In experimental or quasi-experimental studies in which a repeated measures design is used, it is common to obtain scores on several dependent variables on each measurement occasion. Multivariate mixed model (MMM) analysis of variance (Thomas, 1983) is a recently developed alternative to the MANOVA procedure (Bock, 1975; Timm, 1980) for testing multivariate hypotheses concerning effects of a repeated factor (called occasions in this study) and interaction between repeated and non-repeated factors (termed group-by-occasion interaction here). If a condition derived by Thomas (1983), multivariate multi-sample sphericity (MMS), regarding the equality and structure of orthonormalized population covariance matrices is satisfied (given multivariate normality and independence for distributions of subjects' scores), valid likelihood-ratio MMM tests of group-by-occasion interaction and occasions hypotheses are possible. To date, no information has been available concerning actual (empirical) levels of significance of such tests when the MMS condition is violated. This study was conducted to begin to provide such information.
Departure from the MMS condition can be classified into three types— termed departures of types A, B, and C respectively:
(A) the covariance matrix for population ℊ (ℊ = 1,...G), when orthonormalized, has an equal-diagonal-block form but the resulting matrix for population ℊ is unequal to the resulting matrix for population ℊ' (ℊ ≠ ℊ');
(B) the G populations' orthonormalized covariance matrices are equal, but the matrix common to the populations does not have equal-diagonal-block structure; or
(C) one or more populations has an orthonormalized covariance matrix which does not have equal-diagonal-block structure and two or more populations have unequal orthonormalized matrices.
In this study, Monte Carlo procedures were used to examine the effect of each type of violation in turn on the Type I error rates of multivariate mixed model tests of group-by-occasion interaction and occasions null hypotheses. For each form of violation, experiments modelling several levels of severity were simulated. In these experiments: (a) the number of measured variables was two; (b) the number of measurement occasions was three; (c) the number of populations sampled was two or three; (d) the ratio of average sample size to number of measured variables was six or 12; and (e) the sample size ratios were 1:1 and 1:2 when G was two, and 1:1:1 and 1:1:2 when G was three. In experiments modelling violations of types A and C, the effects of negative and positive sampling were studied.
When type A violations were modelled and samples were equal in size, actual Type I error rates did not differ significantly from nominal levels for tests of either hypothesis
except under the most severe level of violation. In type A experiments using unequal groups in which the largest sample was drawn from the population whose orthogonalized covariance matrix has the smallest determinant (negative sampling), actual Type I error rates were significantly higher than nominal rates for tests of both hypotheses and for all levels of violation. In contrast, empirical levels of significance were significantly lower than nominal rates in type A experiments in which the largest sample was drawn from the population whose orthonormalized covariance matrix had the largest determinant (positive sampling).
Tests of both hypotheses tended to be liberal in experiments which modelled type B violations. No strong relationships were observed between actual Type I error rates and any of: severity of violation, number of groups, ratio of average sample size to number of variables, and relative sizes of samples.
In equal-groups experiments modelling type C violations in which the orthonormalized
pooled covariance matrix departed at the more severe level from equal-diagonal-block form, actual Type I error rates for tests of both hypotheses tended to be liberal. Findings were more complex under the less severe level of structural departure. Empirical significance levels did not vary with the degree of interpopulation heterogeneity of orthonormalized covariance matrices.
In type C experiments modelling negative sampling, tests of both hypotheses
tended to be liberal. Degree of structural departure did not appear to influence actual Type I error rates but degree of interpopulation heterogeneity did. Actual Type I error rates in type C experiments modelling positive sampling were apparently related to the number of groups. When two populations were sampled, both tests tended to be conservative,
while for three groups, the results were more complex. In general, under all types of violation the ratio of average group size to number of variables did not greatly affect actual Type I error rates.
The report concludes with suggestions for practitioners considering use of the MMM procedure based upon the findings and recommends four avenues for future research on Type I error robustness of MMM analysis of variance. The matrix pool and computer programs used in the simulations are included in appendices. / Education, Faculty of / Educational and Counselling Psychology, and Special Education (ECPS), Department of / Graduate
|Prosser, Robert James
|University of British Columbia
|University of British Columbia
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