The Annals of Statistics

Assessing Normality in Random Effects Models

Nicholas Lange and Louise Ryan

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When one uses the unbalanced, mixed linear model $\mathbf{y}_i = \mathbf{X}_i\mathbf{\alpha} + \mathbf{Z}_i\mathbf{\beta}_i + \varepsilon_i, i = 1, \cdots, n$ to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume $\varepsilon_i \sim_{\operatorname{ind}} \mathscr{N}(\mathbf{0}, \sigma^2\mathbf{I}_i)$ independent of $\mathbf{\beta}_i \sim_{\operatorname{iid}} \mathscr{N}(\mathbf{0,\Delta})$, where $\sigma^2$ and the elements of an arbitrary $\mathbf{\Delta}$ 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 $\mathbf{\beta}_i$. 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. 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.

Article information

Ann. Statist. Volume 17, Number 2 (1989), 624-642.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62J05: Linear regression
Secondary: 62G30: Order statistics; empirical distribution functions 62F12: Asymptotic properties of estimators 62P10: Applications to biology and medical sciences

Weighted normal plots longitudinal data analysis variance and covariance components empirical Bayes estimation restricted maximum-likelihood estimation adjustments for estimated parameters


Lange, Nicholas; Ryan, Louise. Assessing Normality in Random Effects Models. Ann. Statist. 17 (1989), no. 2, 624--642. doi:10.1214/aos/1176347130.

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