SSCP matrices of order p-2. the smallest value for p that allows for such a test is four. The SSCP matrices for hypothesis and error were: (1) of order-4 or 5 for between-group tests. (2) of order-3 or 4 for overall occasion tests of no trends. and (3) of order-2 or 3 for tests of no trends higher than linear. .Numher_gf_fironps._k. The simulated experiments had simple one-way fixed designs on the independent factor with k I 2. 3. or 6 groups. Group_§ize._n. Small to moderately large experiments were simulated. with n a 10. 20. or 50 in each group. In all cases. groups of equal size were considered. T¥2e_nf_Heferngeneity. The identity matrix. I. was used to model homogeneous populations. For heterogeneity conditions. two populations with covariance matrices equal to I and d1 were used. This type of diffuse structure was chosen for the contaminating matrix because it is the kind of violation that typically produces the most severe departures from nominal significance levels. Degree_of_fleterggeneity._d. This factor relates to the size of the violation (i.e.. to how much more variable one distribution is relative to another). Small to large violations were modeled. with d a 2. 4. or 9. For 49 homogeneity conditions. d I l. fiignifigange_neygl‘_2. The probablity of making a Type I error was considered at the .01. .05. and .10 nominal alpha levels. For a given nominal level. (100a)% of the values in a test statistic's distribution will exceed the appropriate critical value under a true null with no assumption violation. Hence. a dependent variable in the Monte Carlo experiments was the empirical estimate of a statistic's percentage exceedance of its critical value at significance level alpha. given a true null and heterogeneous covariance matrices. (The phrase percentage exceedance is used throughout the thesis to refer to the percentage of replications of a statistic that exceed a critical value). Pager. This is equal to l - PWType II error). Nominal power relates to the percent of values in a test statisticfls distribution that will exceed the critical value under a true alternative with no assumption violation. A second dependent variable in the Monte Carlo experiments was the empirical estimate of actual power (i.e.. the percentage exceedance given a true alternative and heterogeneous covariance matrices). This was conducted at all three nominal alpha levels. Power is a function of the discrepancy between central and noncentral distributions for a test statistic. The MANOVA noncentrality parameter (ncp) is a standardized 50 measure of the distance between group means in the population (Olson. 1974) and may be defined as the sum of the eigenvalues, gj (j - 1.....p). or trace of a matrix G. where G - FV'l. V is the population covariance matrix and k F = 1E1n1(£i - l)(£i “.Ei'. where 11.1 is the population mean vector for the ith Of k groups and u is the grand mean vector in the population. When data are ordered according to time. the ncp for tests of within-group hypotheses incorporates the time dimension. This is done by representing the elements of the covariance matrix and the means in the above equations as functions of time (Morrison. 1972). Since power depends on the common covariance matrix V. no theoretical power values exist under heterogeneous conditions. and the choice for V is open. Therefore. the noncontaminated covariance matrix (I in canonical form) is typically used for V in order to calculate the ncp. In this way. a comparison can be made between a test's ability to detect differences when assumptions are violated to its ability to do so when they they are met. Procedures Monte Carlo techniques were used to generate either 10.000 or 2.000 replications of multivariate data sets 51 distributed N(.Q.I). Each data set represented a particular combination of k and n with p - 5 measures across time. Using these data. critical values were calculated for tests of three multivariate hypotheses using four test statistics at three nominal alpha levels. The data in each set were then transformed seven times to calculate: (1) actual significance levels in three central heterogeneous cases for between-group and within-group tests. (2) nominal power in a noncentral homogeneous case for within-group tests. and (3) actual power in three noncentral heterogeneous cases for within-group tests. All calculations were performed a second time on the same data sets using only the first four measures to simulate conditions with p - 4. Since noncentral situations refer only to tests of within-group differences. in these cases. the null hypotheses of no group by occasion interaction and of no group differences remained true. A FORTRAN V program was written to generate. transform. and analyze the data. A detailed description of the computational procedures appears in Appendix A. These procedures guided the creation of the computer program. which also appears in Appendix A. The remainder of this section describes the determination of critical values. the design for the study. the analysis procedures. and the interpretation of computed significance levels and power values. 52 WWW Critical values for the multivariate test criteria and the combinations of p. k. n. and alpha levels used in the study were not all available in published tables. Also. tabled values have generally been obtained analytically rather than empirically. Therefore. values used in the study were empirically determined via Monte Carlo techniques. Using three nominal significance levels. critical values were calculated such that (1000)% of the N noncontaminated replications (where N - 10.000 or 2.000) under a true null would be judged significant using that critical value. This was accomplished by taking the arithmetic average of the (Na)th and (Na+1)th smallest of the N values for W and the corresponding largest of the N values for R. T. and V. Values thus obtained will be referred to as Monte Carlo critical values to distinguish them from tabled values. War The design for the study is given in Table 4-1. where combinations of k and n used for all levels of p and d are denoted by an x in part (a). Hypotheses tested under central and noncentral conditions with four statistics at three nominal levels are indicated in part an. The matrix in part (c) shows how the two types of conditions from (a) and (b) were combined to create the necessary statistics. 53 Table 4-1 Design for the Study a) b) umber of measures (p). of groups (k). and equal sample sizes (n) under heterogeneity conditions (d) . * k: 2 3 6 n: 10 20 50 10 20 50 10 20 50 Condition d p Hanogeneity 1 5 x x x x X 4 X X x X X Heterogeneity 2 5 x x x x x 4 X X X X X 4 5 X X X X X 4 X X X X X 9 5 X X X X X 4 x x x x x * x indicates conditions replicated 2.000 times. Conditions replicated 10.000 were with k -- 3 and n = 20. Statistics calculated under central and noncentral conditions for various hypotheses at three nominal alpha levels and for every combination of factors indicated in (a). * Alpha: .01 i .05 .10 Statistic: RTVW RTVW RTVW Condition Hypothesis Central B C L Noncentral C I. * Hypotheses tested were: bebveen-group differences. B. within- group test of trends. C. within-group test of trends higher than linear. L; using test statistics: Roy's largest root. R. Hotelling—Lawley trace. T. Pillai-Bartlett trace. V. and Wilks' likelihood ratio. W. 54 Tuflce4€l(Canu) c) Empirical values derived from each replicated data set by crossing elements from conditions on covariance matrices in (a) and cmifltflauson hnxnheafisin a». (Axflutflu:on<2warhmxmaMNUflces Hanxpneuy' Hehumgmudty Central Monte Carlo Actual significance Condition critical values levels on ngmhaum lkmcanxal Nmunalrnwm: AcUrfl.pmnm induce \HUues For the first part of the study. five-variate vector scores from a population distributed N(Q.I) were generated for 10.000 replications of one situation with three equal groups of size 20. Four-variate situations were simulated by using the same data and dropping the fifth measure in each vector score. For the second part of the study. new sets of 2.000 replications from the same population were generated for each of the five combinations of k and :1 indicated in Table 4-l(a). Equal cell sizes were used throughout the study and the same procedures followed for every combination of p. k. and n. regardless of the number of replications. Calculated statistics from the data in each set of N replications under homogeneous conditions were used to 55 determine Monte Carlo critical values for all combinations of multivariate tests. test statistics. and nominal alpha levels shown under the central case of Table 4-1(b). Regardless of the number of groups represented. score vectors for only one group in each case were transformed to simulate data that might arise from populations distributed N(Q.dI). The data sets represented central heterogeneous conditions. and all test statistics were recalculated. Each resulting value was then compared to the corresponding Monte Carlo critical value for the three alpha levels considered. Actual significance levels (i.e.. empirical Type I errors under heterogeneity) were determined by counting the number of values in each replication that were: (1) greater than the corresponding critical value for R. T. and V statistics. and (2) less than the corresponding critical value for W statistic. and then dividing by N. the number of replications. To investigate the power of multivariate within-group tests under true alternatives for the occasions. the original noncontaminated data sets (with d - 1) were transformed to reflect a given curvilinear trend across time. Under homogeneity and an alternative condition for within-group tests only. the above calculations were again performed to determine Monte Carlo nominal power values for tests of within-group differences. 56 The final step in the process was to add the curvilinear trend to the heterogeneous data sets and repeat the calculations to determine actual powers for within- group tests under noncentral heterogeneous conditions. By comparing these values to those for nominal power. the effects of heterogeneity on the power of within-group tests under an alternative hypothesis could be evaluated. CW In order to empirically determine whether between- group and within-group test statistics respond differentially to identical heterogeneity conditions under true null hypotheses. one experimental situation with k - 3 and n - 20 was replicated 10.000 times. The large number of replications was used in order to insure relatively small standard errors. The main interest in a comparison between actual significance levels for the group and occasion tests was examined from two perspectives. First. tests were compared within a given p to simulate practical analyses where tests of both hypotheses are performed on the same data set. However. discrepancies between actual significance levels evidenced here might occur because group and occasion tests are based on SSCP matrices of order-p and p-l. respectively. Therefore. a second comparison was made between the group tests with p = 4 and the occasion tests with p - 5. so that both would be based on order-4 SSCP 57 matrices. W Using the same 10.000 replications. comparisons of actual significance levels were made between the two sets of within-group tests for general trends and for trends higher than linear. With the data modified to reflect true alternatives for within-group hypotheses. the power to reject the null under heterogeneity was also evaluated. The second stage of the research was an attempt to examine the effects of heterogeneity on within-group tests when the number of groups and of equal sample sizes are varied. Both robustness and power were considered with 2.000 replications each for five combinations of k and n. Tests of between-group differences in the central case were also made in order to determine if discrepancies between these tests and within-group tests were sensitive to changes in number of groups and sample size. W The critical values and probability levels for significance (Type I error) and power were obtained via Monte Carlo methods and are therefore subject to sampling error. To take this error into account. the standard error (S.E.) of a proportion for a sample size equal to the number of replications was employed. The S.E. for a proportion depends on the true value of the proportion. P. and is equal to (P(l-P)/N)l/2. where N 58 equals the number of replications. Since the true value of P (i.e.. nominal alpha) is known. this formula may be used to calculate the S.E. at the three nominal alpha levels considered. These are given in Table 4-2. Table 4-2 Standard Errors for Nominal Alpha Levels and Number of Replications Used in the Study Alpha N . 2.000 N . 10.000 .01 .0022 .0010 .05 .0049 .0022 .10 .0067 .0030 .Monte_£arln_Technisues The methods for exploring the issues of robustness in this study involved the use of simulated data generated by computer algorithms. Through the analysis of a large number of samples under known population parameters. one can investigate the properties of statistics by observing their resulting distributions. These empirical distributions obtained under heterogeneity are then compared to the nominal distributions obtained under homogeneity for the statistics in question. The FORTRAN program was used to generate either 10.000 or 2.000 samples of vector observations for each experimental condition and 59 to perform the required data transformations and analyses. The procedures followed are specified in this section. The required data were 5x1 vector observations. normally distributed with known mean vector and covariance matrix. The generation and transformation procedures consisted of three steps: 1) Generate a set of independent random observations uniformly distributed on the interval 0 to 1. 2) Combine the uniform variates to create a set of observations normally distributed with mean vector zero and covariance matrix equal to the identity. 3) Transform these observations to obtain the desired structure with mean A and covariance matrix V. Each step will be considered separately. .Randnm_Numhex_Generation Hammersley and Handscomb (1964) stated that “the essential feature common to all Monte Carlo computations is that at some point we have to substitute for a random variable a corresponding set of actual values. having the statistical properties of the random variable" (p. 25). These values are called random numbers. In practice. what is actually produced via computer programs are a set of pseudo-random numbers calculated sequentialy from a completely specified algorithm. This algorithm is devised in such a way that a statistical test should not detect any significant departure from randomness. The subroutine GGUBS from the International Mathematical and Statistical Library was used to obtain a 60 sequence of uniform random numbers U1.....Un distributed EHO.1). This routine uses a congruential generator based on the following relation Xi - aXi-1 (mod m) where a - 75 and m - 2+31 - 1. Once the procedure is started by an initial seed value. each Xi is determined from the previous value. The constant terms a and m are chosen so as to maximize the period of the generator. since a sequence repeats itself when a value for X1 reappears. The numbers 01 - x1/231 are a pseudo-random sequence in the interval 0 to 1. They are independent of each other and behave as if they were random. We Several approaches are available to create independent normal deviates from uniform random numbers. A simple approach to program is based on the Central Limit Theorem (CLT) and uses a summation of a fixed number of values. where this number may be as low as 12 for reasonable approximations. However. the procedure ”is very slow and it does not adequately sample in the extreme tails of the normal distribution“ (Lehman. 1977. r» 148). The method used in this study for generating normal deviates from independent random numbers. which was devised to be reliable in the tails. was suggested by Box and Muller (1958). They cite a detailed comparison with several other methods. including the Central Limit 61 summation. and state that their approach gives higher accuracy and compares favorably in terms of speed. The procedure uses a pair of random numbers U1 and 02 from the same distribution on the interval (0.1) to generate a pair of normal deviates from the same normal distribution. N(0.1). The following transformations are used: 21 - (-21ogeUl)1/2cos2nU2 22 a (-2logeul)1/Zsin2u02 The resulting values are a pair of independent random variables. normally distributed with zero mean and unit variance. Vectors of five such variables taken together represent 5x1 observational vectors. which are multivariate normal and distributed N(Q.I) (Anderson. 1958. pp. 19-27). Observations of this form were used to simulate the central case with homogeneous covariance matrices. W The first step to determine a vector with specified variances and intercorrelations among the variables is to factor a known covariance matrix V into a lower triangular matrix T such that V =- TT' This is the square root method or Cholesky factorization of a symmetric positive-definite matrix. V (Bock. 1975. p. 85). Then. a transformation of a vector of normal deviates 1. Y a T1 + E 62 produces a normally distributed vector y with the desired characteristics. since Var(y) I T(Var(z))T' I TT' I V when var(z) I I. The only effect due to adding a known vector of means 1; is to change the point of central tendency for the distribution of y. In the present study. where V I dI. T 3 51/21 and therefore. 2 I (61/21); +.u . 51/21 ... 11 was the transformation used for one group to simulate data from heterogeneous populations in the noncentral case. Other transformations used the above equation with (l) u I Q for the central heterogeneous case. and (2) d I l for the noncentral homogeneous case. After generating the data. the program performed the required multivariate tests. calculated the critical values. and tabulated the proportion of times the values of each statistic exceeded its critical value for a given nominal significance level when: (l) a null hypothesis was true. and (2) an alternative hypothesis was true. Obtained proportions were the actual Type I error rates and powers. respectively. for the statistics. Multiplying these obtained values by 100 produces percentage exceedance rates under heterogeneity. 63 CHAPTER V RESULTS The results of the study are presented in this chapter in four sections. The first two sections are based on 10.000 replications of experiments with k I 3 and n I 20 and deal first with comparisons-of multivariate between- group and within-group tests with respect to robustness and then with the power of within-group tests under heterogeneity of group covariance matrices. The latter two sections present the effects of varying sample size and number of groups first on the robustness and then on the power of within-group tests under heterogeneity conditions. Results in these latter sections are based on 2.000 replications for each of five combinations of k and n. Critical values for each set of N replications (where N I 10.000 or 2.000) under central homogeneous conditions were obtained empirically through Monte Carlo methods and are tabled in Appendix B. Actual significance levels under central heterogeneous conditions and powers under noncentral conditions were calculated by determining the number of times obtained test statistics exceeded the corresponding critical values and then dividing by the number of replications. These empirical values were multiplied by 100 and are reported in this chapter in terms 64 of percentage exceedance rates of the Monte Carlo critical values. W The objective for this portion of the study was to determine whether heterogeneity of group covariance matrices produces differential effects for multivariate tests of between-group and within-group differences. 'The question could be phrased: Given no interaction effects and no main effects for either group or occasions in the populations from which the data are sampled. are there differences in the frequency with which rival test statistics indicate a significant effect for tests of between-group and within-group hypotheses under heteroscedastic conditions? A secondary question relates to differences between two within-group hypotheses (i.e. of no trends in the occasion means and of no trends higher than linear). The situation considered was that of three equal groups of size 20 with either five or four measures across occasions. The procedures for central conditions. which were detailed in Chapter IV. were followed. Since the data were randomly generated using computer algorithms. random error in the data must be considered. To insure that this error be small. 10.000 replications were used. Given known parameters (i.e.. nominal alpha levels). the standard error of a proportion with 10.000 65 replications (see Chapter IV) may be used to calculate 95% probability intervals around the known parameters instead of confidence intervals around the sample estimates. This produces the following intervals for the three nominal levels considered: .01 i .0020 .05.: .0043 .10 1 .0059 Expressed in terms of percentages. to correspond with tabled values. the 95% probability intervals are: (0.80. 1.20) at ..01 (4.57. 5.43) at .05 (9.41.10.59) at .10 Thus. obtained percentage values within these intervals may be considered to be within sampling error of nominal percentages. ‘ Critical values were estimated with Monte Carlo methods and. therefore. are subject to error. Since exceedance rates were derived from the same data sets transformed to heterogeneous conditions. the deviations from nominal levels in the following tables reflect only added error due to heterogeneity. As far as possible. parameters used in this part of the study will be discussed separately in terms of their effects on the percentage exceedance of Monte Carlo critical values for the three hypotheses under investigation (i.e.. of no between-group differences. B. of no trend over occasions. C. and of no trend higher than 66 linear. L). Table 5-1 contains the actual percentage exceedance rates (i.e.. empirical Type I error times 100) for central heterogeneous situations. .Significance_negel‘;1. Percentage exceedance rates for all three hypotheses tested increased with larger nominal alpha levels. except where obtained values were within 95% probability intervals of the nominals. Although the patterns were similar. increases in exceedance rates were greatest for the between-group tests. B. and lowest for the within-group tests of no trends higher than linear. L. However. when tests for a given hypothesis are considered with respect to standard errors. which also increase with alpha level. different amounts of heterogeneity showed consistent effects regardless of significance level. For example. at all three alpha levels. departures from the nominal for tests of B ranged from about one standard error with the V statistic at d I 2 to over 50 times the standard error with the R statistic at d I 9. Departures for the within-group tests were typically around one standard error with all test statistics at d I 2. and never exceeded 13 standard errors for tests of C and eight standard errors for tests of L at d I 9. 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Percentage exceedance rates were generally larger with five dependent variates than with four. Whenever this was not the case. discrepancies between corresponding exceedance rates at p I 4 and 5 were less than twice the standard error for a difference of two proportions. The smallest differences between exceedance rates for corresponding tests at the two levels of p occurred for the L tests. This may be due to the relatively small departures from nominal levels for tests of L. regardless of the number of variates. .DeHxee_n£_fietexngeneit¥i_d. In general. tabled values tended to be within 95% probability intervals of nominal values with low heterogeneity and. in all cases. the percentage exceedance rates increased with d. The effects of greater heterogeneity were the most pronounced for the B tests. where actual Type I error departed by as much as .16 from a nominal..10 level. However. discrepancies between actual and nominal values were less than .04 for the C tests and .02 for the L tests at a nominal .10 level. ‘Test_Statistig. Considering low heterogeneity. percentage exceedance rates tended to fall within 95% probability intervals of nominal values with the V or W statistics when testing the between-group hypothesis. while they did so with all four statistics when testing either within-group hypothesis. As expected from previous research on the robustness of between-group tests (e.g.. Olson. 1973). the 69 four tests statistics were ordered V-W-T-R from best to worst when testing B. Differences between actual Type I errors for the V and R statistics for B were always greater than twice the standard error of a difference. reaching as high as .13 with high heterogeneity and five variates. While results for tests of C generally followed the same ordering from best to worst statistic. those for tests of L did not. However. differences in departures from nominal levels among the statistics for tests of both within-group hypotheses were negligible. generally being less than twice the standard error of a difference and only once reaching a .01 difference. Except for tests of L. the effect of greater heterogeneity increased the differences between the best and worst statistic. This increase was considerably more pronounced for tests of between-group differences than for within-group tests of trends. Testa_nfi_Mn1tixariate_flypgtheses. Exceedance rates for within-group tests tended to be within 95% probability intervals of nominal levels only under low heterogeneity. For between-group tests. this tended to be true only when the V or W statistics were used. ‘To evaluate robustness in terms of acceptable Type I error. results were considered too liberal if they exceeded .015. .06. and .12 at nominal levels of .01. .05. and .10. respectively. Using these criteria when k I 3 and n I 20. only the between-group tests with T. V. and W statistics would be considered 70 robust under low heterogeneity. For within-group tests. robustness would extend to all four statistics and to moderate heterogeneity (d I 4). To summarize the differential effects of heterogeneity on tests of the three hypotheses and to examine more explicitly the differences among them. Table 5-2 provides differences in the actual percentage exceedance rates. The first two sets of rows relate to tests with a given p to simulate practical analyses with tests for both B and C performed on the same data sets. In the third set of rows the comparisons between B and C control for the size of the SSCP matrices from which the tests are derived. so that both sets of tests are based on matrices of order-4. In the lower half of the table are presented similar comparisons for the two within-group tests. As before. the first two sets of rows relate to within-group tests with the same initial p. while the third set compares the tests with equal SSCP matrices of order-3. The differences portray the extent to which departure from nominal levels were typically greater for tests on B than on C. Regardless of which set of comparisons was considered. the differences followed similar general patterns. The discrepancies in exceedance rates between tests on B and C tended to be less than two standard errors of a difference of two proportions when d I 2 or when the V statistic was used. 71 4791.503 «0 003.303 mumm 050 .0500 0:» so @003 300» o» 0»0H0u 03:000.: 3300:: 5H3 0093 03:3 .300 33 053 0:» so @003 300» o» 333 .m 63:000.: Hes—00 52: 009.30qu .3 .0303 EHmeHH .82»: 0:0 N> .003» 303331333 5. .003» >0H331§HH0»0: 8 Joe 3033 0&9. “0030330 »m0» H.505 53303330: no 03000 c »c 3 300:: :2» 30:3: 033» @9335“: 6 .0303» 93015233 8 60023033 mach—01:00:»0n Go 000052;: How Tm 033. 5 0030., scum @3333 303 009.30qu a vv.l mm. NH. mN.N mm. v°.1 cm.m mm.H NN. mm.N cm.H co. .3 Hm.1 mm.1 no.1 no.1 cm.1 cH.I Hm. NH. m¢.1 mm.1 me. no. No. HH.H VN. No. HH.I mH. ma. «o.m ov. mm.H vo. NN. wo.N ch.‘ Hm.H vv.N MN. he. Hm. mm.v Hm. Hc.N No.1 «m. >_ B cH. I c NN.I «H.I nc. om. wc.l we. Hm.H oo.H Nv. Hm. Hm. no. cw. HH. on.1 Na. av. mH. hH.N H¢.H Nv. mh.N mm.H vm. Hm.N HN.H hN. _3 NN. Nv. Nc. on. HH. so. an. ab. 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The largest occurred with the R statistic. where differences were as high a 12 percentage points. These patterns reflect that betweenegroup tests tend toward robustness when homogeneity is low or when the V test statistic is used and that actual significance levels for the 8 tests increase considerably from V to R, while they remain relatively stable across the four statistics for the C tests. Differences between the two within-group tests did not follow the patterns of the B and C differences. The discrepancies in percentage exceedance rates between tests on C and L were less than two standard errors of a difference with both d =- 2 and 4, as well as in over half the cases with d - 9. Regardless of the alpha level, these differences were typically negligible and rarely exceeded one percentage point. 12nwer_9f_Hithin:sr9up_Tests_nf_Irends The power of the tests to reject the null was evaluated under a homogeneous (d a l) and three heterogeneous conditions. The original 10,000 data sets for three equal groups of size 20 were transformed to 73 reflect the same quadratic trend over the five time points for each vector score. Since all the groups were equally transformed, this provides a situation with neither interaction nor between-group main effects, but with a within-group main effect. The percentage of rejections for the null hypotheses of no trend (C) and of no trend higher than linear (L) were determined. As shown in Table 5-3, power values were quite stable across the four statistics within a given heterogeneity condition and alpha level. This is fairly consistent with previous findings for power of between-group tests under heterogeneity (e.g., Olson, 1973), where differences across the test statistics, although sometimes present, were relatively minor. Regardless of heterogeneity, power was always larger at larger nominal levels. This trend follows what is expected under general homogeneity conditions, since “u.setting alpha larger makes for relatively more powerful tests 0f 30" (Hays, 1973, p. 359). Within each nominal alpha level, power decreased as heterogeneity increased. For example, with p - 5, power of the C test at nominal .01 went from over 90% under homogeneity to around 30% with a high degree of heterogeneity (d - 9). At .05 and .10, power dropped from 98% and 99% to slightly over 50% and 65%, respectively. This downward trend was remarkably consistent among all 74 S can m u a mom. 2. m. a. v. 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I 5 no. I 5 Ha. I a .mgflnfimui was ~85 Him 0.33. Bums 905155; How mwuom 8:33on wmcucmozm 75 test statistics for both hypotheses using both values of p. The only difference among the four conditions was one of magnitude. With five variates, power for the subsequent L tests tended to be slightly better than for the corresponding C tests. However, with four variates, the reverse was true, with a dramatic loss in power occurring between the C and corresponding L tests (e.g., going from 86% to 48% under homogeneity at the .01 nominal level). Comparing p a- 5 to p - 4, power dropped only slightly for the C test, but significantly for the L test. The reason for the substantial reduction in power for the L test with four variates seems to be due to the nature of the transformation used to create an alternative hypothesis condition. While the curve was strongly curvilinear with five measures, a linear trend serves as a reasonable approximation of the data when only the first four measures were used in the analyses (see Figure 5-1). To test the above hypothesis and further explore effects of heterogeneity on power, a second trend transformation was used that resulted in more pronounced curvilinearity at four time points (see Figure 5-2). Power results for this second curve are presented in Table 5-4. Comparing tables 5-3 and 5-4 shows that both Monte Carlo nominal values and obtained values under homogeneity were quite similar in all cases when p = 5, as 76 12345 1254 occasions occasions Figure 5-1. Trend transformation for power results of Table 5-3 with mean vectors: (0 .4 .8 .5 .1) for p- 5 (0 .4 .8 .5) for p - 4 Means Means ; : : . 0 . ; f : 1 2 3 4 5 l 2 3 4 occasions occasions Figure 5-2. Second trend transformation for power results of Table 5-4 with mean vectors: (0 .6 .7 .2 .05) for p a 5 (0 .6 .7 .2) for par 4 77 .v I a bow .N. h. m. 8 can m u m ...ou 3c. N. h. e. 8 303 muouoo.’ :8: .3 .033 poonHmeHH .mxHHz can »> 603» yumflummleHHE «a. .83... aonslmcHHHmuom E .33 umomumH mxom «onumHumum umou 05m: 3 .umocHH can... .852 8:3... msoumncHfiHa 6 .85.!» mucuEfiHz “mum: @963 8858»: 43358.5: muowHuuu H u E 3353335 no common 0 oops: $398... n my; 8 I c 33 mo mucoum H350 m I x no 233333 28.3 50.5 ... mm.mm mo.om om.mm mm.mm mh.o¢ mm.w¢ mo.w¢ om.ev ov.NN mH.NN mm.NN om.mN m ow.mm mh.mm om.mm mv.mm mm.mn mo.¢h oo.vh ch.mn co.mv om.vv cN.mv om.m¢ v mm.vm om.vm os.vm ch. 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I a me. I 5 Ho. I a .mgfimcuoufi 3:. 83:8: 5... vim anmB momma. ““3515:qu you mwumm 356896 003538 78 well as for tests of C when p - 4. However, exceedance rates for the L test were considerably higher with the second trend transformation than with the first when p =- 4. These results were consistent with those for five variates using the first transformation. There was a slight gain in power from the C to the L test regardless of size of p. Also, as in the first case for the C test, there were slight reductions in power when going from p s 5 to 4 with both tests. B l | n i y . C i'l' Having shown that multivariate tests of between-group and within-group differences respond differently to violations of homogeneity, the second stage of this research was an attempt to evaluate the effects of heterogeneity on within-group tests based on different levels of k and n. The design for this part of the study allows for an assessment of robustness and power under heterogeneity when: (1) sample size is varied (with equal groups of 10, 20, and 50), while holding the number of groups constant at three, and (2) the number of groups is varied (k a 2, 3, and 6), while holding sample size constant at 20 per group. Results tabled in this section deal with tests of the hypothesis of no within-group trends, C. The data in this and the following section are based on 2,000 replications each of five combinations of k and n. 79 With a reduction in the number of generated data sets comes an increase in standard errors. Therefore, using the same procedure as before, 95% probability intervals for the three nominal levels considered now become: .01 i .0044 .05 i .0096 .10 i .0131 In terms of the tabled values, which are expressed in percentage form, these intervals are: (0.56, 1.44) at .01 (4.04, 5.96) at .05 (8.69,ll.31) at .10 W Table 5-5 gives the percentage exceedance rates of within-group tests of trends over occasions, C, for experimental conditions with k = 3 and equal groups of size 10, 20, and 50. ‘With samples of size 10, actual significance levels were within the 95% probability intervals of nominal values only if heterogeneity was low (d a 2). Increasing the sample size to 20 brought improved results (e.g., exceedance rates were also within 95% probability intervals with d = 4 in all cases with four variates and about half the time with five variates). For large samples of 50, values were additionally within these intervals about half the time with d = 9. When outside the confidence intervals, empirical significance levels were all liberal. However, excluding results with d - 9, Type I errors did not exceed .02, .08, 80 .3 .258 82:3: .933 c5 .> .68: fimflfimluza a. .8...» 333.2238: E ....03 $093 9.6% "moflmHuBm an...» mch: .o .855. 905.5:qu 0: Ho no: common. 38509»: SuHmcomououms uo 8.58 c Luce: magnum... m 53 c win no @955 H39 m u x we 2038:me ooo.~ scum .. 8]. cm.oH om.OH mo.cH mm.HH om.m mm.m om.m mN.w cH.H OH.H cH.H cH.H m mh.oH mF.OH on.OH mm.OH oN.m cN.m ON.m o¢.m oH.H OH.H OH.H oc.H ¢ cm.m mw.m om.m om.m om.v mo.v om.v mN.m OH.H cH.H cH.H mm. 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With sample sizes of 50. robustness extends to conditions with high heterogeneity (d a 9) when only four variates were analyzed. For a given sample size, the other factors in this study behaved in the same manner as previously described for 10,000 replications where k . 3 and n = 20. In general, departure from the nominal significance level increased as heterogeneity, number of variates, or nominal significance level increased. The main difference across conditions with different sample sizes was one of degree. With respect to the multivariate test statistics, in only about half the cases did the R statistic produce the greatest exceedance rates and the V statistic the smallest. But, where this was not the case. the two values were typically within sampling error of the nominals and their difference was within one standard error of a difference of two proportions. Even when the R statistic had larger exceedance rates than the V statistic, the differences 82 among the four statistics were considerably less pronounced than is typical for between-group tests. W. For tests of the second within-group hypothesis of no trends higher than linear, L, percentage exceedance rates followed the patterns of the overall within-group tests. Obtained values were either within 95% probability intervals of nominal values or liberal. In most cases, significance levels for the L tests were lower than those for the C tests. However, differences between them were typically negligible. Values for tests of L are tabled in Appendix D. WW3. Percentage exceedance rates obtained in this study under a true null for between- group tests were consistent with previously established results. These values are tabled in Appendix C. The differences between the B and C tests followed the patterns described earlier in this chapter regardless of sample size. The only difference was one of degree. Discrepancies between the tests were generally smallest with large samples and low heterogeneity, or with the V and W statistics. They were greatest with small samples and high heterogeneity, or with the R and T statistics. 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N 3, _> B m z, > B m 3. > B m c ca. I a no. I 6 as. I a «ON u C 5H3 max—Ohm... NO mung HON :52 was s amps: manna mocmcoooxm ommucoodom wlm manna 84 size is held constant at 20 and the number of groups varied, with k - 2. 3, or 6. The values tended to be within 95% probability intervals of the nominal alpha with both low and moderate heterogeneity regardless of number of groups. The major exception was with five variates at .10 alpha. where values tended to fall outside the probability intervals with moderate heterogeneity. When d . 9 values were all liberal. However, Type I errors were .02. .08. and .15 for corresponding .01. .05. and .10 nominal levels. Results would be considered robust with low and moderate heterogeneity in all cases. as well as in almost half the cases with high heterogeneity. An unexpected finding from this set of results was that the impact of heterogeneity did not appear to be greatest with the larger number of groups. In about half the cases, the largest exceedance rates occurred with k - 3. The remaining cases were split with the largest departures occurring about equally with either k s 2 or 6. It might be assumed that this result was due to the high level of robustness of the C tests, since over half of the values in Table 5-6 were within 95% confidence intervals of the nominal value. However, even when only considering values for d s 9, which were outside these intervals. in more than half of the cases the largest departures still occurred with k a 3, while the rest tended to occur with k = 6. 85 It appears that the impact of heterogeneity on within- group tests of trends is lessened by increasing the size of equal samples but that, for a given n. decreasing the number of groups may not help. With a given number of groups, the effects of the other factors being examined were not always evident. This is probably due to the fact that actual values tended to be within sampling error of nominal alpha except for high heterogeneity. Still, some patterns emerged. In general. differences among the four statistics were still relatively minor, never exceeding two percentage points. Departure from nominal alpha typically increased as heterogeneity and alpha levels increased, although the latter reflects larger standard errors at higher alpha levels. The effect of decreasing p was evident only when d = 9. where lower exceedance rates were associated with the smaller number of variates. MWWW As was evident when sample size was varied. differences in actual significance levels between the two within-group tests were minimal. The discrepancies between most of the corresponding exceedance rates rarely exceeded one percentage point. Percentage exceedance rates for tests of non-linearity when number of groups was varied are tabled in Appendix D. 86 .QQmpaLiaQn_1ith_flgtueen:grgnp_mests. ‘Unlike the results for different sample sizes, when number of groups was varied, the B and C tests responded differently not only in terms of degree but also in kind. For the 8 tests, departures from nominal alpha consistently increased when there were more groups (except if heterogeneity was low). It appears that, for this situation with equal groups of size 20, within-group tests under any level of heterogeneity were similar in robustness to between-group tests under low heterogeneity. W The effects of sample size and number of groups on the power of within-group tests of trends under heterogeneity conditions were assessed with the same 2.000 replications used to study robustness. The data were transformed to model the alternative hypothesis situation in Figure 5-1. Power trends for both within-group tests were similar to those previously defined for the same transformation with 10,000 replications. The tests for non-linearity had slightly better power than the overall tests for trends with five variates but had considerably lower power with four variates. Since this was consistent across the experimental conditions considered, only the results for the test of trends will be discussed. Holding other factors constant, power values were extremely stable across the four test statistics for a given condition. With 87 samples of size 10. there was slightly more variability among the test statistics but. even here. the discrepancies were not noteworthy. Therefore. results presented in this section are percentage exceedance rates for tests of trends averaged over the four statistics. Complete tables for both sets of within-group tests are included in Appendix E. W Table 5-7 gives the average percentage exceedance rates for three groups of size 10. 20. and 50 under noncentral conditions. For a given sample size. the results were consistent with those from the first part of the study. With respect to varying the size of equal samples. n had a considerable effect on power. which decreased as n did even under homogeneity (d = 1). This effect was compounded as heterogeneity was introduced. With n - 50. where within-group tests were robust. the effect of heterogeneity was negligible. particularly if alpha was greater than or equal to .05. in which case exceedance rates were still over 90% with d a 9. Although lower. power values with n a 20 were reasonable under low and moderate heterogeneity. where robustness was achieved. However. with only 10 subjects per group. power tended to be poor even with low heterogeneity. where tests were robust. This was particularly so at a .01 nominal level. where power was low even under homogeneity. 88 'hble 5-7 Average Percentage Exceedance Rates Under True Alternatives for Tests of Trends with k - 3* P-5 9'4 alpha: .01 .05 .10 .01 .05 .10 n d 10 l 45.04 70.38 81.61 38.19 64.65 78.01 2 33.10 58.21 71.16 27.69 51.64 67.25 4 21.85 44.53 58.19 18.81 38.89 52.39 9 14.98 30.93 43.00 12.05 26.69 38.95 20 1 91.33 98.26 99.51 85.10 95.75 98.14 2 77.93 92.31 96.46 69.24 87.96 93.86 4 55.71 77.65 86.63 45.81 70.53 82.16 9 28.84 52.58 65.04 23.40 44.61 58.99 50 1 100.00 100.00 100.00 100.00 100.00 100.00 2 99.95 100.00 100.00 99.53 100.00 100.00 4 97.95 99.74 99.94 95.61 99.26 99.65 9 76.15 93.06 96.34 67.25 88.64 93.88 * From 2.000 replications of k = 3 equal groups of size n with p measures under d degrees of heterogeneity «i=sl reflects homogeneity). Hypothesis tested was of no within- group trends. C. Tabled values were averaged over four test statistics: Roy's largest root. R; Hotelling—Lawley trace. T; Pillai-Bartlett trace. V. and Wilks' likelihood ratio. W. 89 W Table 5-8 gives the average percentage exceedance rates for equal groups of size 20 with two. three. and six group designs under noncentral conditions. Effects within a given condition were again consistent with results from the first part of this study. With respect to number of groups. power was best with k a 6 and worst with k a 2. For six groups. empirical power was above 90% in all cases except for d s 9 at .01 alpha with four variates. The powers with six groups and high heterogeneity were consistently higher than those for three groups with low heterogeneity. The effect of heterogeneity was considerable with both two and three groups. particularly at .01 alpha. However. in both these cases. power was reasonable with d = 2 and 4. where tests achieved robustness. as long as a nominal level of.01 is not considered. MW Considering all five combinations of k and n together. power under low and moderate heterogeneity seems to be a function of total sample size. N. .As shown in Figure 5-3 (power curves of values in Tables 5-7 and 5-8. which were averaged over four test statistics). for d s 2 and 4 power increases as total N increases until. with N = 120 and 150. the curves are indistinguishable. However. with d a 9. heterogeneity appears to have a greater impact on power 90 Table 5-8 Average Percentage Exceedance Rates Under True Alternatives for Tests of Trends with n - 20* P'5 ps4 alpha: .01 .05 .10 .01 .05 .10 d 2 1 70.65 87.19 93.01 63.14 82.00 90.04 2 47.31 70.46 80.89 40.58 63.60 75.19 4 26.10 47.96 61.86 22.45 41.84 54.88 9 13.55 27.99 39.78 10.96 24.14 35.03 3 1 91.33 98.26 99.50 85.10 95.75 98.15 2 77.93 92.31 96.46 69.24 87.96 93.86 4 55.71 77.65 86.63 45.81 70.53 82.16 9 28.84 52.58 65.04 23.40 44.61 58.99 6 1 100.00 100.00 100.00 99.88 100.00 100.00 2 99.81 100.00 100.00 99.44 99.90 100.00 4 99.31 99.83 99.94 98.11 99.29 99.75 9 93.04 97.34 98.86 86.84 94.81 97.51 * From 2.000 replications of k equal groups of size n a 20 with p measures under d degrees of heterogeneity (d = 1 reflects homogeneity). Hypothesis tested was of no within- group trends. C. Tabled values were averaged over four test statistics: Roy's largest root. R; Hotelling-Lawley trace. T; Pillai-Bartlett trace. V. and Wilks' likelihood ratio. W. 91 I“. ”4 In. 70. P a. E m R .4 c 30. E a. N u. T o A m. G 9.. E u. 7.. E sol X 5.. C .. E ... E 3.. 2*“ N C c m. E 90. so; ”+ ... 5.. ”a 30. 4 a. n: ‘33 .. .... ::?§::I;‘° «:z‘iégg:;1° '.:r‘:"*v; .;;':"’fi; d d Figure 5-3. Power curves averaged over four test statistics for different total sample sizes N. where: N830 40 so 120 150 H30 B-"El4o k =- 3 2 3 6 3 °'“°60 n-lO 20 20 20 so *-*120 9'9150 92 when a larger percentage of vector scores were heterogeneous. even though the group sizes were larger. This latter result is most likely due to the manner in which heterogeneity was allocated across the groups. coupled with the analysis being performed. The test for within-group main effects assumes that the curves of the k groups are parallel (i.e.. that no group by occasion interaction exists). The test then is used to evaluate whether the curves are constant (i.e.. if there is any trend across the occasions). Hence. this test compares the means of each measure over the total number of subjects. N. In all cases. only one group was drawn from a heterogeneous population. Therefore. since the 120 vector scores came from six groups of size 20. only 20 vectors (17%) were heterogeneous. The 150 vector scores were from three groups of size 50. so that 50 vectors (33%) were heterogeneous. The higher proportion of discrepant vectors in the latter situation may have produced the reverse effect at d = 9 than would be expected based on N. This result was consistent across all four test statistics and three alpha levels (see Appendix B). The identicalphenomenon occurred with small total samples. When N a 30 (k - 3 and n = 10). only 10 (33%) of the vector scores were heterogeneous. while half of the vectors were heterogeneous when N = 40 (k . 2 and n = 20). This reversal of power levels was consistently evidenced 93 across all but the V statistic. where power was slightly higher with N - 40 at the .01 and .05 nominal levels. When the percentage of heterogeneous vectors was held constant at 33% (three situations with k - 3). power decreased steadily as N decreased. Unfortunately. given the present data. the effects on power if total N is held constant while varying the percentage of heterogeneous vectors could not be evaluated. For the conditions examined. the results indicate that. for low or moderate heterogeneity. total N dictates the level of power but that. for high heterogeneity. the percent of discrepant vector scores has the greater impact. 94 CHAPTER VI DISCUSSION The results presented in the previous chapter provide an indication that multivariate tests of between-group and within-group differences are not equally subject to the effects of heterogeneity of covariance matrices. Conclusions based on these results will be presented in this chapter. followed by guidelines for the researcher analyzing repeated measures studies with time ordered data and suggestions for future research. conclusions Under the conditions considered in this study. it appears that multivariate tests for trends over occasions in repeated measures designs with equal groups are not as sensitive to violations of the assumption of homoscedasticity across groups as are tests for between- group differences. In most cases. within-group tests are extremely well behaved. while between-group tests tend to be robust when two groups are involved or if heterogeneity is low. This difference is most likely due to the manner in which the mean square hypothesis (HE‘l) is formed for the tests of the two hypotheses. The elements of the hypothesis matrix. H. for between-group tests consists of 95 sums of squares and cross products. while for within-group tests. these elements consist of squared sums of means and products of means. It is likely that this difference produced the differential effect that heterogeneity had on multivariate tests of the two hypotheses. Within-group tests for general trends tended to be robust even with heterogeneity of d s 4. Although such a level of heterogeneity was considered moderate in this thesis. covariance matrices that differ by a factor of four would indicate a dramatic discrepancy from a practical perspective. Hence. in most practical situations. where groups are equal and heterogeneity is present. multivariate within-group tests should be robust. Conclusions for between-group tests with equal samples uphold previous findings that these tests tend to be robust when covariance matrices differ only by a factor of two. ‘ Differences among the four test statistics considered were evident. with Pillai-Bartlettis trace. V. typically showing the least departure from nominal levels and Royus largest root. R. the most. However. discrepancies between the V and R statistics were relatively minor for within- group tests but pronounced for between-group tests. Even under low heterogeneity (d - 2). the R statistic on between-group tests tended to be liberal unless sample size was at least 50 per group. However. for within-group tests. R was robust at d - 2. as well as in over half the 96 cases at d = 4. For within-group tests. increasing the number of groups did not produce a consistent effect on actual significance level. but changes in the sample size and. to a small degree. the number of occasions did. ‘With four variates the tests were robust even under high heterogeneity'(d - 9) with equal samples of at least 50. With equal samples of 20. robustness was achieved with all four statistics under moderate heterogeneity (d - 2) when p - 4. and also when p - 5 except for Roy's largest root. This remained true for a constant group size of 20 regardless of whether there were three or six groups. Only with two groups of size 20 and four variates did robustness tend to extend to high heterogeneity. Actual significance levels for the within-group tests of trends higher than linear were typically closer to nominal alpha levels than for general tests of trends. although differences were minor. It is expected that. given a larger number of time points. a slight increase in robustness would be achieved with each succeeding multivariate test for higher order trends. However. given the fact that departures from nominal levels were typically more severe with five variates than with four. a word of caution is in order. It is likely that. with a significant increase in the number of time points. the overall test of trends would produce a too liberal test. Therefore. if the 97 initial departure from nominal levels is large enough. no worthwhile gains in robustness may result with successive tests for higher order trends. Also. tests of between- group differences become increasingly more liberal with increases in the number of variates. The combined effect on both tests would therefore need to be considered. The four multivariate test statistics evaluated in this study behaved almost as one with respect to their power to reject a null of no trend in the data. Heterogeneity affected them equally (i.e.. power was reduced as heterogeneity increased). At a given heterogeneity level. power of the second test for trends higher than linear was slightly greater than that of the overall test for trends when a fairly strong curvilinear trend was present. As would be expected. the subsequent test lost power dramatically when the trend tended toward linearity. Decreasing the number of subjects per group compounded the effect of heterogeneity on the power of within-group tests (i.e.. the smaller the group. the lower the power. even under homogeneity). However. power was still reasonable with equal groups of 20 or more. where the within-group tests were robust. W The analysis of RM data may be undertaken with either univariate or multivariate statistical tests. It is known 98 (e.g.. Davidson. 1972) that. as long as there are more subjects per group then there are groups. multivariate tests are the preferred choice when the univariate assumption of uniformity is violated. ‘With RM data from equally spaced time points. this assumption is generally violated. since autocorrelation is typically present. If more than one group is involved and the researcher further suspects that the assumption of a common covariance matrix across groups may be violated. the results from this study would indicate that the problem may not be too serious for within-group tests. even for violations as large as variances differing by a multiple of four. However. previous research has shown that this is not the case for between-group tests. Given this discrepancy. the question arises about how to analyze and interpret repeated measures data when heterogeneity across groups is suspected. Of particular concern are situations where group by occasion interactions exist. Since a test of this hypothesis must precede tests of both main effects hypotheses. the problem may be considerable. The mean square hypothesis for interaction consists of the (p-l)x(p-l) submatrices of the hypothesis and error matrices used for the between-group test. It may therefore be presumed that violations would cause effects similar to the between-group tests and results from between-group robustness studies should apply. 99 Hence. for tests of interactions. Pillai-BartlettIs V statistic should be employed and results interpreted with caution. If the number of measures is relatively small and equal groups have been maintained. then results may be considered valid. However. if this is not the case. an assessment of whether greater heterogeneity was present in a smaller or larger group should be attempted. In the former case. results would be liberal and. in the latter case. conservative. If it can be assumed that there are no group by occasion interactions. then a two-stage analysis would be recommended. The RM dimension may be tested through a multivariate test of within-group trends without excessive concern. An appropriate approach for testing between-group effects would be to use the mean of the RM as the dependent variable in a univariate analysis of group differences. This would eliminate concern for heterogeneity since it has been demonstrated repeatedly that. except when samples are small and unequal. the univariate F-test is robust against this violation. ' W An aspect that needs to be considered is unequal sample sizes. .Although it is quite likely that this would not produce any radical departures from nominal significance levels for within-group tests. it might do so for interaction tests. since the latter are based on 100 submatrices of hypothesis and error matrices of between- group tests. Also. with equal samples. consideration should be given to a study of the effects of varying number of groups while holding total N constant. thereby varying the n/N ratio. This would be particularly useful in further exploring the effects of heterogeneity on the power of within-group tests. Additionally. noncentrality structure would be a relevant factor for inclusion in a study of the power of within-group tests of trends. Different results in terms of power are likely to occur depending on whether noncentrality is concentrated in the first canonical variate. or spread equally over all canonical variates. Since the RM dimension in the present study represented the same measure repeatedly taken over time. a polynomial representation was used to transform the data for within-group tests. This transformation uses the regression model matrix (see Chapter II). which is a matrix of normalized orthogonal polynomial coefficients. If the measures are taken to correspond to a factorial classification. a treatment contrasts and interaction representation would be used. Since the various transformations each partition the source of variation for occasions in a different manner. it is possible that they may react differently to violation of homoscedasticity across groups. 101 The regression model matrix is orthogonal. Design matrices may be orthogonal. such as a Helmert contrast matrix used with hierarchical designs over the measures. or they may be nonorthogonal. such as a paired differences matrix used in profile analyses. It might be assumed that all orthogonal transformations would behave in like manner but that nonorthogonal ones would not. However. this assumption needs to be tested. Although findings from this study are of a preliminary nature. they provide strong evidence that. at least for equal samples. within-group tests of trends in a repeated measures design are fairly robust to violations of homoscedasticity that might occur in practical situations. Furthermore. these tests maintain reasonable power under heterogeneity. except for small sample sizes. 102 APPENDICES APPENDIX A COMPUTATIONAL DETAILS AND COMPUTER PROGRAM The data generation and analysis were performed on a CDC (Control Data Corporation) Cyber 750 computer at Michigan State University. This 60 bit word machine uses the Scope/Hustler operating system. The program. which was coded in FORTRAN V by Jeff Glass. uses some Compass assembly language to decrease field lengths and thereby reduce costs. .All computation was done in double precision. In this appendix. the following are listed or described: (I) the subroutines used from package programs. (2) the actual values input by the user. (3) the steps followed in the computer program. (4) the procedures performed to check the operation of the computer program. and (5) the complete listing of the computer program. W a) Subroutines taken from the International Mathematical and Statistical Library (IMSL. 1979): GGUBS To generate uniform pseudo-random numbers. VMULFF For matrix multiplication. 103 VMULFM For matrix multiplication of the transpose of a matrix A by a matrix B. VMULFP For matrix multiplication of a matrix A by the transpose of a matrix B. LINVZF To compute the inverse of a matrix. b) Subroutines taken from the EISPACK library (The Argonne eigensystem package. 1972): TREDZ To determine eigenvalues of a symmetric tridiagonal matrix. IMTQLZ To reduce a positive-definite matrix to a triadiagonal matrix for input into TRED2. W a) Seed values for data generation. A different value was used for each combination of k and n. These are listed in the Checking Procedures section in this appendix. b) Defining parameters for N. k. n. and p. c) Matrices of normalized othogonal polynomial coefficients for calculations using P(pxp). where p = S or 4. p1 . ".44721 -.63246 .53452 -.31623 .11952‘ .44721 -.31623 -.26726 .63246 -.47809 .44721 0 -.53452 0 .71714 .44721 .31623 -.26726 -.63246 -.47809 L-44721 .63246 .53452 .31623 .119524 P2 8 .5 -.67082 .5 -.22361 .5 -.22361 -.5 .67082 .5 .22361 -.5 -.67082 .5 .67082 .5 .22361 d) A vector of constants to be added to the second through fifth measures for use in power calculations. For the transformation used in all cases. this was: POWER g (.4 .8 .5 .1) 104 For the modified transformation used only with 10.000 replications. this was: POWER 8 (.6 .7 .2 .05) 2£Q§£3m_fitfinfi The program consists of five main components (RM. DATA. SORT. NTABLE. and OTABLE). Rm is the central part of the program and consists of seven subroutines. Additionally. a sixth component STATS was used to output the mean and variance of the generated data and to test the data for normality. Comments throughout the program explain the steps and provide information for its use at other installations. The actual steps used in the computer program are described below. Variable names are the same as those used in the program. except for the following: N - COUNT number of replications K s GROUP number of groups n SUBJECT number of subjects per group p a MEASURE number of repeated measures 2. - ZBAR matrix of group means A. Win 1. Use IMSL subroutine GGUBS to generate N replications of nkp uniform (0.1) pseudo-random numbers using double precision (where p = 5 and N - 2.000 or 10.000). 2. Transform these values to normal deviates N(O.l) using 105 3. Box-Muller transformation (see Chapter IV). Array resulting values in N matrices z of dimension (nk x p) such that FflT q. N(Q.I) with Zj (n x p) 2k 13.9an 1. 2. 3. 4. £cr_p_:_5. compute for each data set: (k x p) matrix of means ’201'.‘ Z. = . ... where z.j' = (z.j(1) z.j(2) z.j(3) 2'1”) z.j(5)) Preliminary SSCP matrices for: Total T = Z'Z Groups G = nz.'z. SSCP for calculations of test statistics for: Constant C = (n/k)z.'UU'z. . where U a unit vector Between B a G - C Error E a T - G Transform C and E matrices by P(pxp). a matrix of normalized orthogonal polynomial coefficients with resulting elements xij (1.2) = 1.---.P)- The transformed matrices are: 106 7. CTRAN = P'CP ETRAN ' P'EP Take the lower right (p~1)x(p-l) submatrices from (B-4) to be used for tests of occasion trends (elements at”. i.j - 2.....p). and label: CRM ERM Take the lower right (p-l)x(p-l) submatrices of (B-4) to be used for tests of non-linearity (elements xij' i.j - 3.uu.p). and label:‘ CL EL Compute HE”1 for tests of: group differences HE - BE-l occasion trends HC - (CRMHERID’1 non-linearity an - (CL)(EL)'1 Compute the eigenvalues for each test in (B-7) and label: EIGB (p values) EIGC (p-l values) EIGL (p-2 values) Using the formulas for the R. T. V. and W test statistics (see Chapter II) and the eigenvalues from (B-8). compute each statistic for the C. B. and L tests. Each resulting list of either 10.000 or 2.000 values is labeled according to the combination of test and test statistic used: 107 RB TB VB WB RC TC VC WC RL TL VL WL EDI—2.2.4: 10. Drop the last row and column of the B. C. and E matrices in (B-3) and repeat steps (8.1-9). C. W 1. Sort the 24 lists of test statistics in (B-9) (12 for p - S and 12 for p - 4). placing the values in rank order. 2. Using a =- .01. .05. and .10 and N a COUNT. calculate for each list: a) the average of the (Na)th and (Na+l)th largest values to be the critical values for the R. T. and V statistics (result - S4 critical values). b) the average of the (Na)th and (Na+l) smallest values to be the critical values for the W statistics (result - 18 critical values). D. W Repeat this step three times for d - 2. 4.. and 9. l. Transform the 2 matrix from (A-3) so that one group has larger variances. with resulting (pxl) score vectors: 113 3 61/2213. N N(Q'dI) i. g 1.....n j = 1.....p 21' ‘ 21. N N(QII) 1}: n+1'ooo’nk J J j 8 1.....p 108 2. Repeat (B.l-10) to get 24 lists of test statistics under a true null and a given heterogeneity condition. Compare each value in each list to the corresponding critical values from (C-2) at each alpha level and count the number of times each value is: a) greater than its corresponding critical value for R. T. and V statistics. b) less than its corresponding critical value for W statistic. For each list and alpha level. divide the resulting values from (D-3) by the number of repetitions (COUNT) to get actual Type I error rates. S.ELsminaticmLMentsJarlLNnminaLPcwers 1. Transform the 2 matrix from (A-3) so that all groups reflect a polynomial trend across the time points by adding a constant to each measure. The resulting (pxl) score vectors are: zij a Zij + 1 N N(1.I) § : 1:::::Bk Repeat (B.l-lO) to get 24 lists of test statistics under a true alternative and no violation to homogeneity. Repeat.(D.3-4) to get Monte Carlo nominal power values. 109 KW Repeat this step three times for d s 2. 4. and 9: 1. of the generated data. the following results were obtained for 1.000 replications of 300 data points with six randomly Transform the 2 matrix from (A-3) so that one group has larger variances and all groups reflect a polynomial trend across timepoints. The resulting (pxl) score vectors are: an . dl/zzij + g 6. N(LdI) 1 - 1.....n J . lpooo'p Zij . Zij + 11. N N(E'I) 1. 3 n+1.....nk j . lpoooyp Repeat (B.l-10) to get 24 lists of test statistics under a true alternative and a given heterogeneity condition. Repeat (D-3.4) to get actual powers under heterogeneity conditions. W Using the IMSL subroutine GTNOR to test for normality choses initial seeds: of the generated N(0.l) data points were calculated. Seed Chi-square Probability 444.852.461 5.78 .76 9.458.577.882 6.20 .72 11.261.152.461 14.68 .10 2.344.743.849 5.18 .82 2.341 5.08 .83 112.623.455 9.06 .43 For each experimental condition. the mean and variance 110 The initial seed used to generate the data for the study with the corresponding means and variances were as follows: COUNT k n Seed Mean Variance 10,000 3 20 739.604.919 1.1x10-4 .99901 2,000 3 10 740.848.519 -3.0x10'3 .99707 2 20 837.616.087 -1.6x10-3 .99895 3 20 739.604.919 -1.5x1073 .99870 6 20 344.148.214 -1.4x10-4 1.00005 3 50 203,577,315 -9.8x10-6 1.00000 A final check on the calculation of the SSCP matrices was performed by outputing the B, C. E, CTRAN. ETRAN. CRM. ERM. CL and EL matrices from one replication with k - 3 and n a 20. The results were hand checked to ascertain that the program was correctly calculating these matrices. To check the results of the Monte Carlo critical values. the parameters needed to find tabled values were determined. Tabled values were found in 92 cases (21.3%L. These were in fairly close agreement with the calculated values. 111 IDENT RM ASSEMBLY'LANGUAGE DRIVER PROGRAM ENTRY RM RM BSS 0 RJ -XXRM ENDRUN ENTRY ABORT ERROR TERMINATION - ABORT 855 I RJ -XENDOUT CLOSE OUTPUT FILES IN CASE OF ERROR ABORT END RM SUBROUTINE XRM CW C* RM - REPEATED MEASURES TESTING. C! C* THIS PROGRAM IS THE HEART OF THE RM PROGRAM SET. IT CALCULATES TEST C* STATISTICS R. T. V. AND W. WRITING THE RESULTS TO FILES FOR FURTHER C* PROCESSING BY OTHER PROGRAMS.‘ Ct CW C* TRANSPORTABILITY NOTE: C* THESE PROGRAMS WERE WRITTEN AS CLOSE TO STANDARD FTNS THAT ACTUAL C* CONSIDERATIONS OF COST WOULD ALLOW. C’ I) SEVERAL PROGRAMS (THE DATA GENERATION AND STATISTICS PROGRAMS. C* THE SORT PROGRAM. AND THIS ONE) HAVE ASSEMBLY-LANGUAGE MAIN 6* PROGRAMS TO REDUCE EXECUTION COST. THESE ASSEMBLY-LANGUAGE 6* ROUTINES MAY DE REPLACED BY FTNS PROGRAMS IF NEED SE. C: 2) STANDARD PTNs I/o HAS JUDGED TO BE TOO EXPENSIVE FOR MULTIPLE CA RUNS or IOOOO CASES. so A NON-STANDARD I/o PACKAGE CALLED FASTIO 6* WAS USED IN THE AFOREMENTIONED PROCRAHS. CA CONVERSION TO STANDARD FTNS I/O HOULD BE STRAIGHTFORWARD. FOR 6* THIS PROGRAM. ROUTINES SETOUT. OUTPUT. ENDOUT. AND CETDATA HANDLE Ct ALL THE I/O. AND ONLY THOSE ROUTINES HOULD NEED MODIFICATION. C* 3) THE CETREC SUDROUTINE AND THE CCL RECISTERS RI AND R2 6* (USED IN ROUTINE CETDATA) ARE HSU SYSTEM FUNCTIONS HHICH ALLOH Ce FTNS PROGRAMS TO COMMUNICATE WITH THE USER (AND OTHER PROGRAMS) Ct BY A NEANS OTHER THAN THROUGH FILES. SHOULD THESE PROCRAHS NEED C: To DE TRANSPORTED. THE USER COULD PROVIDE HIS OWN SUDROUTINE 6* CETREC HHICH NOULD RETURN THE SAHE VALUES. BY SOHE OTHER IFTNS Ct STANDARD) DEVICE. CW C* INPUT CONDITIONS: C* I) THE TEST DATA RESIDES ON LOCAL FILE DFILE . C* 2) CCL REGISTER RI IS SET TO THE VALUE OF D FOR AN ACTUAL-VALUE RUN: C* OR TO I FOR A NOMINAL-VALUE RUN. C* 3) CCL REGISTER R2 IS NON-ZERO FOR A POWERS RUN: ZERO OTHERWISE. Ct C* OUTPUT CONDITIONS: C* I) DFILE. RI. AND R2 ARE UNCHANGED. C* 2) LOCAL FILES TAPEI THROUGH TAPEZA CONTAIN THE TEST STATISTICS: CO C* R T V W C* P-5 D TAPEI TAPEZ TAPE} TAPEh C* C TAPES TAPES TAPE7 TAPES C* L TAPES TAPEIO TAPEII TAPEIZ C* P'A B TAPEI3 TAPEIA TAPEIS TAPEI6 C* C TAPEI7 TAPEID TAPEI9 TAPEZO C* L TAPEZI TAPEZZ TAPEZ} TAPEZA CW C* OPERATION OF RM PROGRAMS: C3 I) THE DATA-GENERATION PROGRAM DATA IS RUN. C* 2) THE DATA-STATISTICS PROGRAM STATS IS RUN. C* 3) CCL REGISTER RI IS SET TO I: R2 IS SET TO 0. C* A) THIS PROGRAM RM IS RUN. 1512 CA 5) FILES TAPEI THROUCH TAPEzb ARE SORTED: THE SORTED FILES ARE C* PLACED ON FILES TAPEzs THROUCH TAPEua IN THE SAME ORDER 6* ( TAPEI Is SORTED ONTO TAPE25: TAPEI7 IS SORTED ONTO TAPEAI: Ct TAPE IS SORTED ONTO TAPE ). 6* 6) OUTPUT PROCRAH NTADLE IS RUN. 6* 7) RI Is SET TO D. 6* 8) THIS PROCRAH RM Is RUN. 6* 9) OUTPUT PROCRAH OTADLE IS RUN. 6* I0) STEPS 7.8.9 ARE REPEATED FOR AS MANY VALUES OF 0 AS ARE DESIRED. CA II) RI Is SET TO I: R2 Is SET NON-zERO. CA I2) THIS PROCRAN RM Is RUN. 6* I3) OUTPUT PROCRAH OTADLE IS RUN. CA IA) STEPS 7.8.9 ARE REPEATED FOR AS MANY VALUES OF D AS ARE DESIRED. a: CA RISCELLANEOUS INFORNATION: CA I) LOCAL FILES DFILE AND NVALUE SHOULD *NEVER* BE RETURNED. CA 2) LOCAL FILE STATFIL MAY BE RETURNED AFTER STEP 2. -~ CA 3) LOCAL FILES TAPEI THROUCH TAPEzh ARE NOT NEEDED AFTER STEP 5. CA AND MAY DE RETURNED. CA A) LOCAL FILES TAPEzs THROUCH TAPEbB ARE NOT NEEDED AFTER STEP 6. Ct AND MAY DE RETURNED. Ct s) TAPEIOO Is USED FOR DEDUC PURPOSES. CW CA PARAMETERS COMMON AHONC THE RM PROCRAHS: Ct CA COUNT - THE NUHDER OF CASES IN THE TEST Ca CROUP - THE NUHDER OF CROUPS PER CASE Ct SUBJECT - THE NUNDER OF SUBJECTS PER CROUP CA NEASURE - THE NUHDER OF TESTS OR HEASURES PER SUDJECT cmam Ct COOINC CONVENTIONS: CA COHHENT LINES DECINNINC HITH 'Ca' DENOTE INFORMATIONAL COHHENTS. 6* THIS. COHHENT LINES EECINNINC HITH 'C ' DENOTE DEDUCCINC CODE THAT 6* MAY DE USEFUL IN THE FUTURE. ETC. cmm 6* ROUTINES USED: 6* VMULFF. VMULFM. VMULFP. LINV2F - FROM IMSL. CA TRED2. IMTQLZ - FRON EISPACK. CW IMPLICIT REALIA-l) INTECER COUNT. MEASURE. SUDJECT. CROUP PARAHETER ( COUNT-IOOOO. MEASURE-5. SUBJECT-20. GROUP-3 ) LOCICAL FIRST.SECOND PARAMETER ( FIRST-.TRUE.. SECOND-.FALSE. ) INTECER ITERATE. I. J. N. IERR COMMON IOATA/ Z(GROUP*SUBJECT.MEASURE). zDARICROUP.NEASURE). T(HEASURE.HEASURE). GIMEASURE.MEASURE).. C(MEASURE.MEASURE). B(MEASURE.MEASURE). EIMEASURE.MEASURE). CTRAN(MEASURE.MEASURE). ETRAN(NEASURE.NEASURE). CRM(MEASURE-I.MEASURE-I). ERM(MEASURE-I.MEASUR£-I). CLIN(MEASURE-2.MEASURE-2). ELIN(MEASURE-2.MEASURE-2). HB(MEASURE.MEASURE). HC(HEASURE-I.HEASURE-I). SCRI(MEASUR£.MEASURE). SCRZ(MEASURE*MEASURE+3*MEASURE). EICD(HEASURE). ElGC(M£ASURE-I). EICLIHEASURE-z) _ REAL PI(HEASURE.NEASURE). P2(MEASURE-I.MEASURE-I). U(CROUP.I) COHNON /ITERATE/ ITERATE ' DATA (PI(I.I).I-I.NEASURE) + /.AA72I. -.632h6. .53A52. -.3I623. .11952/ DATA (PI(2.I).I-I.MEASURE) + /.AA72I. -.31623. -.26726. .63256. -.A7809/ DATA (PI(3.I).I-I.MEASURE) ++++++++ 1J13 + /.bb72I. 0.0. -.53552. 0.0. .7IJIH/ DATA (PI(5.I).I-I.MEASURE) + /.bh72I. .3I623. -.25726. '.632h6. -.h7809/ DATA (PI(5.I).I-I.MEASURE) + /.Bh72l. .632b5. .53h52. .31623. .II952/ DATA (P2(I.|).I-I,MEASURE-I)/.S. -.67082. .5. -.2236I/ DATA (P2(Z.I).I-I.MEASURE-I)/.5. -.2236I. -.S. .67082/ DATA (P2(3.I).I-I.MEASURE-I)/.5. .2236I. -.5. -.67082/ DATA (P2(h.l).I-I.MEASURE-I)/.5. .67082. .5. .2236I/ C* SET THE UNIT VECTOR. DO 5 l-I.GROUP 5 U(I.I)-I.O C* INITIALIZE OUTPUT FILES CALL SETOUT CA BECINI 00 I00 ITERATE-I.COUNT CALL GETDATAI z ) C* COMPUTE ZBAR -- THE MEAN OF MEASURES ACROSS GROUPS DO IO K-I.MEASURE ' DO 20 l-I.GROUP sun-0.0 DO 30 J-I.SUBJECT 30 SUM-SUM+Z( (I-I)*SUBJECT+J. K ) ZBARII.K)-SUM/$UBJECT 20 CONTINUE IO CONTINUE C* DEBUG PRINT... C WRITEIIOO.*) ' ZBAR-' C WRITEIIOO.'(IX.5FI5.5)') ((ZBAR(I.J).J-I.MEASURE).I-I.GROUP) CA T - z'z CALL VHULFH( z. z. GROUP*SUBJECT. MEASURE. HEASURE. + GROUP*SUBJECT. GROUPisuaJECT. T. HEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINTA. 'ERROR - IN z"z - IERR-‘. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENOIF C* G I ZBAR'ZBAR (MULTIPLICATION BY SUBJECT TO FOLLOW) CALL VMULFM( ZBAR. ZBAR. GROUP. MEASURE. MEASURE. GROUP. + CROUP. C. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN zBAR"zBAR - IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENOIF C* C I ZBAR'U (MORE TO FOLLOW) CALL VMULFM( ZBAR. U. GROUP. MEASURE. I. GROUP. GROUP. SCRZ. + HEASURE. IERR ) IF (IERR .NE. 0) THEN - C PRINT*. ‘ERROR - IN ZBAR"U - IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT 1J14 ENDIF (ZBAR'U)U' (MORE TO FOLLON) CALL VMULFP( SCRz. U. MEASURE. I. CROUP. MEASURE. CROUP. SCRI. + MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN (ZBAR"U)U" - IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF C* C C* C (ZBAR'U'U)ZBAR (MULTIPLICATION BY SUBJECT/CROUP TO FOLLOMn CALL VMULFF( SCRI. ZBAR. NEASURE. CROUP. MEASURE. MEASURE. + CROUP. C. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN (ZBAR"JJ")ZBAR - IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF C* G SUBJECT*6 : C - SUBJECT/CROUPAC : B - C-C : E - T-C D0 to J-I.MEASURE D0 1.0 I-I.MEASURE C(I.J)-FLOAT(SUBJECT) * C(I.J) c(I.J)-FLOAT(SUBJECT)/CROUP A C(I.J) B(I.J)-C(I.J) - C(I.J) E(IOJ)-T(IBJ) ' G(I..I) ho CONTINUE C* DEBUG PRINT... WRITE(IOO.*) ' C" WRITEII00.'(IX.SFI5-5)') (ICII.J).J'I.5).|'I.5) WRITE‘I00.*) ' B" ' RITE “000' ('xOSEISOS) ') ((3‘) DJ) BJ-IBS) B '-‘05) WRITE(I00.*) ' E" HRITE('OOO.('x05FIS'5).) ((E(IOJ)IJ-‘OS)OI-105) nnnnnn CALL COMPUTE( PI. HEASURE ) CALL RESULT( EICB. MEASURE. RB. TB. VB. NB ) CALL RESULT( EICC. MEASURE-I. RC. TC. VC. WC ) CALL RESULT( EICL. MEASURE-2. RL. TL. VL. WL ) C* WRITE THE EIGENVALUES TO THE (UNSORTED) OUTPUT FILES. CALL OUTPUT(FIRST.RB.TB.VB.WB.RC.TC.VC.WC.RL.TL.VL.WL) CALL COMPUTE( P2. MEASURE-I ) CALL RESULT( EICB. MEASURE-I. RB. TB. VB. NB ) CALL RESULT( EICC. MEASURE—2. RC. TC. VC. NC ) CALL RESULT( EICL. MEASURE-3. RL. TL. VL. WL ) C* WRITE THE EIGENVALUES TO THE (UNSORTED) OUTPUT FILES. CALL OUTPUT(SECOND.RB.TB.VB.WB.RC.TC.VC.WC.RL.TL.VL.WL) IOO CONTINUE Ct CLOSE OUTPUT FILES. CALL ENDOUT C* MAKE SURE RM HASN'T OVERWRITTEN ITSELE: CLOSE DEBUG OUTPUT FILE. C WRITE(IOO.*) ' PI" c WITE(IOOI. (IxostS-S) .) ((P| (I OJ) OJ-‘OS’ B '-105) ILLS C HRITE(IOO.A) ' P2-' C HRITEIIOO.'(Ix.5FI5.5)') ((P2(I.J).J-I.D).I-I.h) C NRITE(IOO.A) ' u-' C WRITE(IOO.'(IX.5FI5.5)') U C REHIND(IOO) RETURN END SUBROUTINE COMPUTE( P. LENGTH ) CW CA COMPUTE PERFORMS SEVERAL REPETITIVE COMPUTATIONS. THE ONLY CA DIFFERENCE AMONG THE REPETITIONS IS THE VALUE OF THE ARRAY CA P AND THE VALUE OF LENGTH. RESULTS ARE RETURNED THROUGH /DATA/ . CWW CA COMPUTATIONAL NOTE: CA IF AN ERROR IS DETECTED IN INVERTING A MATRIX. THE INVERSE CA Is SET TO 0. THIS FORCES ALL THE EIGENVALUES COMPUTED LATER CA TO BE 0 ALSO. CW IMPLICIT REAL(A-z) INTEGER MEASURE. SUBJECT. GROUP PARAMETER ( MEASURE-5. SUBJECT-20. GROUP-3 ) INTEGER ITERATE. I. J. K. IERR. LENGTH. OPT. IDIGIT COMMON /DATA/ 2(GROUPASUBJECT.MEASURE). zBAR(GROUP.MEASURE). T(MEASURE.MEASURE). G(MEASURE.MEASURE). C(MEASURE.MEASURE). B(MEASURE.MEASURE). E(MEASURE.MEASURE). CTRAN(MEASURE.MEASURE). ETRAN(MEASURE.MEASURE). CRM(MEASURE-I.MEASURE-I). ERMIMEASURE-I.MEASURE-I). CLIN(MEASURE-2.MEASURE-2). ELIN(MEASURE-2.MEASURE-2). HB(MEASURE.MEASURE). HC(MEASURE-I.MEASURE-I). SCRI(MEASURE.MEASURE). SCRZ(MEASURE*MEASURE+3*MEASURE). EIGB(MEASURE). EIGCIMEASURE-I). EIGL(MEASURE—2) REAL P(LENGTH.LENGTH) CHARACTERAIO CI.C2 COMMON /ITERATE/ ITERATE ++++++++ C* CTRAN - P'C (MORE TO FOLLOW) CALL VMULFM( P. C. LENGTH. LENGTH. LENGTH. LENGTH. MEASURE. + SCRI. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN P"C - LENGTH-'. LENGTH. ' IERRO'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF CA CTRAN - (P'C)P CALL VMULFF( SCRI. P. LENGTH. LENGTH. LENGTH. MEASURE. LENGTH. + CTRAN. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN (P"C)P - LENGTH-'. LENGTH. ' IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF CA ETRAN - P'E'IMORE TO FOLLOW) CALL VMULFM( P. E. LENGTH. LENGTH. LENGTH. LENGTH. MEASURE. + SCRI. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINTA. 'ERROR - IN P"E - LENGTH-'. LENGTH. . IERR-I. IERR C PRINT*.'ON ITERATION I. ITERATE IJLB CALL ABORT ENDIF CA ETRAN - (P'E)P CALL VMULFF( SCRI. P. LENGTH. LENGTH. LENGTH. MEASURE. LENGTH. + ETRAN. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN (P"E)P - LENGTH-'. LENGTH. ' IERR-'. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF CA DEBUG PRINT... c HRITE(I00.A) ' CTRAN-' C HRITE(IOO.'(Ix.5Fls.5)') ((CTRAN(I.J).J-I.LENGTH).I-I.LENGTH) C WRITE (IOO.A) ' ETRAN-' C HRITE(IOO.'(Ix.5FIs.5)') ((ETRAN(I.J).J-I.LENGTH).I-I.LENGTH) DO 20 J-2.LENGTH DO 20 I-2.LENGTH CRM(I-I.J-I)-CTRAN(I.J) ERM(I-I.J-I)-ETRAN(I.J) 20 CONTINUE CA HB - B E-INVERSE - SCRI CONTAINS E-INVERSE IDIGIT-o CALL LINV2F( E. LENGTH. MEASURE. SCRI. IDIGIT. SCR2. IERR ) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. C2 ) CALL INT2CHR( IERR. CI ) CALL REMARKI'ERROR - E-INV - IERR-'IICI/l' ITERATE-'l/Cz) C CALL ABORT DO 50 J-I.MEASURE L DO 50 I-I.MEASURE 50 SCRI(I.J)-O.O ENDIF CALL VMULFF( B. SCRI. LENGTH. LENGTH. LENGTH. MEASURE. MEASURE. + HB. MEASURE. IERR ) IF (IERR .NE. 0) THEN C PRINT*. 'ERROR - IN B"E - LENGTH-'. LENGTH. ' lERR-‘. IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF CA HC - CRM ERM-INVERSE - SCRI CONTAINS ERM-INVERSE IDIGIT-O . CALL LINv2F( ERM. LENGTH-I. MEASURE-I. SCRI. IDIGIT. SCR2. IERR ) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. CI ) CALL INT2CHR( IERR. C2 ) CALL REMARKI'ERROR - ERM -INv - IERR-'llczll' ITERATE-'IICI) C CALL ABORT D0 A0 J-I.MEASURE D0 D0 I-I.MEASURE ho SCRI(I.J)-0.0 ENDIF CALL VMULFF( CRM. SCRI. LENGTH-I. LENGTH-I. LENGTH-I. MEASURE-I. + MEASURE-I. HC. MEASURE-I. IERR ) 1137 IF (IERR .NE. 0) THEN C PRINT*.'ERROR - IN CRM ERM-INV - LENGTH-'.LENGTH.' IERR-'.IERR C PRINT*.'ON ITERATION '. ITERATE CALL ABORT ENDIF CA CALL EISPACK ROUTINES TRED2 AND IMTOLz TO DO EIGENVALUES. CA AFTER TRED2. SCRI HILL CONTAIN z CA EIGB HILL CONTAIN D CA SCR2 HILL CONTAIN E CALL TREDZ( MEASURE. LENGTH. HB. EIGB. SCR2. SCRI ) CALL IMTQL2( MEASURE. LENGTH. EIGB. SCR2. SCRI. IERR ) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. CI ) CALL INT2CHR( IERR. C2 ) CALL REMARN('ERROR - HB IMTQL2_- IERR-'//C2//' ITERATE-'I/CI) CALL ABORT ENDIF CALL TRED2( MEASURE-I. LENGTH-I. HC. EIGC. SCR2. SCRI ) CALL IMTOL2(MEASURE-I. LENGTH-I. EIGC. SCR2. SCRI. IERR) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. CI ) CALL INT2CHR( IERR. C2 ) CALL REMARNI'ERROR - HC IMTQL2 - IERR-'//C2//' ITERATE-'IICI) CALL ABORT ENDIF C* DEBUG PRINT... C WRITE(IOO.*) ' EIGCO' C WRITEIIOO.*) (EIGC(I).I-I.LENGTH-I) C* C* PERFORM LINEAR TRENDS STATISTICS. CR DO IO J-3oLENGTH DO IO I-3.LENGTH CUIH (I'ZoJ‘Z) 'CTRAN (l OJ) ELIN(I'Z.J-2)'ETRAN(I.J) IO CONTINUE CA HLIN - CLIN ELIN-INVERSE - SCRI CONTAINS ELIN-INVERSE CA STORE HLIN IN THE HC ARRAY. SINCE THE DATA IN HC HILL NOT CA BE REUSED. CA CA DEBUG PRINT... C HRITE(IOO.A) ' CLIN-' C HRITE(IOO.'(Ix.3FIs.5)') ((CLIN(I.J).J-I.LENGTH-2).I-I.LENGTH-2) C HRITE(IOO.A) ' ELIN-' - C HRITE(IOO.'(IA.3F15.5)') ((ELIN(I.J).J-I.LENGTH-2).I-I.LENGTH-2) IDIGIT-o CALL LINV2F( ELIN. LENGTH-2. MEASURE-2. SCRI. IDIGIT. SCR2. IERR ) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. CI ) CALL INT2CHR( IERR. C2 ) CALL REMARK('ERROR - ELIN -INV - IERR-'l/CZ/l' ITERATE-'//CI) C CALL ABORT DO 30 I-I.MEASURE DO 30 J-I.MEASURE 30 SCRI(J.I)-0.0 ILLS ENDIF CALL VMULFF( CLIN. SCRI. LENGTH-2. LENGTH-2. LENGTH-2. MEASURE-2. + MEASURE-2. HC. MEASURE-2. IERR ) IF (IERR .NE. 0) THEN C PRINT*.'ERROR°IN CLIN ELIN-INV. LENGTH-'.LENGTH.' IERR-'.IERR C PRINT*.'ON ITERATION '. ITERATE ' CALL ABORT ENDIF CALL TRED:( MEASURE-2. LENGTH-2. HC. EIGL. SCR2. SCRI ) CALL IMTOL2(MEASURE-2. LENGTH-2. EIGL. SCR2. SCRI. IERR) IF (IERR .NE. 0) THEN CALL INT2CHR( ITERATE. CI ) CALL INT2CHR( IERR. C2 ) CALL REMARK('ERROR — HL IMTQLZ - IERR-'//C2//' ITERATE-'l/CI) CALL ABORT ENDIF C* DEBUG PRINT... C WRITE(IOO.*) ' EIGL-' C WRITE(IOO.*) (EIGLII).I-I.LENGTH-2) RETURN END SUBROUTINE RESULT( EIGEN. LENGTH. R. T. V. H ) CW CA RESULT CALCULATEs SEVERAL STATISTICS (R. T. V. AND H) BASED CA ON THE EIGENVALUES IN ARRAY EIGEN . CW IMPLICIT REAL(A-Z) INTEGER LENGTH. I. ITERATE DIMENSION EIGEN(LENGTH) COMMON /ITERATE/ ITERATE C INTEGER DEBUG C DATA DEBUG/IO/ V-T-0.0 W-I.O DO IO I-I. LENGTH VALUE-EIGEN(I) T-T + VALUE v-V + VALUE/(1.0 + VALUE) H-H/(I.O + VALUE) IO CONTINUE R-EIGENILENGTH)/(I.O + EIGEN(LENGTH)) C IF (DEBUG .GT. 0) THEN C DEBUG-DEBUG-I C HRITE(IOO.*) 'EIGENVALUE5-'.EIGEN C NRITE(IOO.*) 'STATSO'.R.T.V.W C ENDIF RETURN END SUBROUTINE GETDATA( z ) (:qu C* GETDATA RETURNS THE NEXT SET OF VALUES TO BE ANALYZED BY RM. CA 1J19 CA THE DATA Is READ FROM LOCAL FILE DFILE. HHICH Is INITIALLY CA REHOUNO. cmmrttm IMPLICIT REAL(A-z) INTEGER MEASURE. SUBJECT. GROUP PARAMETER ( MEASURE-5. SUBJECT-20. GROUP-3 I REAL Z(GROUPASUBJECT.MEASURE). POHER(2:MEASURE) INTEGER FET(8). BUF(2OA9). EDP INTEGER I. J. O. POHERON LOGICAL FIRST CHARACTER DCA3 DATA FIRST/.TRUE./. POHER/.A. .8. .5. .I/ CA IF THIS IS THE FIRST CALL TO GETDATA. INITIALIZE THE DATA FILE.. CA CHECA CCL REGISTER I FOR THE D PARAMETER. AND CHECK R2 FOR THE CA POHER PARAMETER. IF (FIRST) THEN CALL FILEC( 'DFILE'. FET. 8. BUF. 20kg ) CALL REHINOF( FET ) CA CA FTN5 STANDARD CODE - IF RM NEEDS TO BE TRANSPORTED. CA DELETE THE PRECEDING FILEC AND REHINDF CALLS. AND USE CA THE FOLLOHING CODE: Ct C OPEN(999.FILE-'DFILE') C REHIND(999) CA FIRST-.FALSE. CALL GETREG( 'RI'. D ) IF (D .NE. I) THEN DSORT-SORT(FLOAT(D)) CALL INT2CHR( D. DC ) CALL REMARN( 'RM CALLED HITH D-'//DC ) ENDIF CALL GETREG( 'Rz'. POHERON ) IF (POHERON .NE. 0) THEN CALL REMARK( ' CALCULATING POHERS' ) ENDIF ENDIF CALL READH( FET. z. GROUPAMEASUREASUBJECT. EDP ) :1: CA FTNs STANDARD CODE - IF RM NEEDS TO BE TRANSPORTED. CA DELETE THE PRECEDING READH CALL. AND USE CA THE FOLLOHING CODE: u C READ(999.A.IOSTAT-EOP) 2 CA IF (EOP .EQ. 0) THEN C* MULTIPLY ONLY THE FIRST GROUP OF 2 BY THE SQUARE ROOT OF D. IF (D .NE. I) THEN DO IO J-I.MEASURE DO IO I-I.SUBJECT IO Z(|.J)-OSQRT * Z(I.J) ENDIF C* ADD CONSTANTS TO ALL GROUPS. JJZO IF (POHERON .NE. 0) THEN DO 20 J-2.MEASURE DO 20 I-I.SUBJECT*GROUP 20 Z(I.J)'Z(I.J) + POWER(J) ENDIF ELSE CALL REMARK( 'UNEXPECTED *EOP ON READING DATA.‘ ) CALL ABORT ENDIF C HRITE(IOO.A) ' DATA-' C ‘HRITE(I00.'(IX.5FI5-5)') ((Z(I.J).J-I.MEASURE).I-I.GROUPASUBJECT) RETURN . END SUBROUTINE SETOUT CW CA SETOUT INITIALIZES THE OUTPUT FILES THAT THE UNSORTED TEST CA STATISTICS HILL BE HRITTEN TO. CA FILES USED ARE 'TAPEI' THROUGH 'TAPE2A' . CW IMPLICIT INTEGER(A-z) COMMON IIO/ FET( 8. 25 ) DIMENSION BUF( 5I3. 2A ) CHARACTER UNITLFNA7 UNITLFN-‘TAPE’ CA RETURN EACH FILE BEFORE ANY FURTHER PROCESSING. DO IO I-I.2A CALL INT2CHR( I. UNITLFN(s:) ) CALL FILEC( UNITLFN. FET(I.I). 8. BUF(I.I). 5I3 ) CALL RETF( FET(I.I) ) CALL FILEC( UNITLFN. FET(I.I). 8. BUF(I.I). 5I3 ) IO CONTINUE CA CA FTNs STANDARD CODE - IF RM NEEDS TO BE TRANSPORTED. CA DELETE THE DO LOOP. AND USE THE FOLLOHING CODE: 0 C 00 IO I-I.2L C OPEN( I ) C CLOSE( I. STATUS-'DELETE‘ ) C OPEN( I ) C IO CONTINUE cA RETURN END SUBROUTINE OUTPUT( FIRST. RB. TB. VB. HB. RC. TC. VC. HC. + RL. TL. VL. HL ) cammt CA OUTPUT HRITES THE EIGENVALUES TO THE OUTPUT FILES. CA HILL BE HRITTEN TO. CA FILES USED ARE 'TAPEI' THROUGH 'TAPEzb' . CA CA FIRST - .TRUE. IFF THIS SET OF EIGENVALUES HAS OBTAINED HITH CA LENGTH - MEASURE: FIRST - .FALSE. IFF LENGTH - MEASURE-I . cmm IMPLICIT REAL(A-z) INTEGER BEGIN 1J21 LOGICAL FIRST COMMON /IO/ FET( 8. 2h ) IF (FIRST) THEN BEGIN-0 ELSE BEGIN-I2 ENDIF CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CALL CA HRITEH( FET(I.I+BEGIN). RB. HRITEH( FET(I.2+BEGIN). TB. HRITEH( FET(I.3+BEGIN). VB. HRITEH( FET(I.h+8EGIN). HB. HRITEH( FET(I.5+BEGIN). RC. HRITEH( FET(I.6+BEGIN). TC. HRITEH( FET(I,7+BEGIN). VC.’ HRITEH( FET(I.8+BEGIH). HC. HRITEH( FET(I.9+BEGIN). RL. HRITEH( FET(I.IO+BEGIN). TL. I ) HRITEH( FET(I.II+BEGIN). VL. I ) HRITEH( FET(I.Iz+BEGIN). HL. I ) dd‘dddddd vvvvvvvvv C* FTNS STANDARD CODE - IF RM NEEDS TO BE TRANSPORTED. C* DELETE THE HRITEW CALLS ABOVE. AND USE THE FOLLOWING CODE: :1: C HRITE( BEGIN+I. A ) RB C HRITE( BEGIN+2. * ) TB C HRITE( BEGIN+3. A ) VB C HRITE( BEGIN+A. * ) HB C HRITE( BEGIN+5. A ) RC C HRITE( BEGIN+6. * ) TC C HRITE( BEGIN+7. A ) VC C HRITE( BEGIN+8. * ) HC C HRITE( BEGIN+9. A ) RL C HRITE( BEGIN+IO.* ) TL C HRITE( BEGIN+II.A ) VL C HRITE( BEGIN+I2.* ) HL Ct RETURN END SUBROUTINE ENDOUT cmm CA ENDOUT CLOSES THE FILES THAT THE EIGENVALUES HERE HRITTEN TO. cmm IMPLICIT INTEGER(A-z) COMMON /IO/ FET( 8. 2h ) IO CALL HRITEOR( FET(I.I) ) CA C* FTNS STANDARD CODE - IF RM NEEDS TO BE TRANSPORTED. C* DELETE THE DO LOOP CALL ABOVE. AND USE THE FOLLOHING CODE: CA C DO IO I-I.2h C IO 'REWIND( I ) CA RETURN END *EOSOO LINE-672 SEC-I 1J22 IDENT DATA AAAAAAAAAAAAAA DATA-GENERATION PROGRAM. CREATES COUNT COLLECTIONS OF DATA. EACH WITH MEASURE¢GROUP*SUBJECT ITEMS. USING THE BOX-MUELLER (SP?) METHOD. SEE THE COMMENT SECTION FOR PROGRAM RM FOR DETAILED INFORMATION A A A A DATA IS HRITTEN TO LOCAL FILE DFILE . A A A ABOUT THE OPERATION OF THESE PROGRAMS. AAAAAAAAAAAAAA ENTRY DATA DATA ass 0 RJ -XXDATA ENDRDN END DATA SUBROUTINE XDATA CAAAAAAAAAAAA 6* ROUTINES USED: ca ct GGUBS - FROM IMSL. CAAAAAAAAAAAA IMPLICIT INTEGER(A-z) PARAMETER (COUNT-IOOOO. MEASURE-5. GROUP-3. SUBJECT-20) REAL ARRAY(MEASUREAGROUPASUBJECT) REAL R. THETA. AVERAGE. PIxz DIMENSION FET(8). BUF(2OA9) DOUBLE PRECISION OSEED DATA DSEED/73960A919.OD0/ PIx2-8.OAATAN (I .0) CALL FILEC( 'DFILE'. FET. 8. BUF. 20A9 ) CALL RETF( FET ) CALL FILEC( 'DFILE'. FET. 8. BUF. 2059 ) cA CA FTH5 STANDARD CODE - DELETE THE PRECEDING FILEC AND RETF CALLS. CA AND USE THE FOLLOHING CODE: CA C OPEN(I.FILE-'DFILE‘) C CLOSE(I.STATus-'DELETE') C OPEN(I.FILE-'DFILE') CA AVERAGE-0.0 DO IO J-I. COUNT CALL GGUBS( OSEED. MEASUREAGROUPASUBJECT. ARRAY ) DO 20 I-I.MEASUREAGROUPASUBJECT/z R-SORT( -2.0 A LOG(ARRAY(2AI-I)) ) THETA-PIx2 A ARRAY(2AI) ARRAY(2AI-I)-R A COS( THETA ) ARRAY(2AI)-R A SIN( THETA ) AVERAGE-AVERAGE+ARRAY(2AI)+ARRAY(2AI-I) 20 CONTINUE CALL HRITEH( FET. ARRAY. MEASUREAGROUPASUBJECT ) ca C* FTNS STANDARD CODE - DELETE THE PRECEDING HRITEW CALL. 1J23 'CA CA c1: )0 CA CA CA CA CA CA CA CA CA CA CA CA CA nnnnnn *EDSOO AND USE THE FOLLOHING CODE: HRITE().A) ARRAY CONTINUE AVERAGE-AVERAGE/(GROUPAMEASUREASUBJECTACOUNT) CALL HRITEOR( FET ) FTN5 STANDARD CODE - DELETE THE PRECEDING NRITEOR CALL. AND USE THE FOLLOWING CODE: RENIND I NRITE THE AVERAGE OF THE DATA GENERATED TO LOCAL FILE STATFIL . THIS AVERAGE WILL BE NEEDED IN ORDER TO COMPUTE THE VARIANCE OF THE DATA. CALL HHBF( FET ) CALL FILEC( 'STATFIL'. FET. 8. BUF. 65 ) CALL RETF( FET ) CALL HRITEH( FET. AVERAGE. I ) CALL HRITEOR( FET ) FTNS STANDARD CODE - DELETE THE PRECEDING FILEC AND RETF CALLS. AND USE THE FOLLOHING CODE: OPEN(2.FILE-'STATFIL') CLOSE(2.STATU$-'DELETE') OPEN(2.FILE-'STATFIL') HRITE(I.*) AVERAGE REHIHO I RETURN END LINE-I03 SEC-I 1J24 IDENT STATS AAAAAAAAAAAAAA A DATA-STATISTICS PROGRAM. COMPUTES THE VARIANCE OF THE DATA. USING A THE AVERAGE ALREADY CALCULATED BY THE DATA-GENERATION PROGRAM. A BOTH THE AVERAGE AND THE VARIANCE ARE HRITTEN TO LOCAL FILE STATFIL . A A INPUT CONDITIONS: A DATA IS ON FILE DFILE : THE AVERAGE Is ON LOCAL FILE STATFIL . A A SEE THE COMMENT SECTION OF PROGRAM RM FOR MORE DETAILED EXPLANATION A OF THE FUNCTIONING OF THESE PROGRAMS. AAAAAAAAAAAAAA‘ ENTRY STATS STATS ass 0 RJ -xxSTATS ENDRUN END STATS SUBROUTINE XSTATS IMPLICIT INTEGER(A-Z) . PARAMETER (COUNT-IOOOO. MEASURE-5. GROUP-3. SUBJECT-20) REAL ARRAY(MEASUREAGROUPASUBJECT). AVERAGE. VAR DIMENSION FET(8). BUF(2OA9) C REAL OBSC(IOO). STAT(3). CSOBS(IOO) CA READ THE DATA AVERAGE ALREADY COMPUTED. CALL FILEC( 'STATFIL’. FET. 8. BUF. 65 ) CALL REHINDF( FET ) CALL READH( FET. AVERAGE. I. EDP ) CALL HHBF( FET ) ‘ CALL REHINDF( FET ) C OPEN(I.FILE-'RAHDATA') C REHIND I C DO 5 I-I,IOO C 5 CSOBS(I)-OBSC(I)-0.0 VAR-0.0 DO IO J-I. COUNT CALL READH( FET. ARRAY. MEASUREAGROUPASUBJECT. EDP ) IF (EDP .LT. 0) THEN CALL REMARK( 'UNEXPECTED AEOP ON DATA FILE.’ ) RETURN ENDIF DO 20 I-I.MEASUREAGROUPASUBJECT 20 VAR-VAR + (ARRAY(I)-AVERAGE)A(ARRAY(I)-AVERAGE) C IF (J .NE. COUNT) THEN c STAT(3)-o C ELSE C STAT(3)-I C ENDIF C K-IOO C CALL GTNOR(ARRAY.GROUP*MEASURE*SUBJECT.K.STAT.OBSC.CSDDS.IERR) C IF (IERR .NE. 0) HRITE(I.A) ' IERR-'.IERR I0 CONTINUE VAR-VAR/(COUNT*GROUP*MEASURE*SUBJECT) C WRITE(I.*) STAT JJZS C REWIND I CA CA HRITE THE AVERAGE AND VARIANCE OF THE DATA GENERATED TO LOCAL CA FILE STATFIL . (:1: CALL HHBF( FET ) CALL FILEC( 'STATFIL'. FET. B. BUF. 65 ) CALL RETF( FET ) CALL HRITEH( FET. AVERAGE. I ) CALL HRITEH( FET. VAR. I ) CALL HRITEOR( FET ) RETURN END *EOSOO LINE-7D SEC-I JJZG IDENT SORT AAAAAAAAAAAAAAA SORT - SHELL-METZNER SORT OF TEST STATISTICS. SORT.INLFN.OUTLFN. A A * CALLING SEQUENCE: A A A * SEE THE COMMENT SECTION OF PROGRAM RM FOR A DETAILED EXPLANATION A OF THE FUNCTIONING OF THESE PROGRAMS AND THEIR INTERACTION. “WA ENTRY SORT SORT ass 0 SAI PLIST PLIST CONTAINS INLFN AND DUTLFN RJ -AXSORT ENDRUN PLIST ass 0 CON 2 CON 3 DATA 0 END SORT SUBROUTINE XSORT( INLFN. DUTLFN ) IMPLICIT IHTEGER(A-z) PARAMETER (COUNT-IOOOO) REAL ARRAY(COUNT). T DIMENSION FET(8). BUF(5I3) CALL FILEc( INLFN. FET. 8. BUF. 5I3 ) CALL REHINDF( FET ) CALL READH( FET. ARRAY. COUNT. EDP. LEVEL. NGET ) CALL HHBF( FET ) C* FTNS STANDARD CODE - DELETE THE PRECEDING FILEC. REHINDF. READW. C* AND WNBF CALLS. CHANGE INLFN AND DUTLFN TO CHARACTER VARIABLES. C* AND USE THE FOLLOWING CODE: IO OPEN(I.FILE-INLFN) REWIND I READ(I.*.IOSTAT-EOP) ARRAY IF (EDP .NE. D) THEN CALL REMARK( 'UNEXPECTED *EOP ON READING.‘ ) ENDIF M-NGET CONTINUE HIM/2 IF (M .EQ. 0) THEN CALL FILEC( DUTLFN. FET. 8. BUF. 513 ) CALL RETF( FET ) CALL FILEC( DUTLFN. FET. 8. BUF. 5I3 ) CALL HRITEH( FET, ARRAY. NGET ) CALL HRITEOR( FET ) ‘ CA FTNS STANDARD CODE - DELETE THE PRECEDING FILEC. RETF. NRITEN. C* AND WRITEOR CALLS. AND USE THE FOLLOWING CODE: OPEN(2.FILE-OUTLFN) REHIND 2 HRITE(2.*) ARRAY REHINO 2 1J2? ' cA RETURN ENDIF C* ELSE... K-NGET-M J-I 20 CONTINUE I-J 30 CONTINUE L-I+M IF (ARRAY(I) .GT. ARRAY(L)) THEN T-ARRAY(I) ARRAY (I) «mm (L) ARRAY(L)-T III-M IF (I .GE. I) GOTO 30 ENDIF J-J+I IF (.I .GT. K) GOTO IO GOTO 20 END *EOSOO LINE-93 SEC-I JJZB PROGRAM NTABLE( OUTPUT ) cAAAAAAAAAAAAAAAAAA C* TABLE READS THE SORTED LISTS OF TEST STATISTICS. AND CALCULATES C* THE VALUES FOR ALPHA I 0.0I. 0.05. AND O.IO . cA C* FOR THE EXPECTED SIZE OF I0.000 . THESE VALUES ARE CALCULATED C* BY AVERAGING THE IOOTH AND IOIST. THE SOOTH AND THE SOIST. AND C* THE IOOOTH AND THE IOOIST ELEMENTS. RESPECTIVELY. FOR H. AND C* THE SSOOTH AND 990IST. 9500TH AND 950IST. AND SOOOTH AND SOOIST C* ELEMENTS FOR THE R. T. AND V TESTS. cA C* CALLING SEQUENCE: cA C* NTABLE.OUTLFN. cA C* WHERE DUTLFN IS THE FILE TO WHICH THE TABLED OUTPUT IS TO BE WRITTEN. CI DEFAULT: OUTPUT cA C* TAPEZS THROUGH TAPEBB CONTAIN THE SORTED TEST STATISTICS: cA C* R T V N CA CI PIS B TAPEZS TAPEZB TAPE27 TAPEZD C* C TAPE29 TAPEBO TAPE3I TAPE32 C* L TAPE33 TAPEBA TAPE35 TAPE36 CI PIA B TAPE37 TAPE38 TAPE39 TAPEBO C* C TAPEAI TAPEB2 TAPEA} TAPEhb C* L TAPEAS TAPEBS TAPEN] TAPEAD cA cAAAAAAAAAAAAAAAAAA , CI NTABLE IS USED TO PRODUCE THE OUTPUT FOR THE NOMINAL-VALUE RUN. CA IT ALSO HRITES THE NOMINAL VALUES TO FILE NVALUE FOR USE BY THE C* OBSERVED-VALUE TABLE'GENERATION PROGRAM. cAAAAAAAAAAAAAAAAAA C* SEE THE COMMENT SECTION OF PROGRAM RM FOR A DETAILED EXPLANATION OF C* THE INTERACTION OF THESE PROGRAMS. CAAAAAAAAAAAAAAAAAA CA CA CA CA CA 5 IMPLICIT INTEGER(A-Z) PARAMETER (COUNT-IOODO) _ . PARAMETER (AI-COUNT/IOO. Az-COUHT/zo. A3-COUNT/IO) PARAMETER (BI-COUNT-AI.,Bz-COUNT-Az. B3-COUNT-A3) REAL ARRAY(COUNT) DIMENSION FET(8). BUF(5I3) CHARACTER UNITLFN*7. TITLEABO REAL NI(2A). N2(2A). N3(zh) CALL NOMSG CALL FILEC( 'ZZZZIN'. FET. 8. BUF. 65 ) CALL CONNECF( FET. O ) CALL HRITEH( FET. ' NTABLE - PLEASE ENTER A TITLE -'. A ) CALL READH( FET. TITLE. 8. EOP ) FTNS STANDARD CODE - DELETE THE PRECEDING FILEC. CONNECF. NRITEH. AND READH CALLS. AND SUBSTITUTE SOME OTHER METHOD OF READING IN A TITLE FROM THE USER. L-LNB (TITLE) DO 5 I-L.I.-I P-(BO-L)/2+I TITLE(P:P)-TITLE(I:I) CONTINUE 1J29 TITLE(:(DO-L)/2)" ' UNITLFNI'TAPE' DO IO III.2A UNITII+ZB CALL INT2CHR( UNIT. UNITLFN(5:) ) CALL FILEC( UNITLFN. FET. 8. BUF. sI3 ) CALL REHINDF( FET ) CALL READH( FET. ARRAY. COUNT. EOP ) CALL HHBF( FET ) CI FTNS STANDARD CODE - DELETE THE PRECEDING FILEC. REWINDF. READH. C* AND WNBF CALLS. AND USE THE FOLLOWING CODE: cA C OPEN( I.FILE-UNITLFN ) C REHIND I C READ(I.A.IOSTAT-EOP) ARRAY cA IF (EOP .Eq. 0) THEN CA CA I MOD A - O INDICATES A H TEST: CA OTHERHISE IT IS AN R. T. OR V TEST. CA IF (MOD(I.A) .Eq. 0) THEN NI(I)-( ARRAY(AI)+ARRAY(AI+I) )/2 N2 (I)-( ARRAY (A2) +ARRAY (A2+I) ) I2 N3(I)-( ARRAY(A3)+ARRAY(A3+I) )/2 ELSE NI(I)-( ARRAY(BI)+ARRAY(BI+I) )/2 N2(I)-( ARRAY(BZ)+ARRAY(BZ+I) )/2 N3(I)-( ARRAY(33)+ARRAY(BJ+I) )l2 ENDIF ELSE CALL REMARK( 'UNEXPECTED *EOP ON READ OF '//UNITLFN ) NI(I)IN2(I)IN3(I)IO.O ENDIF IO CONTINUE NRITE(*.'(A)') 'I' WRITE(*.'(IX.A)') TITLE HRITE(A.A) ' . HRITE(A.'(T6.A.TI5.A.2x.A.T3I.A(A.I3XI') ... 'ALPHA'. IPI. OTESTI. URI. 8T8. IVI' I": NRITE(*.R) ' ' HRITE(..IOO) * 'O-OI'. '5'. '3'. (NI(I).I'I.3). 'C'. (HI(|).I'5.3). T 'L'. (NI(I):I'9:I2) WRITE(‘:IID) + '~.D 'B.o ("‘(l)o'-‘3o‘6)o 'c.o (N‘(|)O|.l7ozo)o A 'L'. (NI(I).II2I.ZN) WRITE(*.IOO) * '0-05'. '5'. ‘3'. (N2(I).III.A). 'C'. (N2(|).II5.3). + 'L'. (NZIII.|'9.IZI 1530 WRITE(*.IIO) + 'A'. '8'. (N2(I).I-13.16). 'C'. (N2(I).I-17.20). + 'L'. (N2(I).I-2I.2A) HRITE(*.IOO) + '0.10'. '5'. '8'. (H3(I).I-I.A). 'C'. (N3(I).I-5.8). + 'L'. (NBII).|-9.12) HRITE(*.IIO) + 'II'. '8'. (N3(|).|-I3.16). 'C'. (N3(I).l-17.20). + 'L'. (N3(I).I-2I.2A) CA CA HRITE THE NOMINAL VALUES TO NVALUE. CA CALL FILEC( 'NVALUE'. FET. 8. BUF. 513 ) CALL RETF( FET ) , CALL FILEc( ‘NVALUE'. FET. 8. BUF. 513 ) CALL HRITEH( FET. NI. 2A ) CALL HRITEH( FET. N2. 2A ) CALL HRITEH( FET. N3. 2A ) CALL HRITEOR( FET ) CA CI FTNS STANDARD CODE - DELETE THE PRECEDING FILEC. RETF. WRITEW. C* AND WRITEOR CALLS. AND USE THE FOLLOHING CODE: CA C OPEN( 2.FILEI'NVALUE' ) C REHIND 2 C WRITE(2.*) NI C HRITE(2.*) N2 C HRITE(2.*) N3 C REHIND 2 CA STOP IOO FORMAT( T6.A.TI$.A.3(T19.A.T20.AFIA.5./) ) IIO FORMAT( TI5.A.3(TI9.A.T20.AFIA.5./) ) END AEOSOO LINE-I6I SEc-I 13]. PROGRAM OTABLEI OUTPUT ) CAAAAAAAAAAAAAAAAAA CA OTABLE READS THE LISTS OF TEST STATISTICS AND CALCULATES THE C* ACTUAL VALUES FOR ALPHA I 0.0I. 0.05. AND O.IO BY EMPIRICALLY Ct FINDING THE PROPORTION OF STATISTICS EXCEEDING THE NOMINAL VALUES. C* . C* FOR THE EXPECTED SIZE OF I0.000 . THESE VALUES ARE CALCULATED C* BY AVERAGING THE IOOTH AND IOIST. THE SOOTH AND THE SOIST. AND C* THE IOOOTH AND THE IOOIST ELEMENTS. RESPECTIVELY. CA C* CALLING SEQUENCE: CA C* OTABLE.OUTLFN. CA C* WHERE DUTLFN IS THE FILE TO WHICH THE TABLED OUTPUT IS TO BE HRITTEN. CI DEFAULT: OUTPUT CAAAAAAAAAAAAAAAAAA CI REFER TO THE COMMENT SECTION OF PROGRAM RM FOR A DETAILED EXPLANATION CA CA CA OF THE INTERACTION OF THESE PROGRAMS: REFER TO THE COMMENT SECTION OF PROGRAM NTABLE FOR INFORMATION ON MAKING THIS PROGRAM TRANSPORT- ABLE (THE PROCEDURE IS ALMOST EXACTLY THE SAME AS FOR NTABLE). CAAAAAAAAAAAAAAAAAA IMPLICIT INTEGER(A-z) PARAMETER (COUNT-IOOOO) REAL ARRAY(COUNT) DIMENSION FET(8). BUF(20A9) CHARACTER UNITLFN*7. TITLEA8O. AHSHER REAL NI(2A). N2(2A). N3(2A). OI(2A). 02(2A). 03(2A) LOGICAL SHOHB CALL NOMSG CALL FILEC( 'ZZZZIN'. FET. 8. BUF. 65 ) CALL CONNECF( FET. O ) CALL HRITEH( FET. ' OTABLE - PLEASE ENTER A TITLE -'. A ) CALL READH( FET. TITLE. 8. EDP ) ' L-LNB(TITLE) DO 5 l.LO'O-I P-(BO-L)/2+I TITLE(P:P)-TITLE(I:I) CONTINUE TITLE(:(8O-L)/2)-' ' CALL HRITEH( FET. ' OTABLE - PRINT B TEST?’. 3 ) CALL READH( FET. AHSHER. I. EDP ) SHOHB-ANSHER(:I) .EQ. 'Y' CALL FILEC( 'NVALUE'. FET. 8. BUF. 20A9 ) CALL REHINDF( FET ) CALL READH( FET. NI. 2A. EOP ) CALL READH( FET. N2. 2A. EOP ) CALL READH( FET. N3. 2A. EDP ) IF (EOP .LT. 0) THEN CALL REMARK('ONEXPEETED AEOP ON NVALUE FILE.‘ ) CALL ABORT ENDIF UNITLFNI'TAPE' OPEN(I.FILEI'RAHDATA') REWIND I DO IO III.2A 1J32 CALL INT2CHR( I. UNITLFN(5:) ) CALL FILEC( UNITLFN. FET. 8. BUF. 5I3 ) CALL REHINDF( FET ) CALL READH( FET. ARRAY. COUNT. EDP ) CALL HHBF( FET ) IF (EDP .EQ. 0) THEN WRITE(I.*) UNITLFN. ARRAY(I) IIII2II3IO CA . CA I MOD A - O INDICATES THAT THE FILE CONTAINS VALUES FROM THE CA H TEST. SO THE TEST IS REVERSED: OTABLE MUST CHECK FOR VALUES CA THAT ARE LESS THAN THE NOMINAL VALUE. NOT GREATER THAN. CA IF (M0D(I.A) .EQ. 0) THEN DO 20 J-I.COUNT IF ( ARRAY(J) .LT. N3(I) ) THEN I3Il3+l IF ( ARRAY(J) .LT. N2(I) ) THEN I2-I2+I IF ( ARRAY(J) .LT. NI(I) ) THEN IIIII+I ENDIF ENDIF ENDIF ZO CONTINUE C* ' C* ELSE THE FILE CONTAINS VALUES FROM A R. T. OR V TEST. CA ELSE DO 30 JII.COUNT IF ( ARRAY(J) .GT. N3(l) ) THEN I 3II 3+I IF ( ARRAY(J) .GT. N2(I) ) THEN I2II2+I IF ( ARRAY(J) .GT. NI(I) ) THEN II-II+I ENDIF ENDIF ENDIF 30 CONTINUE ENDIF OI(I)-II/FLOAT(COUNT) 02(I)-I2/FLOAT(COUNT) 03(I)-I3/FLOAT(COUNT) ELSE CALL REMARK( 'UNExPECTED AEOP ON READ OF '//UNITLFN ) OI (I)-02(I)-O3(I)-0.0 ENDIF 1J33 IO CONTINUE NRITE(I.*) .0)-'.DI WRITE(I.*) '02-..02 WRITE(I.*) 'D3".03 WRITE (*.' (A) ') 'I' WRITE(*.'(IX.A)') TITLE NRITE(*.*) ' ' HRITE(*.'(T5.A.TI5.A.2X.A.T3I.H(A.I3X)') + IALPH‘I. 8P8. ITESTI’.IRI. ITI' IVI’ I". WRITE(*.*) ' ' . IF (SHOWB) THEN WRITE(*.IOO) 4' 'O-OI'. '5‘. '8'. (OI(I).I-I.II). ’C'. (CHILI-5.8). + 'L'. (OI(I).I-9.Iz) HRITE(A.IIo) 4' 'A'. '8'. (DINA-13.16). 'C'. (CHILI-17.20). + 'L‘. (OI(I).I-2I.2A) NRITE(*.IOO) + '0-05'. '5'. '8'. (CHILI-IA). 'C'. (CHILI-5.8). + 'L'. (02(I).I- 9.12) WRITE(*.IIO) + .k'o .... (02(l)9'-‘3016)0 'C'. (02")...17020)o + 'L'. (02(I).I-2I.2A) WRITE(*.IOO) + '0.IO'. '5'. 'B'. (03(l).|-I.h). 'C'. (”Ind-5.8). + 'L'. (03(I)o" 9912) HRITE(A.IIO) + 'A'. '8'. (03(I).I-I3.16). 'C'. (03(I).I-I7.20). + 'L'. (03(I).I-2I.2A) ELSE HRITE(A.IOO) A '0.0I'. '5'. 'C'. (CHILI-5.3)- + 'L'. (OI(I).I-9.12) HRITE(A.IIO) + "0'. 'C'. (OI(I).I-17.20). + 'L'. (OI(I).I-zI.2A) HRITE(A.Ioo) + I0.05'. '5'. 'C'. (02(I).I-5.8). + 'L'. (02(I).I- 9.12) HRITE(A.IIO) + '5'. 'C'. (02(I).I-I7.20). + 'L'. (02(I).I-2I.2A) HRITE(*.IOO) + .0010.0 .5.! .c.0 (03(')||-508)I + 'L'. (03(I).I- 9.12) HRITE(*.IIO) + 'II'. 'C'. (03(I).|-I7.20). + 'L'. (03(I).I-21.2A) ENDIF STOP IOO FORMAT( T6.A.TIS.A.3(TI9.A.TZO.hFIh.S./) ) 1J34 IIO FORMAT( TI5.A.3(TI9.A.TZO.IIFIII.S./) ) ' END *EOSOO LINEII87 SECII 135 APPENDIX B MONTE CARLO CRITICAL VALUES The values in the following tables were determined under conditions of homogeneity and true null hypotheses. The tables were generated by the computer program written for this study. Values in the first table were used in determining actual significance levels and powers for 10,000 replications with k = 3 equal groups of size n s 20 and p measures. Values in the remaining tables were used in determining actual significance levels for 2,000 replications of the corresponding five combinations of k equal groups of size n. The hypotheses tested at three nominal alpha levels were: B = between-group differences C - within—group trends L a within-group trends higher than linear The test statistics used were: Roy's largest root Hotelling-Lawley trace Pillai-Bartlett trace Wilks' likelihood ratio S.m nm.NH no.Ha ov.o~ no.nH oo.~H n¢.oa oo.NH ch.HH ON.oH nh.NH no.HH nh.oH OH. nN.oH om.nn on.od oo.nH no.~H o~.ou no.m~ nn.~H on.oa om.ou oo.ma nm.oH no.nH no.NH on.ou nn.¢~ nv.NH ON.HH nH.vm nH.om no.NH nm.om n¢.va oN.~H oo.nH nH.MH nn.oH om.om oo.HN ON.NH oa.n~ no.5H oo.HH nN.nH nN.MH om.oa oh.HH no.0 nm.n cm.n on.h no.n no.5 no.n nn.n nn.NH no.5 nv.n nh.ou oo.o Oh.n nm.o co.n nv.n oN.m on.o nn.n ov.n nn.o no.n nn.n no.o o¢.n nn.n _> no. nm.nN no.NH nN.o nm.¢~ OH.OH no.n no.o nH.n nm.n nH.Hm no.nH om.n nh.hd nm.HH nN.b no.o ch.o on.n m nn.n o~.N on.H oN.v on.~ nN.H no.~ on.~ no. no.v 3838 one com th O O O O O O HHH HHN HHM onn >. do. o¢.h nh.N om.u n~.n nv.~ nm.H nn.H oo.H nH.va cn.v nn.H nv.o nh.¢ no.N no.H n¢.H nc.H oo.od o¢.n oo.H on.o oo.v nn.a nm.N ch.H nH.H Am NQ'O'I have: NVO’I '0 NVOA NVO’I NQ'OT com A a 5:3 Boon. moougofioo now NIU manna :52 033 a 30.5 @35— moan—ooOOAo omficoouom JJAS APPENDIX D SIGNIFICANCE LEVELS FOR WITHIN-GROUP TESTS OF NON-LINEARITY The following tables are actual significance levels expressed as percentage exceedance rates of Monte Carlo critical values for multivariate within-group tests of the null hypothesis of no trends higher than linear. L. calculated under heterogeneity levels, d. values are based on 2,000 replications of five combinations of k equal groups of size n with p I 4 or 5 measures. In the first table k I 3 and sample size varies while in the second table n I 20 and the number of groups varies. The test statistics used were: R I Roy's largest root T I Hotelling-Lawley trace V I Pillai-Bartlett trace W I Wilks' likelihood ratio 146 o: d :0 Common. Sag—mu Dougaxm A nN.HH nn.oH no.0H co.HH nm.HH no.0H on.- no.on on.o~ nN.~H nH.HH no.0H no.~a ov.HH ov.om on.v~ oH.NH oo.ou 0N.HH nn.on nm.on ch.a ov.HH ou.oH on.~H 0H.HH no.oH nN.NH n~.HH oo.ou ch.NH nv.HH on.oH on.nn nn.HH nn.oH ca. nN.HH nn.ou ov.oa oo.HH ov.HH no.oa nn.NH oo.oa nn.oa nm.NH nH.HH no.o~ no.o~ on.- on.on oh.vn nN.NH no.0d n¢.HH nh.oa oH.OH nH.NH n¢.HH nH.oH nv.NH nn.on od.oa on.NH nn.~H nh.oH nh.ma co.HH on.oa nn.nH ov.~H on.o~ ow.n oo.n ov.n no.5 oo.n om.n om.o no.n on.n In In:3In Inc: ”"3. no.n nh.n om.n oa.o on.n om.n no. 86 fl.» 2.... 86 mm... 9...... and 86 8.... 86 8.... mm... 36 36 23 8:... mm... 8.... mm... 3.» 8A 26 mm... min 2..» 3;. mm...“ 8.. 8.... 8.... $6 85 8.» £6 85 36 a a no. no. no. nh.H on.H oc.H ch.H nv.H oo.H co.H co. no. nn.H ov.~ nN.H nv.N on.n nn.H no. on. no. nn.H om.H oo.H on.H o¢.H oo.H co.~ co. co. ov.H o¢.H nN.H no.~ on.a nH.A .> do. no. no. co. nh.d nv.H no.H oo.H nm.H no. oo.H no. no. nn.H n¢.H nN.H nn.N nn.H om.H no. as. no. ON.N ch.n nN.H oo.~ nn.H no.H o~.H oc.H oc.H nn.~ n~.H no. nh.N on.H on.H m wvo Nfl'm NV‘Q '0 “I‘d: Nvo NVO‘ on on OH on om OH Am A x 8:3 3235752 «0 Bums 905L353 uow :5 man. u Amoco moumm 850896 $3.80qu HID OHQNB 1J47 n: .o co Amman min—Eu boumcmmeo A oo.HH oo.oH nn.oH oo.HH nm.HH no.oH no.HH on.HA no.oH on.HH oo.o oo.o no.oa o¢.HH ov.oH on.NH ov.HH oo.oH oH.HH oo.oH nn.oa oh.H ov.HH oN.oH oo.~H nn.HH oh.oH n¢.aH oo.o no.o oh.NH nv.HH on.oH nn.NH nH.HH oo.oH oH. oo.oa oo.oH on.oH oo.HH ov.AH no.ou no.HH nn.HH oh.oH on.HH no.o oo.o no.na on.HH on.o~ nn.NH on.HH oo.HH nH.HH oo.o~ nn.oH nA.NH nv.aH nH.oH om.NH no.HH oo.9H oo.HH no.oH nh.o nn.nn oo.HH 8.3. oo.na no.HH no.oH ono O COLD O O nnn ADMIN ADMVD if} 3 o oo.n no. oo.n onn nazII 3. n on n 6:0: O O nnn l-l'Il-I'Il~ nnn 3 E-fl o no.n oh.n nH.n ov.n oo.n nn.n oH.n (3:: FIG] O O InIn gon IDOL“ . .NC‘ who O MIDIO “DION MIDID onn GRID o: nn.H oo.H oo. nh.H oo.n oo.H oH.H no.H no.a no.H nN.H oo. nn.H ov.H nn.n nv.H oo. no. nn.H oo.H oo. nn.H om.a oo.H oa.H nH.H no.H no.H nN.H oo. ov.H ov.H nN.H ov.H oN.H no. > nn.H oo.H oo. nh.H nv.H no.H oN.H oo.H oo.H no.H nN.H oo. nn.H no.n nN.H oo.n no. no. no nn.H nH.H no. oN.~ oh.H nN.A ov.H nH.H oo.H nH.N nn.n no.H nn.a nN.H no. oN.H no. oo.H m NQO) NVQ NVOI '0 cu<¢¢m ¢V B m 3, >. a m 3, > a m o : oH. no. Ho. .m u o. .33 85.9 «o 3.8. ASSESS: .3 moan—p.532 was 395 moumm oocmcoooxm 33538 HIm «Hams oz .9 :0 Common mots—mu Doug—max? 1151 nn.no nn.No on.ho no.no on.oo oo.oo oh.vo nH.no no.no no.no oo.no nN.oo o nN.oo nN.oo nN.oo nN.oo on.oo on.oo oo.oo nN.oo nH.oo no.oo nH.oo oH.oo v oo.ooH oo.ooH oo.ooH oo.ooH oo.oo oo.oo oo.oo oo.oo nv.oo nv.oo nv.oo oo.oo N oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH no.oo no.oo oo.oo oo.oo H nN.on no.on oo.on no.on oo.vv no.ov oo.nv oo.no nH.oN nH.oN oN.MN oH.oN o nn.No oN.No no.No nn.Ho no.Hh oN.Hh oH.Hh nN.on oo.nv nN.oo oo.nv on.ov I oo.vo oo.vo no.no on.oo no.oo no.oo oH.oo no.no oo.on ov.on oo.on no.on N nH.oo nH.oo nH.oo nH.oo. no.no oH.no oo.no nH.no no.no oH.no oo.vo no.no H oh.vm on.vm no.oo no.nm oo.oN oo.MN nv.vN om.eN no.oH on.oH no.HH nN.HH o nN.on oo.nn oo.vn no.vn nn.Ho oo.Ho no.Ho no.No on.NN on.NN nn.NN nH.NN v nn.nn oo.nn nN.nb nN.on oo.nn nn.nn nN.nn on.oo on.Hv no.ov no.oo on.oo N nH.oo no.oo no.oo on.oo nN.No oo.No on.No no.Ho oo.nn no.Nn on.oo ov.Nn H oo.oo oo.oo no.oo oo.oo nN.ho nH.ho nN.No oh.ho oo.no oo.no no.No oo.no o no.oo no.oo no.oo oo.oo no.oo no.oo oo.oo oo.oo nN.oo nN.oo nN.oo on.oo v oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.oo oo.oo oo.oo no.oo N oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH oo.ooH H nn.nn nH.no oN.nn n¢.vn nn.Nn no.Nn no.Nn nv.Nn no.oN oh.hN on.oN oo.on o oo.No nN.oo oo.no no.no no.oo nn.Nh nN.on nn.nn oN.nn oo.vn nN.nn on.Nn q nn.no on.oo oo.no oo.no on.No oo.No on.No nN.Ho oo.No nn.Nn nn.Nh on.oo N on.oo on.oo on.oo on.oo no.oo oo.oo on.oo oo.oo nn.Ho oo.Ho no.Ho oo.Ho H no.on on.oo no.on om.Hv no.NN oH.oN nH.oN nN.oN oo.oH nn.NH oo.oH no.oH o oo.No nn.Ho oh.Hn oH.Nn no.oo nn.nv oN.o¢ nn.Nv oo.nN oo.nN nN.nN no.nN v no.oo no.oo on.oo nn.Ho oN.Hh oo.ob oo.on no.on oo.oo oo.no oN.No nN.nv N oN.mo oH.no oH.no no.No no.no oo.no no.No no.no nn.Nn nn.on oo.Hh ov.on H .3 _> B. m 3, .> .3 m 3. H> B. a o oH. no. Ho. . AoN N c 5:3 8:39 no 3mg 93:5 3 Lou Sauna—Lou: 039 Hoes: mouno 85885 0 3:038 NIm mHnNa a: d :0 Human flung—mu boa—38.35.. mh. aw 152 cp.mo o~.mm mo.mw om.pm nn.Nn co.~m co.am mm.~m mm.h~ oa.e~ oa.an m cH.Hm mv.Hm c¢.Hm nH.om mm.mc ow.na on.mo no.Hm oo.~m mm.om mm.om mm.om H no.HH mo.om mo.am mm.~m oo.mm ce.mm oo.na om.ma nH.om no.oo cm.om nN.on m c~.ma o~.mm oh.am mw.mm mm.om ma.om mm.ma co.ma mm.~m om.~m mm.~m cm.~a H on cv.mm m¢.um mq.om oo.on om.m~ om.m~ mm.m~ op.m~ mm.a mm.” mH.m oH.OH a mo.mm mH.mm oc.mm cm.vm om.oq em.OH mh.cq oo.c. cm.hH mc.~H mm.>H nH.mH H ov.H> om.Hh me.H~ cH.Hh mm.mm om.mm nn.nn c~.mm om.m~ mo.m~ mm.o~ oo.on N mm.~m nH.no mm.~o oo.nm mp.~h mp.~p m~.~h co.~p cm.Hc om.H¢ cm.H¢ op.ev H o~ mm.p~ nn.n” m¢.p~ m~.~n nN.HH OH.~H m~.~H cm.hH mo.m cm.m m~.m cc.» a nH.om nH.om oo.wm on.mm mH.v~ nH.H~ oo.Ha om.m~ av.» mH.o o¢.a mm.o e o~.ee cm.o¢ mm.mc OH.mv oq.Hm o~.Hm om.Hm mo.Hn oe.~H o~.~H oo.HH oc.~H N oo.Hm ma.mm oa.vm mm.mm ms.mn mw.an om.an m~.am oH.oH oo.HH mo.oH mm.uH H cH oH.oa oa.om m¢.om m>.om A oo.Ha oo.HH oo.Hm o~.qa nH.oa on.om on.oo me.cc H mm.mm mm.ma mm.mm mm.mm om.ma oa.oa oa.am no.HH oq.om mvnom o..am m¢.mm e cc.oOH oo.ooH oc.OOH cc.ooH oo.ooH oo.ocH co.ooH co.ocH mm.mm mm.a¢ mm.am mm.aa a oo.oOH oc.OOH oc.ooH oo.OOH oo.ooH oo.ooH o°.ooH oo.oOH co.oOH oo.ooH co.QOH oo.ooH H on mm.>o om.pm mm.~o cH.>m no.Hm o~.¢m oc.¢m o¢.¢m nN.on o¢.m~ OH.om oo.m~ m c~.mo mH.mm nH.mo oH.oo cp.oa om.om cm.om oo.oo mo.am mu.hm om.pm nN.Hm H ma.pa eo.ma co.ma c~.~m o~.vm ou.¢m om.¢a OH.cm oH.Hm oc.Ho oa.oo o~.m~ N cm.mm cm.mm om.ma mo.mm oo.nn om.mm cm.mm mm.mm mH.mm om.~m mm.~m oH.~m H c~ ca.¢¢ oh.mv m~.mq mm.H¢ c¢.Hm o~.~n o~.«m m~.~m no.oH m¢.mH nH.oH ca.mH a c~.~o nn.Ho cm.Hm nH.ou nn.Hm nH.om o°.Hm oo.Hq cm.p~ mw.u~ m¢.m~ co..~ . mm.mh mm.ms mv.mn °~..p mo.ee c>.mo mh.co om.Ho am.~e oH.HH HH.H¢ nH.nm H on.oo cm.mm om.mm m~.qm on.os no.n» m~.h~ om.v~ co.mm mo.¢m mm.mm oo.om H OH 2 _> a m 3. > a m z, > a. m c : cH. mo. Ho. .3 a x .33 bHuaocHHEoz no 389 9.8 .33 3H mmfiumfimfld was H85 8%»: 8538.6 w 3:038 aim mHan. 3H d co ~39? 9:99. bougmema, ON.om nN.oo o~.oo mn.mw mo.mw no.No no.on no.on oH.o¢ mc.oc ON.mv mm.hv m oo.Nm cm.Nm nn.Na mm.Nm nn.nn oH.nm cm.hm o¢.>c cm.Hn mN.Hh mm.Hh mH.Nh c mm.nm nn.NN mm.nm OH.hm mH.¢m nN.Hm nH.oN mN.¢m mN.mm oH.no on.mm cw.mm N oN.NN on.om ch.mm om.mm cH.hm oH.nm mH.hm no.NN nN.om om.om no.Hm cc.Hm H oc.wm mv.on mw.wm om.mm om.mN om.mN mm.mN oh.mN mc.m no.o mH.a oH.cH m mo.mm mH.mm oo.nm cm.vm om.ov om.ov mh.oq on.ov on.NH no.NH mm.>H mH.mH v ov.HN nn.Hh mq.Hb OH.H5 mm.mm on.om mh.om oN.nn on.NN no.nN mm.mN oo.on N mm.Nm mH.mm no.No om.Nm mN.Nh nn.Nh nn.Nh om.Nh co.Hv cm.Hv om.Hv oh.vv H om.mN om.MN mw.MN mv.vN om.mH mN.mH om.mH mm.MH mn.e mm.v mN.¢ mw.c m om.wm mh.mm oo.on nn.nn mc.NN mv.NN cp.NN ch.NN oo.n om.m mN.m mH.m v mm.mm mm.mm co.vm cv.mm oo.mm oo.on mo.wm OH.wm nN.oH oN.oH nH.oH nN.oH N cm.mw mm.mm mm.mm ch.mm oN.nn mH.vm mh.mm ON.mm mv.mN .mm.oN oH.mN mm.mN H oo.ma oo.Nm oo.Na 9H.mm no.NN oo.Nm mm.hm mh.hm mo.mm cN.mm mo.mm ms.Na m mm.aa mm.mm mm.mm mm.mm mm.mm mm.mm mm.mm mm.mm ov.am om.mm oc.mm mv.mm v oo.ooH oo.oOH oo.ocH on.ocH cc.OOH cc.oOH oo.ooH oc.ooH mm.mm mm.mm mm.mm mm.am N oo.ocH oo.ooH oo.ccH cc.ooH co.ooH oo.ooH co.ocH cc.ooH oo.:cH oo.ooH oo.ooH oo.ocH H mm.no ow.hw mm.~m OH.ho no.Hm oN.vm ov.vm cv.vm nN.on ov.mN oH.cm cc.mN m oN.No mH.mm mH.mo OH.mm on.oo om.om on.om oo.no no.on mm.Nm oN.nn mN.mm w mm.ha co.mm oo.oa on.NN on.vm ow.va ow.cm oH.vm oH.Hm cc.Hm o¢.om oN.nn N ow.mm on.NN ow.mm mm.mm om.oo om.mm oo.om mm.wm mH.mm oN.Nm mm.Nm oH.Na H mw.Nv on.Nv mm.N¢ oo.H¢ no.oN oc.aN no.oN om.mN mv.mH mm.MH mN.MH oc.MH a ow.mw mm.mo mm.mc mw.mm o».Nm cv.mm OH.Nm cm.om ON.>N ms.hN mh.wN mm.mN q co.¢m mv.vo mm.vo mm.mm mo.mh ov.mn oh.¢h nn.nn nN.Hm no.Hm on.om ca.m¢ N mN.mm mv.mm mH.mm ov.vm mm.om co.om no.om oH.mm oN.mh ov.mb mo.vh nN.nn H 3 > .H. m 3 > .H. m 3 > .H. m O OH. mo. Hc. h.éN u : 53 3:357:02 uo 338 9.3 53 How mm>HumcumuH< ways yucca «mama wocmcomoxm m mucmonm fiLm mHnt 1153 BIBLIOGRAPHY BIBLIOGRAPHY Anderson. '1'.W. (1958) An lnttoduttien to W Statistical Analysis. New York: John Wiley and Sons. Bock. RJL (1963) Multivariate analysis of repeated measures. In C.W. Harris (ed.) Emblems in W ‘Change (pp. 85-103). Madison. Wisconsin: University of Wisconsin Press. Book. RD. (1975) Multilariate Statistical Methods in Behaxigral Sciences. ‘New YOrk: McGraw-Hill Book Company. Box, G.E.P. (1954) Some theorems on quadratic forms applied in the study of analysis of variance problems, II. Effects of inequality of variance and of correlation between errors in the two-way classification. Annals of. Mathematical Statistics 25. 484-498. Box, S.E.P. and Muller, M.E. (1958) A note on the generation of random normal deviates. IAnnals oi Mithfimfliigfll‘5£§§ifitiflfil 22: 610-511. Ceurvorst, R.W. (1980) Robustness of MANOVA under heterogeneity of variance and correlation. Unpubl shed doctoral dissertation. Arizona State University. Collier, 12.0., Baker, F.B., Mandeville. G.K.. and Hayes, 159. (1967) Estimates of test size for several test procedures based on conventional variance ratio in the repeated measures design. ‘Bsyghgmetrika, 32, 339- 353. Davidson, M.L. (1972) Univariate versus multivariate tests in repeated measures experiments. .EsxsthQQQQAl Bulletin. 11: 445-452. Finn. J.D. (1974) A General Model tor Mnltixariate Analysis. New York: Holt, Rinehart and Winston, Inc. 154 Glass. G.V., Peckham. P.D.. and Sanders. J.R. (1972) Consequences of failure to meet assumptions underlying the fixed-effects analysis of variance and covariance. Series of Educational Research. .42. 237-288. Greenhouse. S.W.. and Geisser. S. (1959) On methods in the analysis of profile data. Zaxchomcttika. 2.4.. 95-112. Hakstian. A.R i' Roed. J.C. and Lind. J.C. (1979) Two- sample '1‘ procedure and the assumption of homogeneous covariance matrices. Psxcholooical Bulletin. so. 1255-1263. Hammersley. J.M. and Handscomb. D.C. (1964) Monte Carlo Methods. New York: Barnesand Noble. Inc. Harris. R.J. (1975) A Brimer of Multilariate Statistics. New York: Academic Press. 36178. w.L. (1973) Statistics for the Social Sciences. New York: Holt. Rinehart. and Winston. 1973. Holloway. L. N. and Dunn, O. J. (1967) The robustness of Hotelling' 8 T2. Journal of the American Statistical Association. 5.2. 124-136. Hopkins. J.W. and Clay. P.P.E‘. (1963) Some empirical distributions of bivariate T2 and homoscedasticity criterion M under unequal variance and leptokurtosis. Journal of the Americal Statistical Association. 53. 1048-1053. Huynh. H. and Feldt. L.S. (1970) Conditions under which mean square ratios in repeated measuresments designs have exact F- -distributions. Journal of the American Statistical Association. 5.5. 1582- 1589. Ito. K. (1962) A comparison of the powers of two multivariate analysis of variance tests. siemettika. .49.. 455-462. Ito. K. (1969) On the effect of heteroscedasticity and nonnormality upon some multivariate test procedures. In P.R. Krishnaiah (ed. ). Multitariate Analysis 11 (pp.87-120) New York: Academic Press. Ito. P.R. (1980) Robustness of ANOVA and MANOVA test procedures. In P.R. Krishnaiah (ed.) Handbook of. Statistics... 291. I (pp. 199-236) North Holland Publishing Company. 155 Ito. K. and Schull. W.J. (1964) On the robustness of the To2 test in multivariate analysis of variance when variance-covariance matrices are not equal. Biometrika. 51. 71- 82. Korin. B.P. (1972) Some comments on the homoscedasticity criterignW M and the multivariate analysis of variance tests T W. and R. .siemettiks. 52, 215-216. Lehman. R.S. (1977) Comouter Simulation and Modeling. Hillsdale. New Jersey. Mendoza. J.L.. Toothaker. L.E. and Nicewander. W.A. (1974) A Monte Carlo comparison of the univariate and multivariate methods for the groups by trials repeated measures design. Multitariate Behatioral Research. 2' 165- -1770 Morrison. ELF. (1972) The analysis of a single sample of repeated measurements. .Biemetties. 23. 55-71. Morrison. v.3. (1976) Multixariate Statistical Methods. New York: McGraw Hill Book Company. 1976. Olson. CJL (1973) A Monte Carlo investigation of the robustness of multivariate analysis of variance. Unpublished doctoral dissertation. University of Toronto. Olson. C. L. (1974) Comparative robustness of six tests in. multivariate analysis of variance. .Jenrnal of the American Statistical Assiciation. 69.. 894- 908. Pillai. KJLS. and Sudjana. (1975) Exact robustness of tests of two multivariate hypotheses based on four criteria and their distribution problems under violations. The Annals of Statistic. Ii. 617-636. Potthoff. R.P. and Roy. S.N. (1964) A generalized multivariate analysis of variance model useful especially for growth curve problems. ‘siemettiks. 51. 313-326. Ramsey. P.R. (1980) Exact Type I error rates for robustness of Student's t test with unequal variances. Journal of Educational Statistics. 5. 337-349. Scheifley. V; (1974) Analysis of repeated measures data: A simulation study. Unpublished doctoral dissertation. Michigan State University. 156 Scheifley. V. and Schmidt. W. (1978) Analysis of repeated measures data: A simulation study. Moltisariate Eehauioral Research. 1.1. 347-362. Scheffé. H. (1959) The Analxsis of Manse. New York: John Wiley and Sons. Tatsuoka. 14.11. (1971) Multixariate Analysisi Techniques for Educational and Rszcholocioal Research. New York: John Wiley and Sons. Timm. mm (1975) Multiuariate Analxsis saith Applications in Education and Rsmhologxsesearch . Monteray. California: Brooks Cole Publishing Company. Timm. N.H. (1980) Multivariate analysis of variance of repeated measures. In P.R. Krishnaiah (ed.) Handbook of Statistics... In]. 1 (pp. 41-87). New York: North Holland Publishing Company. 157