Two slice-EM algorithms for fitting generalized linear mixed models with binary response

The celebrated simplicity of the EM algorithm is somewhat lost in its common use for generalized linear mixed models (GLMMs) because of its analytically intractable E-step. A natural and typical strategy in practice is to implement the E-step via Monte Carlo by drawing the unobserved random effects from their conditional distribution as specified by the E-step. In this paper, we show that further augmenting the missing data (e.g., the random effects) used by the M-step leads to a quite attractive and general slice sampler for implementing the Monte Carlo E-step. The slice sampler scheme is straightforward to implement, and it is neither restricted to the particular choice of the link function (e.g., probit) nor to the distribution of the random effects (e.g., normal). We apply this scheme to the standard EM algorithm as well as to an alternative EM algorithm which treats the variance-standardized random effects, rather than the random effects themselves, as missing data. The alternative EM algorithm does not only have faster convergence, but also leads to generalized linear model-like variance estimation, because it converts the random-effect standard deviations into linear regression parameters. Using the well-known salamander mating problem, we compare these two algorithms with each other, as well as with a variety of methods given in the literature in terms of the resulting point and interval estimates.