On the effect of the number of quadrature points in a logistic random-effects model: an example

Although generalized linear mixed models are recognized to be of major practical importance, it is also known that they can be computationally demanding. The problem is the evaluation of the integral in calculating the marginalized likelihood. The straightforward method is based on the Gauss-Hermite technique, based on Gaussian quadrature points. Another approach is provided by the class of penalized quasi-likelihood methods. It is commonly believed that the Gauss-Hermite method works relatively well in simple situations but fails in more complicated structures. However, we present here a strikingly simple example of a logistic random-intercepts model in the context of a longitudinal clinical trial where the method gives valid results only for a high number of quadrature points (Q). As a consequence, this result warns the practitioner to examine routinely the dependence of the results on Q. The adaptive Gaussian quadrature, as implemented in the new SAS procedure NLMIXED, offered the solution to our problem. However, even the adaptive version of Gaussian quadrature needs careful handling to ensure convergence.