Generalized p value tests for variance components in a class of linear mixed models

Abstract

The problems of testing hypotheses on variance components in linear mixed effects models have been addressed by various workers, although existing methodology is still restricted to a narrow range of models. To overcome this difficulty we develop new general p value tests in general settings. The p values are motivated by a useful matrix inequality. It is shown that the proposed test is invariant under the group of location-scale transformations. Numerical results show that the test can control the Type I errors satisfactorily, and it also exhibits good power properties. Most importantly, the new methods are simple and easy to apply.

Appendix Generalized test variable and generalized p values

Here we introduce the method provided by Tsui and Weerahandi (1989) with their extended definition of p values. Let X be random variable whose distribution depends on the parameters $(\theta ,\eta )$ assuming values in a parameter space H, where $\theta$ is the parameters of interest and $\eta$ represents nuisance parameters. Let $\mathcal {X}$ be the sample space and x be the observed value of X.

Definition A.1

Assume we want to test the null hypothesis $H_{0}: \theta \leqslant \theta _{0}$ versus the alternative hypothesis $H_{1}: \theta > \theta _{0}$, where $\theta _{0}$ is a specified value. Define the $generalized\ test\ variable$ $T({X}; {x}, \theta , \eta )$, which depends on the sample, the observed value of the sample, and the parameters $(\theta ,\eta )$, and satisfies the following conditions:

• (1) The distribution of $T({X}; {x}, \theta _{0}, \eta )$ is free of the nuisance parameter $\eta$.

• (2) The observed value of $T({X}; {x}, \theta _{0}, \eta )$, i.e., $T({x}; {x}, \theta _{0}, \eta )$, is free of $\eta$.

• (3) For fixed x and $\eta$, and for all t, $\mathrm {Pr}\{T( {X}; {x}, \theta , \eta )> t\}$ is nondecreasing in $\theta$, that is, T is stochastically increasing in $\theta$.

Definition A.2

For a generalized test variable $T({X}; {x}, \theta , \eta )$ satisfying the conditions given in Definition A.1, the set of possible sample X that are more extreme than or as extreme as the observed sample x in the sense of the test variable is called the generalized extreme region $C_{{x}}( \theta , \eta )$. More precisely, $C_{{x}}( \theta , \eta )=\{{X}\ \mid \ T({X}; {x}, \theta , \eta )\geqslant T({x}; {x}, \theta , \eta )\}.$ The generalized p-value is defined by

$$\begin{aligned} p({x})=\mathrm {Pr}\{{X}\in C_{{x}}( \theta , \eta )\ \mid \ \theta =\theta _{0}\}. \end{aligned}$$