The gradient function as an exploratory goodness-of-fit assessment of the random-effects distribution in mixed models

Inference in mixed models is often based on the marginal distribution obtained from integrating out random effects over a pre-specified, often parametric, distribution. In this paper, we present the so-called gradient function as a simple graphical exploratory diagnostic tool to assess whether the assumed random-effects distribution produces an adequate fit to the data, in terms of marginal likelihood. The method does not require any calculations in addition to the computations needed to fit the model, and can be applied to a wide range of mixed models (linear, generalized linear, non-linear), with univariate as well as multivariate random effects, as long as the distribution for the outcomes conditional on the random effects is correctly specified. In case of model misspecification, the gradient function gives an important, albeit informal, indication on how the model can be improved in terms of random-effects distribution. The diagnostic value of the gradient function is extensively illustrated using some simulated examples, as well as in the analysis of a real longitudinal study with binary outcome values.