Vertex Exchange Method for non-parametric estimation of mixing distributions in logistic mixed models

The conventional normality assumption for the random effects distribution in logistic mixed models can be too restrictive in some applications. In our data example of a longitudinal study modelling employment participation of Australian women, the random effects exhibit non-normality due to a potential mover-stayer scenario. In such a scenario, the women observed to remain in the same initial response state over the study period may consist of two subgroups: latent stayers-those with extremely small probability of transitioning response states-and latent movers, those with a probability of transitioning response states. The similarities between estimating the random effects using non-parametric approaches and mover-stayer models have previously been highlighted. We explore non-parametric approaches to model univariate and bivariate random effects in a potential mover-stayer scenario. As there are limited approaches available to fit the non-parametric maximum likelihood estimate for bivariate random effects in logistic mixed models, we implement the Vertex Exchange Method (VEM) to estimate the random effects in logistic mixed models. The approximation of the non-parametric maximum likelihood estimate derived by the VEM algorithm induces more flexibility of the random effects, identifying regions corresponding to potential latent stayers in the non-employment category in our data example.