Diagnosing misspecification of the random-effects distribution in mixed models.

It is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution, making likelihood‐based inferences more feasible theoretically and computationally. However, this assumption does not necessarily hold in practice which may lead to biased and unreliable results. We introduce a novel diagnostic test based on the so‐called gradient function proposed by Verbeke and Molenberghs (2013) to assess the random‐effects distribution. We establish asymptotic properties of our test and show that, under a correctly specified model, the proposed test statistic converges to a weighted sum of independent chi‐squared random variables each with one degree of freedom. The weights, which are eigenvalues of a square matrix, can be easily calculated. We also develop a parametric bootstrap algorithm for small samples. Our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models, including linear mixed models, generalized linear mixed models, and non‐linear mixed models, with univariate as well as multivariate random effects. Both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis