Robust tests for multivariate repeated measures with small samples

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Multivariate repeated measures data naturally arise in clinical trials and other fields such as biomedical science, public health, agriculture, social science and so on. For data of this type, the classical approach is to conduct multivariate analysis of variance (MANOVA) based on Wilks' Lambda and other multivariate statistics, which require the assumptions of multivariate normality and homogeneity of within-cell covariance matrices. However, data being analyzed nowadays show marked departure from multivariate normality and homogeneity. This paper proposes a finite-sample test by modifying the sums of squares matrices to make them insensitive to the heterogeneity in MANOVA. The proposed test is invariant to affine transformation and robust against nonnormality. The proposed method can be used in various experimental designs, for example, factorial design and crossover design. Under various simulation settings, the proposed method outperforms the classical Doubly Multivariate Model and Multivariate Mixed Model proposed elsewhere, especially for unbalanced sample sizes with heteroscedasticity. The applications of the proposed method are illustrated with ophthalmology data in factorial and crossover designs. The proposed method successfully identified and validated a significant main effect and demonstrated that univariate analysis could be oversensitive to small but clinically unimportant interactions.

Original languageEnglish
Pages (from-to)555-580
Number of pages26
JournalJournal of Applied Statistics
Issue number3
StatePublished - 2024

Bibliographical note

Publisher Copyright:
© 2022 Informa UK Limited, trading as Taylor & Francis Group.


  • Wilks' lambda
  • affine invariance
  • finite sample approximation
  • heteroscedasticity
  • nonnormality

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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