Gauss-Hermite Cubature Method to estimate parameters of a multivariate GLMM

Abstract

In this paper, the multivariate generalized linear mixed model (GLMM) was studied when there are three response variables, distributed as Normal, Bernoulli, and Poisson. And because there is a multiple integration in the likelihood function for the model under study, it is necessary to use mathematical methods to solve this integration, and because it is not possible to obtain the result of this integration by the well-known methods of integration, numerical methods have been used, the Gauss-Hermite Cubature (GHC) algorithm, which is one of the most common numerical integration methods. Then the estimates were obtained by maximizing the resulting likelihood function with respect to the parameters, and thus, estimates were obtained for the parameters. On the practical side, we have used real data representing the effect of potassium as a fixed effect, and referring patients were considered as a random effect on three response variables: calcium, creatinine, and urea as they follow Normal, Bernoulli, and Poisson distributions taken from the records of the Vajin Hospital in Dohuk city. The regression coefficients showed that the effect of potassium is positive on both calcium and urea because the values of the coefficients are positive, while its effect is negative on the creatinine because the value of the coefficient is negative. Based on the results, the researchers recommended several recommendations, including estimating the standard error of the estimates in their light, in order to construct hypotheses for a significance test about regression coefficients.