Nonlinear quantile mixed models

In regression applications, the presence of nonlinearity and correlation among observations offer computational challenges not only in traditional settings such as least squares regression, but also (and especially) when the objective function is non-smooth as in the case of quantile regression. In this paper, we develop methods for the modeling and estimation of nonlinear conditional quantile functions when data are clustered within two-level nested designs. This work represents an extension of the linear quantile mixed models of Geraci and Bottai (2014, Statistics and Computing). We develop a novel algorithm which is a blend of a smoothing algorithm for quantile regression and a second order Laplacian approximation for nonlinear mixed models. To assess the proposed methods, we present a simulation study and two applications, one in pharmacokinetics and one related to growth curve modeling in agriculture.Comment: 26 pages, 8 figures, 8 table

Penalized additive regression for space-time data: a Bayesian perspective

We propose extensions of penalized spline generalized additive models for analysing space-time regression data and study them from a Bayesian perspective. Non-linear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spatial effects follow a Markov random field prior. This allows to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference can be performed either with full (FB) or empirical Bayes (EB) posterior analysis. FB inference using MCMC techniques is a slight extension of own previous work. For EB inference, a computationally efficient solution is developed on the basis of a generalized linear mixed model representation. The second approach can be viewed as posterior mode estimation and is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to smoothing parameters, are then estimated by using marginal likelihood. We carefully compare both inferential procedures in simulation studies and illustrate them through real data applications. The methodology is available in the open domain statistical package BayesX and as an S-plus/R function

General Design Bayesian Generalized Linear Mixed Models

Linear mixed models are able to handle an extraordinary range of complications in regression-type analyses. Their most common use is to account for within-subject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when treated in full generality, mixed models can also handle spline-type smoothing and closely approximate kriging. This allows for nonparametric regression models (e.g., additive models and varying coefficient models) to be handled within the mixed model framework. The key is to allow the random effects design matrix to have general structure; hence our label general design. For continuous response data, particularly when Gaussianity of the response is reasonably assumed, computation is now quite mature and supported by the R, SAS and S-PLUS packages. Such is not the case for binary and count responses, where generalized linear mixed models (GLMMs) are required, but are hindered by the presence of intractable multivariate integrals. Software known to us supports special cases of the GLMM (e.g., PROC NLMIXED in SAS or glmmML in R) or relies on the sometimes crude Laplace-type approximation of integrals (e.g., the SAS macro glimmix or glmmPQL in R). This paper describes the fitting of general design generalized linear mixed models. A Bayesian approach is taken and Markov chain Monte Carlo (MCMC) is used for estimation and inference. In this generalized setting, MCMC requires sampling from nonstandard distributions. In this article, we demonstrate that the MCMC package WinBUGS facilitates sound fitting of general design Bayesian generalized linear mixed models in practice.Comment: Published at http://dx.doi.org/10.1214/088342306000000015 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org