Gauss-Hermite quadrature approximation for estimation in generalized linear mixed models (Q1424630)

The response \(y_{ij}\) is a result of the \(i\)-th observation in the \(j\)-th subject. It is assumed that given the random effects \(b_i\), \(y_{ij}\) have a distribution from a regular exponential family and \(E(y_{ij}\mid b_i)=h(x_{ij}'\beta+z_{ij}'b_i)\), where \(x_{ij}\) is a vector of covariates, \(\beta\) is the vector of regression coefficients, \(z_{ij}\) is an explanatory vector associated with random effects \(b_i\), and \(h\) is the known inverse link function. The random effects \(b_i\sim N(0,\Sigma)\), \(\Sigma\) being unknown. The authors describe the likelihood function as an integral of \(b_i\) and propose to use the Gauss-Hermite quadrature technique for its evaluation. The covariance of the obtained approximate ML-estimator is investigated. Results of simulations and analysis of biological data are presented.