On testing variance components using permutation tests

Testing zero variance components is challenging under linear mixed models. Permutation tests are successfully used to approximate the distribution of test statistics when the usual asymptotic chi-square approximations fail. We propose a permutation test to approximate the distribution of a new test statistic, which represents a generalization of the intra-cluster correlation in regression models with one variance component. Generally, the test statistic is formulated as the ratio between the trace of the covariance matrix measuring the between-cluster variability and the trace of the model covariance matrix. Significantly positive values of the test statistic provide signs for departure from the null hypothesis of zero variance components. Extensive simulation studies show that the proposed test outperforms the existing permutation test under various forms of violations of the standard distributional assumptions on the error components. A real data example is provided.