One-sided tests in shared frailty models

Tests for the presence of heterogeneity in frailty models use an alternative hypothesis in which the heterogeneity parameter is subject to an inequality constraint. As a result, the classical likelihood ratio asymptotic chi-square distribution theory is no longer valid. Our main result states the limiting distribution of the likelihood ratio and score statistic for the one-sided testing problem. The resulting distribution is a mixture of chi-square distributed random variables. The results are shown for gamma and positive stable frailty distributions, and hold when covariate information is present. A data example illustrates the tests. We also assess, in a simulation study, the performance of the tests regarding the significance level and power.