Testing Variance Components in Linear Mixed Modeling Using Permutation

Linear mixed modeling (LMM) is widely used to deal with repeated measures, clustered subjects, or both in practice. In LMM, inference of variance components provides evidence of heterogeneity between individuals or clusters. When only nonnegative variances are allowed (constrained estimation), there is a boundary (i.e., 0) in the variances' parameter space. With the boundary issue, regular statistical procedures for inferring such a parameter could be problematic. This dissertation aims to introduce a practically feasible permutation method to make inferences about variance components while considering the boundary issue in linear mixed modeling. The permutation tests with different settings (i.e., constrained estimation vs. unconstrained estimation, specific test vs. generalized test, different ways of calculating p -values, and different ways of permutation) were examined with both normal data and nonnormal data. In addition, the permutation tests were compared with the likelihood ratio test with mixtures of chi-squared distributions as reference distributions. In testing a subset of the variance components and testing all the variance component, the permutation tests and the likelihood ratio tests have their specific strengths and limitation respectively in different scenarios in terms of Type I error rates, statistical power, and availability of the methods. An example about the development of verbal IQ and performance IQ for 204 children based on the Wechesler Intelligence Scale for Children is used to illustrate the application of the permutation tests.