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Research article - Peer-reviewed, 2022

Graphical analysis of residuals in multivariate growth curve models and applications in the analysis of longitudinal data

Hamid, Jemila S.; Huang, Wei Liang; von Rosen, Dietrich


Statistical models often rely on several assumptions including distributional assumptions on outcome variables as well as relational assumptions representing the relationship between outcomes and independent variables. Model diagnostics is, therefore, a crucial component of any model fitting problem. Residuals play important roles in model diagnostics and checking assumptions. In multivariate models, residuals are not commonly used in practice, although approaches have been proposed to check multivariate normality and other model assumptions. When done, ordinary residuals are often used. Nevertheless, it has been shown that ordinary residuals in the analysis of longitudinal data are correlated and are not normally distributed. Under sufficiently large sample size, a transformation of residuals were previously proposed to check the normality assumption. The transformation solely focuses on removing the correlation. In this paper, we show that the ordinary residuals in the analysis of longitudinal data are not normally distributed and should not be used for checking the normality assumption. Via extensive simulations, we also show that the transformed (de-correlated) residuals fail to provide accurate model validation, in particular in the presence of model misspecification. We consider decomposed residuals from the multivariate growth curve model, provide practical interpretations, examine their property analytically as well as via simulations, and show how the different components can be used to examine model misspecification and distributional assumptions. Extensive simulations are performed to evaluate and compare performances for normal and non-normal data. Analysis of real data sets are presented as illustrations.


Cholesky decomposition; Decomposed residuals; Decomposition of linear spaces; Growth curve model; Longitudinal data; Model assumptions; Model diagnostics; Ordinary residuals; Small's graphical method; Transformation

Published in

Communications in Statistics - Simulation and Computation
2022, volume: 51, number: 10, pages: 5556-5581

Authors' information

Hamid, Jemila S.
Univ Ottawa
Hamid, Jemila S.
McMaster Univ
Huang, Wei Liang
McMaster Univ
Swedish University of Agricultural Sciences, Department of Energy and Technology
von Rosen, Dietrich (Von Rosen, Dietrich)
Linköping University

UKÄ Subject classification

Probability Theory and Statistics

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