Negative Binomial Loglinear Mixed Model  

Model descriptionThe negative binomial distribution can be used instead of the Poisson distribution to investigate whether there is overdispersion in the data, that is whether the variance of the observations is greater than that which would be expected for a Poisson distribution. Parameter estimation for such models is generally claimed to be difficult. See for example Rhelp the mailing list archives of the statistical modeling language R. The data used in this example are the epilepsy data considered in Venables and Ripley Modern applied statics with S 4th edition. and by Booth et al. Negative Binomial Loglinear Mixed Models.Implementation in ADMBRE callable from RWe coded up the model in ADMBRE (nbmm.tpl) with flexible linear predictors for both fixed and random effects. The program was then compiled into a DLL that can be called from R via the Rfunction glmm.admb(). Examples of how to use this function are given below. Currently glmm.admb() only allows negative binomial, but implementing other distributions like Bernoulli and Poisson is just a question of adding a few lines of code to nbmm.tpl.Comparison with SAS NLMIXEDBooth et al. attempt to fit two negative binomial loglinear mixed models to the data. They refer to these models as the full model and a simpler model. For the full model they report:The fit of the full negative binomial model using NLMIXED was very unstable. Different starting values led to different estimates and very different standard errors. Booth et al also apply a Monte Carlo EM algorithm (MCEM) to the full model and report: Application of the MCEM algorithm in this problem suggest that the random slope is 0. The MCEM algorithm was run for a large number of iterations with all of the estimates except for slope variance and the covariance converging quickly. These latter two estimates appear to be slowly converging toward 0.
The full model of Booth et al. is specified as: 