Rejoinder to "Comments about Joint Modeling of Cluster Size and Binary and Continuous Subunit‐Specific Outcomes"
Zhen Chen, David B. Dunson · Biometrics · 2005
We wish to thank Dr. Gueorguieva for her attention to our work and for suggesting a SAS procedure to obtain maximum likelihood estimates of our joint model of cluster size and subunit-specific outcomes. For researchers who prefer a frequentist approach and for those who rely on existing software, her procedure provides a practical alternative to our fully Bayesian approach. In implementing a Bayesian approach, the suggested procedure can serve as a useful tool for obtaining reasonable starting values for the MCMC algorithm and for model building and program debugging. Conceptually, the suggested procedure can be used to obtain maximum likelihood estimates for most of the models in our framework. However, it is important to note that the estimation procedure used by SAS PROC NLMIXED may experience problems with stability and convergence when there are more than a few latent variables in the model. Dr. Gueorguieva obtained reasonable results for a variety of models in the developmental toxicity example. However, in other cases, the algorithm may not reliably converge to true maximum likelihood estimates, and, to our knowledge, the general performance remains to be evaluated. In addition, it seems questionable to rely on asymptotically justified frequentist inferences when sample sizes are small to moderate, which is clearly the case in developmental toxicity studies. In latent variable analyses of this type, it is not at all clear how large the sample size needs to be, both in terms of numbers of clusters and subunits within clusters, before asymptotic approximations are justifiable. The Bayesian approach of placing prior distributions on unknown parameters to stabilize estimation and place high probability on plausible ranges of the parameters would seem to have major advantages. It is typically the case in the toxicological applications, which motivated our approach, that abundant prior information is available from historical studies of the same design.