Empirical likelihood based goodness-of-fit testing for generalized linear mixed models

In this paper, we propose a bias-corrected empirical likelihood (BCEL) ratio to construct a goodness-of-fit test for generalized linear mixed models. BCEL test maintains the advantage of empirical likelihood that is self scale invariant and then does not involve estimating limiting variance of the test statistic to avoid deteriorating power of test. Furthermore, the bias correction makes the limit to be a process in which every variable is standard chi-squared. This simple structure of the process enables us to construct a Monte Carlo test procedure to approximate the null distribution. Thus, it overcomes a problem we encounter when classical empirical likelihood test is used, as it is asymptotically a functional of Gaussian process plus a normal shift function. The complicated covariance function makes it difficult to employ any approximation for the null distribution. The test is omnibus and power study shows that the test can detect local alternatives approaching the null at parametric rate. Simulations are carried out for illustration and for a comparison with existing method.