Modelling spatial disease rates using maximum likelihood

Abstract

This paper concerns maximum likelihood estimation for a generalized linear mixed model (GLMM) useful for modelling spatial disease rates. The model allows for log-linear covariate adjustment and local smoothing of rates through estimation of spatially correlated random effects. The covariance structure of the random effects is based on a recently proposed model which parameterizes spatial dependence through the inverse covariance matrix. A Markov chain Monte Carlo algorithm for performing maximum likelihood estimation for this model is described. Results of a computer simulation study that compared maximum likelihood (ML) and penalized quasi-likelihood (PQL) estimators are presented. Compared with PQL, ML produced less biased estimates of the intercept but the ML estimates were slightly more variable. Estimates of the other regression coefficients were unbiased and nearly identical for the two methods. ML estimators of the random effects standard deviation and spatial correlation were more biased than the corresponding PQL estimators. The conclusion is that ML estimators for GLMMs cannot be expected to perform better than PQL for small samples.