The Extension of REML Algorithm for Hierarchical Generalized Linear Models

The restricted maximum likelihood procedure is useful for inferences about variance components in mixed linear models. However, its extension to hierarchical generalized linear models has encountered some difficulties. Numerical integration such as Gauss-Hermite quadrature is generally not recommended when the dimensionality of the integral is high. Approximate methods such as penalized quasi-likelihood estimators may have severe biases when analysing binary data. In this paper we introduce the hierarchical likelihood (or h-likelihood) algorithm which resolves these difficulties. Numerical studies show how the proposed method overcomes them. We also discuss how the restricted maximum likelihood estimating equations for mixed linear models can be modified in more general models.