Testing Repeated Measures Hypotheses When Covariance Matrices are Heterogeneous

For balanced designs, degrees of freedom-adjusted univariate F tests or multivariate test statistics can be used to obtain a robust test of repeated measures main and interaction effect hypotheses even when the assumption of equality of the covariance matrices is not satisfied. For unbalanced designs, however, covariance heterogeneity can seriously distort the rates of Type I error of either of these approaches. This article shows how a multivariate approximate degrees of freedom procedure based onWelch (1947 , 1951)-James (1951 , 1954), as simplified byJohansen (1980), can be applied to the analysis of unbalanced repeated measures designs without assuming covariance homogeneity. Through Monte Carlo methods, we demonstrate that this approach provides a robust test of the repeated measures main effect hypothesis even when the data are obtained from a skewed distribution. The Welch-James approach also provides a robust test of the interaction effect, provided that the smallest of the unequal group sizes is five to six times the number of repeated measurements minus one or provided that a reduced level of significance is employed.

[1]  Scott E. Maxwell,et al.  Designing Experiments and Analyzing Data: A Model Comparison Perspective , 1990 .

[2]  G. S. James.  TESTS OF LINEAR HYPOTHESES IN UNIVERIATE AND MULTIVARIATE ANALYSIS WHEN THE RATIOS OF THE POPULATION VARIANCES ARE UNKNOWN , 1954 .

[3]  J. Algina,et al.  Type I Error Rates for Yao’s and James Tests of Equality of Mean Vectors Under VarianceCovariance Heteroscedasticity , 1988 .

[4]  M K Kaiser,et al.  MANOVA method for analyzing repeated measures designs: an extensive primer. , 1985, Psychological bulletin.

[5]  S. Johansen,et al.  The Welch - James approximation to the distribution of the residual sum of squares in a weighted lin , 1980 .

[6]  H. J. Keselman,et al.  Analysing unbalanced repeated measures designs , 1990 .

[7]  H. Huynh,et al.  Estimation of the Box Correction for Degrees of Freedom from Sample Data in Randomized Block and Split-Plot Designs , 1976 .

[8]  Eric R. Ziegel,et al.  Multiple Comparisons, Selection and Applications in Biometry , 1992 .

[9]  R. Blair,et al.  A more realistic look at the robustness and Type II error properties of the t test to departures from population normality. , 1992 .

[10]  G. Duncan,et al.  Multivariate Analysis: With Applications in Education and Psychology. , 1977 .

[11]  G. S. James.  THE COMPARISON OF SEVERAL GROUPS OF OBSERVATIONS WHEN THE RATIOS OF THE POPULATION VARIANCES ARE UNKNOWN , 1951 .

[12]  Juliet Popper Shaffer,et al.  Multiple pairwise comparisons of repeated measures means under violation of multisample sphericity , 1991 .

[13]  Michael L. Davidson.  Univariate versus multivariate tests in repeated-measures experiments. , 1972 .

[14]  B. L. Welch.  ON THE COMPARISON OF SEVERAL MEAN VALUES: AN ALTERNATIVE APPROACH , 1951 .

[15]  S. Geisser,et al.  An Extension of Box's Results on the Use of the $F$ Distribution in Multivariate Analysis , 1958 .

[16]  Welch Bl.  THE GENERALIZATION OF ‘STUDENT'S’ PROBLEM WHEN SEVERAL DIFFERENT POPULATION VARLANCES ARE INVOLVED , 1947 .

[17]  Huynh Huynh,et al.  Some approximate tests for repeated measurement designs , 1978 .

[18]  T. C. Oshima,et al.  Robustness of Yao’s, James’, and Johansen’s Tests Under Variance-Covariance Heteroscedasticity and Nonnormality , 1991 .

[19]  T. Micceri.  The unicorn, the normal curve, and other improbable creatures. , 1989 .

[20]  Jorge L. Mendoza,et al.  Analysis of repeated measurements. , 1979 .

[21]  B. Lecoutre.  A Correction for the ε̃ Approximate Test in Repeated Measures Designs With Two or More Independent Groups , 1991 .