Some properties of Likelihood Ratio Tests in Linear Mixed Models

We calculate the finite sample probability mass-at-zero and the probability of underestimating the true ratio between random effects variance and error variance in a LMM with one variance component. The calculations are expedited by simple matrix diagonalization techniques. One possible application is to compute the probability that the log of the likelihood ratio (LRT), or residual likelihood ratio (RLRT), is zero. The large sample chi-square mixture approximation to the distribution of the log-likelihood ratio, using the usual asymptotic theory for when a parameter is on the boundary, has been shown to be poor in simulations studies. A large part of the problem is that the finite-sample probability that the LRT or RLRT statistic is zero is larger than 0.5, its value under the chi-square mixture approximation. Our calculations explain these empirical results. Another application is to show why standard asymptotic results can fail even when the parameter under the null is in the interior of the parameter space. This paper focuses on LMMs with one variance component because we have developed a very rapid algorithm for simulating finite-sample distributions of the LRT and RLRT statistics for this case. This allows us to compare finite-sample distributions with asymptotic approximations. The main result is the asymptotic approximation are often poor, and this results suggests that asymptotics be used with caution, or avoided altogether, for any LMM regardless of whether it has one variance component or more. For computing the distribution of the test statistics we recommend our algorithm for the case of one variance component and the bootstrap in other cases. Short title: Properties of (R)LRT