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Zbl 0672.62081
Lange, Nicholas; Ryan, Louise
Assessing normality in random effects models. (English)
[J] Ann. Stat. 17, No.2, 624-642 (1989). ISSN 0090-5364

Summary: When one uses the unbalanced, mixed linear model $$ y\sb i=X\sb i\alpha +Z\sb i\beta\sb i+\epsilon\sb i,\quad i=1,...,n $$ to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume $\epsilon\sb i\sim\sb{ind}{\cal N}(0,\sigma\sp 2I\sb i)$ independent of $\beta\sb i\sim\sb{iid}{\cal N}(0,\Delta)$, where $\sigma\sp 2$ and the elements of an arbitrary $\Delta$ are unknown variance and covariance components. In this paper, we describe a method for checking model adequacy and, in particular, the distributional assumption on the random effects $\beta\sb i.$ \par We generalize the weighted normal plot to accommodate dependent, nonidentically distributed observations subject to multiple random effects for each individual unit under study. One can detect various departures from the normality assumption by comparing the expected and empirical cumulative distribution functions of standardized linear combinations of estimated residuals for each of the individual units. \par Through application of distributional results for a certain class of estimators to our context, we adjust the estimated covariance of the empirical cumulative distribution function to account for estimation of unknown parameters. Several examples of our method demonstrate its usefulness in the analysis of longitudinal data.

MSC 2000:: *62J05 Linear regression
62J10 Analysis of variance, etc.
62F12 Asymptotic properties of parametric estimators
62P10 Appl. of statistics to biology

Keywords: empirical Bayes estimation; restricted maximum-likelihood; estimation; adjustments for estimated parameters; growth curve; models; unbalanced, mixed linear model; continuous outcomes; checking model adequacy; weighted normal plot; dependent, nonidentically distributed observations; multiple random effects; linear combinations of estimated residuals; empirical cumulative distribution; longitudinal data

Cited in: Zbl 1076.62077

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