A CONDITIONAL LIKELIHOOD APPROACH TO RESIDUAL MAXIMUM-LIKELIHOOD-ESTIMATION IN GENERALIZED LINEAR-MODELS

Residual maximum likelihood (REML) estimation is often preferred to ma ximum likelihood estimation as a method of estimating covariance param eters in linear models because it takes account of the loss of degrees of freedom in estimating the mean and produces unbiased estimating eq uations for the variance parameters. In this paper it is shown that RE ML has an exact conditional likelihood interpretation, where the condi tioning is on an appropriate sufficient statistic to remove dependence on the nuisance parameters. This interpretation clarifies the motivat ion for REML and generalizes directly to non-normal models in which th ere is a low dimensional sufficient statistic for the fitted values. T he conditional likelihood is shown to be well defined and to satisfy t he properties of a likelihood function, even though this is not genera lly true when conditioning on statistics which depend on parameters of interest. Using the conditional likelihood representation, the concep t of REML is extended to generalized linear models with varying disper sion and canonical link. Explicit calculation of the conditional likel ihood is given for the one-way lay-out, A saddlepoint approximation fo r the conditional likelihood is also derived.