Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional prob...
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Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional probability distribution. The Gibbs sampler, however, is Often justifiably criticized for its sometimes slow convergence, especially when it is used to fit highly structured complex models. The recently proposed Partially Collapsed Gibbs (PCG) sampler offers a new strategy for improving the convergence characteristics of a Gibbs sampler. A PCG sampler achieves faster convergence by reducing the conditioning in some or all of the component draws of its parent Gibbs sampler. Although this strategy can significantly improve convergence, it must be implemented with care to be Sure that the desired stationary distribution is preserved. In some cases the set of conditional distributions sampled in a PCG sampler may be functionally incompatible and permuting the order of draws can change the stationary distribution of the chain. In this article. we draw in analogy between the PCG sampler and certain efficient EM-type algorithms that helps to explain the computational advantage of PCG samplers and to suggest when they might be used in practice. We go on to illustrate the PCG samplers in three substantial examples drawn front our applied work: a multilevel spectral model commonly used in high-energy astrophysics. a piecewise-constant multivariate time series model, and a joint imputation model for nonnested data. These are all useful highly structured models that involve computational challenge,, that can be solved using PCG samplers. The examples illustrate [lot only the computation advantage of PCG samplers but also how they should be constructed to maintain the desired stationary distribution. Supplemental materials for the examples given in this article are available online.
In modeling disease transmission there is much emphasis on how many people are infected but less attention is paid to when the infections occur, and the unrealistic assumption of constant infectiousness is often made....
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In modeling disease transmission there is much emphasis on how many people are infected but less attention is paid to when the infections occur, and the unrealistic assumption of constant infectiousness is often made. We propose a missing data formulation that enables us to apply the ecme algorithm to estimate a discretized intensity function of an inhomogeneous Poisson process. This approach requires interval-censored data only, but known infection times can be incorporated as well. We apply the proposed method to transmission data on severe respiratory syndrome (SARS) collected in Singapore and Hong Kong. The resulting estimates show that the rate of infection as a function of time may have more than one peak. By fitting a two-environment proportional intensity model to the Singapore data, we estimate that the rate of infection in an (unisolated) hospital environment is almost ten times greater than occurs in a nonhospital environment. This lends support to the theory that the SARS epidemic in Singapore was mainly driven by hospital-acquired infections. Estimates of individual infectivity reveal that three persons commonly regarded as "superspreaders" actually do not have unusually high individual infectiousness. The observed superspreading events seem to have been caused by environmental rather than biological factors.
This paper extends the classical linear mixed model by considering a multivariate skew-normal assumption for the distribution of random effects. We present an efficient hybrid ecme-NR algorithm for the computation of ...
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This paper extends the classical linear mixed model by considering a multivariate skew-normal assumption for the distribution of random effects. We present an efficient hybrid ecme-NR algorithm for the computation of maximum-likelihood estimates of parameters. A score test statistic for testing the existence of skewness preference among random effects is developed. The technique for the prediction of future responses under this model is also investigated. The methodology is illustrated through an application to Framingham cholesterol data and a simulation study. Copyright (C) 2007 John Wiley & Sons, Ltd.
Linear mixed-effects models are an important class of statistical models that are used directly in many fields of applications and also are used as iterative steps in fitting other types of mixed-effects models, such ...
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Linear mixed-effects models are an important class of statistical models that are used directly in many fields of applications and also are used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood or restricted maximum likelihood. In general, there is no closed-form solution for these estimates and they must be determined by iterative algorithms such as EM iterations or general nonlinear optimization. Many of the intermediate calculations for such iterations have been expressed as generalized least squares problems. We show that an alternative representation as a penalized least squares problem has many advantageous computational properties including the ability to evaluate explicitly a profiled log-likelihood or log-restricted likelihood, the gradient and Hessian of this profiled objective, and an ecme update to refine this objective. (C) 2004 Elsevier Inc. All rights reserved.
This paper deals with the matrix rate of convergence of the ecme algorithm, a simple extention of EM and ECM algorithms proposed recently by Liu and Rubin. We establish a general formula for the matrix rate of converg...
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This paper deals with the matrix rate of convergence of the ecme algorithm, a simple extention of EM and ECM algorithms proposed recently by Liu and Rubin. We establish a general formula for the matrix rate of convergence of ecme which is a generalization of the result of Liu and Rubin. (C) 1998 Elsevier Science B.V.
In recent years numerous advances in EM methodology have led to algorithms which can be very efficient when compared with both their EM predecessors and other numerical methods (e.g., algorithms based on Newton-Raphso...
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In recent years numerous advances in EM methodology have led to algorithms which can be very efficient when compared with both their EM predecessors and other numerical methods (e.g., algorithms based on Newton-Raphson). This article combines several of these new methods to develop a set of mode-finding algorithms for the popular mixed-effects model which are both fast and more reliable than such standard algorithms as proc mixed in SAS. We present efficient algorithms for maximum likelihood (ML), restricted maximum likelihood (REML), and computing posterior modes with conjugate proper and improper priors. These algorithms are not only useful in their own right, but also illustrate how parameter expansion, conditional data augmentation, and the ecme algorithm can be used in conjunction to form efficient algorithms. In particular, we illustrate a difficulty in using the typically very efficient PXEM (parameter-expanded EM) for posterior calculations, but show how algorithms based on conditional data augmentation can be used. Finally, we present a result that extends Hobert and Casella's result on the propriety of the posterior for the mixed-effects model under an improper prior, an important concern in Bayesian analysis involving these models that when not properly understood has lead to difficulties in several applications.
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