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June, 1989 Assessing Normality in Random Effects Models
Nicholas Lange, Louise Ryan
Ann. Statist. 17(2): 624-642 (June, 1989). DOI: 10.1214/aos/1176347130


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.


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Nicholas Lange. Louise Ryan. "Assessing Normality in Random Effects Models." Ann. Statist. 17 (2) 624 - 642, June, 1989.


Published: June, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0672.62081
MathSciNet: MR994255
Digital Object Identifier: 10.1214/aos/1176347130

Primary: 62J05
Secondary: 62F12, 62G30, 62P10

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 2 • June, 1989
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