Summary. We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h-likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.

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Il Do HA, Jianxin PAN, Seungyoung OH, and Youngjo LEE Variable selection methods using a penalized likelihood have been widely studied in various statistical models. However, in semiparametric frailty models, these methods have been relatively less studied because the marginal likelihood function involves analytically intractable integrals, particularly when modeling multicomponent or correlated frailties. In this article, we propose a simple but unified procedure via a penalized h-likelihood (HL) for variable selection of fixed effects in a general class of semiparametric frailty models, in which random effects may be shared, nested, or correlated. We consider three penalty functions (least absolute shrinkage and selection operator [LASSO], smoothly clipped absolute deviation [SCAD], and HL) in our variable selection procedure. We show that the proposed method can be easily implemented via a slight modification to existing HL estimation approaches. Simulation studies also show that the procedure using the SCAD or HL penalty performs well. The usefulness of the new method is illustrated using three practical datasets too. Supplementary materials for the article are available online.

The restricted maximum likelihood (REML) procedure is useful for inferences about variance components in mixed linear models. However, its extension to hierarchical generalized linear models (HGLMs) is often hampered by analytically intractable integrals. Numerical integration such as Gauss-Hermite quadrature (GHQ) is generally not recommended when the dimensionality of the integral is high. With binary data various extensions of the REML method have been suggested, but they have had unsatisfactory biases in estimation. In this paper we propose a statistically and computationally efficient REML procedure for the analysis of binary data, which is applicable over a wide class of models and design structures. We propose a bias-correction method for models such as binary matched pairs and discuss how the REML estimating equations for mixed linear models can be modified to implement more general models.

The possibility of breeding for uniform individuals by selecting animals expressing a small response to environment has been studied extensively in animal breeding. Bayesian methods for fitting models with genetic components in the residual variance have been developed for this purpose, but have limitations due to the computational demands. We use the hierarchical (h)-likelihood from the theory of double hierarchical generalized linear models (DHGLM) to derive an estimation algorithm that is computationally feasible for large datasets. Random effects for both the mean and residual variance parts of the model are estimated together with their variance/covariance components. An important feature of the algorithm is that it can fit a correlation between the random effects for mean and variance. An h-likelihood estimator is implemented in the R software and an iterative reweighted least square (IRWLS) approximation of the h-likelihood is implemented using ASReml. The difference in variance component estimates between the two implementations is investigated, as well as the potential bias of the methods, using simulations. IRWLS gives the same results as h-likelihood in simple cases with no severe indication of bias. For more complex cases, only IRWLS could be used, and bias did appear. The IRWLS is applied on the pig litter size data previously analysed by Sorensen & Waagepetersen (2003) using Bayesian methodology. The estimates we obtained by using IRWLS are similar to theirs, with the estimated correlation between the random genetic effects being -0·52 for IRWLS and -0·62 in Sorensen & Waagepetersen (2003). © Cambridge University Press 2013. Estimation of breeding values for mean and dispersion, their variance and correlation using double hierarchical generalized linear models SummaryThe possibility of breeding for uniform individuals by selecting animals expressing a small response to environment has been studied extensively in animal breeding. Bayesian methods for fitting models with genetic components in the residual variance have been developed for this purpose, but have limitations due to the computational demands. We use the hierarchical (h)-likelihood from the theory of double hierarchical generalized linear models (DHGLM) to derive an estimation algorithm that is computationally feasible for large datasets. Random effects for both the mean and residual variance parts of the model are estimated together with their variance/covariance components. An important feature of the algorithm is that it can fit a correlation between the random effects for mean and variance. An h-likelihood estimator is implemented in the R software and an iterative reweighted least square (IRWLS) approximation of the h-likelihood is implemented using ASReml. The difference in variance component estimates between the two implementations is investigated, as well as the potential bias of the methods, using simulations. IRWLS gives the same results as h-likelihood in simple cases with no severe indication of bias. For more comple...