Overdispersion and quasi distributions

Dear All,

I am new to R (coming from GenStat) and to this discussion list. In postings in the third quarter of 2008 there were some queries about GLMMs for overdispersed data and problems with LME4 (see e.g. https://stat.ethz.ch/pipermail/r-sig-mixed-models/2008q3/001430.html and https://stat.ethz.ch/pipermail/r-sig-mixed-models/2008q3/001404.html). In the second posting Douglas Bates writes that "If someone can suggest what the formula for the scale factor should be, I ...". 

I am having trouble understanding the GLMM algorithm in lme4, but most estimation methods, among which Breslow and Clayton (1993), use a first order Laplace approximation of the likelihood. The iterative algorithm employed, is based on a linear mixed model for an adjusted dependent variable which is updated in each iteration step. The linear mixed model is fitted with REML resulting in estimates for the fixed effects and predictions for the random effects. Then a new adjusted dependent variable (using the fitted values and the predictions) is calculated and the procedure is repeated. This algorithm involves additional weights that are associated with an added residual variance component (fixed at value 1). The over-dispersed (or quasi-likelihood) version of this estimation method comprises an unknown residual variance, i.e. not fixed at value 1, which is estimated by REML. This unknown residual variance component is the scale or overdispersion factor. In the GLMM routine in GenStat the difference in the specification of the random model is (e.g. for a simple block design, "residual" is a factor with a separate level for each observation and "value" is the estimate from the previous iteration)
   without overdispersion:  random=block+residual ; initialvalue=value,1 ; constraint=none,fixed
   with overdispersion:  random=block+residual ; initialvalue=value,value ; constraint=none,none

Note that in an ordinary GLM SEs are multiplied by the square root of the overdispersion factor, and that the estimates of the fixed effects themselves do not change. This is because the overdispersion factor is estimated from Pearson's chi-squared or the deviance after the GLM model has been fitted. However in a GLMM this might work differently as the overdispersion factor can and should be estimated in every iteration possibly leading to different estimates for both variance components and fixed effects. When there is considerable overdispersion there will in general be a difference between the estimated variance components for the model without and the model with overdispersion. This is because in the model without overdispersion the extra variation at the residual stratum will partly be absorbed by the random effects in the linear predictor. 

As a final remark note that in the algorithm above there is already a residual random effect in the estimation method, so there is no room for a residual random effect in the linear predictor. This can also be viewed in the following way: the algorithm employs predictions for the random effects. A residual random effect does not have replications and therefore predictions for such a random effect are notoriously unstable. Note that this a property (or one might say a defect) of the estimation method. That is why the documentation of the GLMM routine in GenStat states that "The random model must exclude the bottom stratum" Maximum likelihood does not have this restriction, but ML requires that all random effects are integrated out and this is generally not feasible. 

Best regards,

Paul Goedhart