Weighted Normal Plots

Abstract

Abstract Weighted normal plots are proposed as graphical checks on the normality of random effects in Gaussian linear models. The technique is illustrated using the one-way comparisons model Yi = μ i + ϵ i , where the (μ i , ϵ i ), are independent pairs with μ i and ϵ i , independent N(0, σ2) and N(0, σ2 i ), respectively, for i = 1, …, n. When the variance components σ2 and σ2 i are known, an unweighted normal plot of the standardized Zi = Yi (σ2 + σ2 i )-1/2 provides a check of the overall adequacy of the model. Weighted normal plots involve a modification that gives the ith observation a sample weight of Wi = (σ2 + σ2 i )-1. Under the null hypothesis, the sample size must be larger by a factor of (1 + v/m 2), where m and v are the mean and variance of the weights, to produce a weighted plot with approximately the same sampling variance as an unweighted normal plot. Despite this higher variability, we show that weighted plots are more sensitive than unweighted plots to several departures from the assumed distribution on the random effects, μ i . Several numerical examples are included and the effects of substituting maximum likelihood estimates for the parameters σ2 and σ2 i are considered briefly.