A conditional likelihood approach to REML in Generalized Linear Models

Residual maximum likelihood estimation (REML) is often preferred to maximum likelihood estimation as a method of estimating covariance parameters in linear models because it takes account of the loss of degrees of freedom in estimating the mean and produces unbiased estimating equations for the variance parameters. In this note it is shown that REML has an exact conditional likelihood interpretation, where the conditioning is on an appropriate suucient statistic to remove dependence on the nuisance parameters. This interpretation clariies the motivation for REML and generalizes directly to non-normal models in which there exists a low dimensional suucient statistic for the tted values. The conditional likelihood is shown to be well deened and to satisfy the properties of a likelihood function, even though this is not generally true when conditioning on statistics which depend on parameters of interest. Using the conditional likelihood representation, the concept of REML is extended to generalized linear models with varying dispersion and canonical link. Explicit calculation of the conditional likelihood is given for the oneway layout. A saddle-point approximation for the conditional likelihood is also derived.