Gradient test under non-parametric random effects models

Abstract

The gradient test proposed by Terrell (2002) is an alternative to the likelihood ratio, Wald and Rao tests. The gradient statistic is the result of the inner product of two vectors — the gradient of the likelihood under null hypothesis (hence the name) and the result of the difference between the estimate under alternative hypothesis and the estimate under null hypothesis. Therefore the gradient statistic is computationally less expensive than Wald and Rao statistics as it does not require matrix operations in its formula. Under some regularity conditions, the gradient statistic has χ2 distribution under null hypothesis. The generalised linear model (GLM) introduced by Nelder & Wedderburn (1972) is one of the most important classes of statistical models. It incorporates the classical regression modelling and analysis of variance either for continuous response and categorical response variables under the exponential family. The random effects model extends the standard GLM for situations where the model does not describe appropriately the variability in the data (overdispersion) (Aitkin, 1996a). We propose a new unified notation for GLM with random effects and the gradient statistic formula for testing fixed effects parameters on these models. We also develop the Fisher information formulae used to obtain the Rao and Wald statistics. Our main interest in this thesis is to investigate the finite sample performance of the gradient test on generalised linear models with random effects. For this we propose and extensive simulation experiment to study the type I error and the local power of the gradient test using the methodology developed by Peers (1971) and Hayakawa (1975). We also compare the local power of the test with the local power of the tests of the likelihood ratio, of Wald and Rao tests.