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Bayesian inference for generalized additive mixed models based on markov random field priors
 C
, 2001
"... Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in lo ..."
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Cited by 63 (19 self)
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Summary. Most regression problems in practice require ¯exible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a uni®ed approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with ®xed effects, metrical covariates with nonlinear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random ®eld priors with different forms and degrees of smoothness. We applied the approach in several casestudies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.
Inference in Generalized Additive Mixed Models Using Smoothing Splines
, 1999
"... this paper, we propose generalized additive mixed models (GAMMs), which are an additive extension of generalized linear mixed models in the spirit of Hastie and Tibshirani (1990). This new class of models uses additive nonparametric functions to model covariate effects while accounting for overdispe ..."
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Cited by 45 (4 self)
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this paper, we propose generalized additive mixed models (GAMMs), which are an additive extension of generalized linear mixed models in the spirit of Hastie and Tibshirani (1990). This new class of models uses additive nonparametric functions to model covariate effects while accounting for overdispersion and correlation by adding random effects to the additive predictor. GAMMs encompass nested and crossed designs and are applicable to clustered, hierarchical and spatial data. We estimate the nonparametric functions using smoothing splines, and jointly estimate the smoothing parameters and the variance components using marginal quasilikelihood. This marginal quasilikelihood approach is an extension of the restricted maximum likelihood approach used by Wahba (1985) and Kohn, et al. (1991) in the classical nonparametric regression model (Kohn, et al. 1991, eq 2.1), and by Zhang, et al. (1998) in Gaussian nonparametric mixed models, where they treated the smoothing parameter as an extra variance component. In view of numerical integration often required by maximizing the objective functions, double penalized quasilikelihood (DPQL) is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the proposed method is that it allows us to make systematic inference on all model components of GAMMs within a unified parametric mixed model framework. Specifically, our estimation of the nonparametric functions, the smoothing parameters and the variance components in GAMMs can proceed by fitting a working GLMM using existing statistical software, which iteratively fits a linear mixed model to a modified dependent variable. When the data are sparse (e.g., binary), the DPQL estimators of the variance components are found to be subject t...
Geoadditive Models
, 2000
"... this paper is a recent article on modelbased geostatistics by Diggle, Tawn and Moyeed (1998) where pure kriging (i.e. no covariates) is the focus. Our paper inherits some of its aspects: modelbased and with mixed model connections. In particular the comment by Bowman (1998) in the ensuing discussi ..."
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Cited by 33 (1 self)
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this paper is a recent article on modelbased geostatistics by Diggle, Tawn and Moyeed (1998) where pure kriging (i.e. no covariates) is the focus. Our paper inherits some of its aspects: modelbased and with mixed model connections. In particular the comment by Bowman (1998) in the ensuing discussion suggested that additive modelling would be a worthwhile extension. This paper essentially follows this suggestion. However, this paper is not the first to combine the notions of geostatistics and additive modelling. References known to us are Kelsall and Diggle (1998), Durban Reguera (1998) and Durban, Hackett, Currie and Newton (2000). Nevertheless, we believe that our approach has a number of attractive features (see (1)(4) above), not all shared by these references. Section 2 describes the motivating application and data in detail. Section 3 shows how one can express additive models as a mixed model, while Section 4 does the same for kriging and merges the two into the geoadditive model. Issues concerning the amount of smoothing are discussed in Section 5 and inferential aspects are treated in Section 6. Our analysis of the Upper Cape Cod reproductive data is presented in Section 7. Section 8 discusses extension to the generalised context.We close the paper with some disussion in Section 9. 2 Description of the application and data
A survey of Monte Carlo algorithms for maximizing the likelihood of a twostage hierarchical model
, 2001
"... Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternati ..."
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Cited by 10 (4 self)
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Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo NewtonRaphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo NewtonRaphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS0072827. 1 1
Fitting Nonlinear Mixed Models with the New NLMIXED Procedure
"... Statistical models in which both fixed and random effects enter nonlinearly are becoming increasingly popular. These models have a wide variety of applications, two of the most common being nonlinear growth curves and overdispersed binomial data. A new SAS/STAT procedure, NLMIXED, fits these models ..."
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Cited by 8 (0 self)
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Statistical models in which both fixed and random effects enter nonlinearly are becoming increasingly popular. These models have a wide variety of applications, two of the most common being nonlinear growth curves and overdispersed binomial data. A new SAS/STAT procedure, NLMIXED, fits these models using likelihoodbased methods. This paper presents some of the primary features of PROC NLMIXED and illustrates its use with two examples. INTRODUCTION The NLMIXED procedure fits nonlinear mixed models, that is, models in which both fixed and random effects are permitted to have a nonlinear relationship to the response variable. These models can take various forms, but the most common ones involve a conditional distribution for the response variable given the random effects. PROC NLMIXED enables you to specify such a distribution by using either a keyword for a standard form (normal, binomial, Poisson) or SAS programming statements to specify a general distribution. PROC NLMIXED fits the ...
General design Bayesian generalized linear mixed models.” Statistical Science, 21(1), 35–51. Hadfield 17 A. Appendix A.1. Updating the latent variables l The conditional density of l is given by: P r(liy, θ, R, G) ∝ fi(yili)fN(eiriR −1 /i e /i, ri − ri
 P r(ljy, θ, R, G) ∝ ∏ pi(yili)fN(ej0, Rj
, 2006
"... Abstract. Linear mixed models are able to handle an extraordinary range of complications in regressiontype analyses. Their most common use is to account for withinsubject correlation in longitudinal data analysis. They are also the standard vehicle for smoothing spatial count data. However, when t ..."
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Cited by 6 (0 self)
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Abstract. Linear mixed models are able to handle an extraordinary range of complications in regressiontype analyses. Their most common use is to account for withinsubject 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 splinetype 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 SPLUS 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