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Assessing normality in random effects models. (English) Zbl 0672.62081

Summary: When one uses the unbalanced, mixed linear model

y i =X i α+Z i β i +ϵ i ,i=1,···,n

to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume ϵ i ind 𝒩(0,σ 2 I i ) independent of β i iid 𝒩(0,Δ), where σ 2 and the elements of an arbitrary Δ 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 β 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.


MSC:
62J05Linear regression
62J10Analysis of variance and covariance
62F12Asymptotic properties of parametric estimators
62P10Applications of statistics to biology and medical sciences