A new model selection procedure for finite mixture regression models

Abstract

In this article, we propose a new penalized-likelihood method to conduct model selection for finite mixture of regression models. The penalties are imposed on mixing proportions and regression coefficients, and hence order selection of the mixture and the variable selection in each component can be simultaneously conducted. The consistency of order selection and the consistency of variable selection are investigated. A modified EM algorithm is proposed to maximize the penalized log-likelihood function. Numerical simulations are conducted to demonstrate the finite sample performance of the estimation procedure. The proposed methodology is further illustrated via real data analysis.     

Multilevel modelling of complex survey data

Summary.  Multilevel modelling is sometimes used for data from complex surveys involving multistage sampling, unequal sampling probabilities and stratification. We consider generalized linear mixed models and particularly the case of dichotomous responses. A pseudolikelihood approach for accommodating inverse probability weights in multilevel models with an arbitrary number of levels is implemented by using adaptive quadrature. A sandwich estimator is used to obtain standard errors that account for stratification and clustering. When level 1 weights are used that vary between elementary units in clusters, the scaling of the weights becomes important. We point out that not only variance components but also regression coefficients can be severely biased when the response is dichotomous. The pseudolikelihood methodology is applied to complex survey data on reading proficiency from the American sample of the 'Program for international student assessment' 2000 study, using the Stata program gllamm which can estimate a wide range of multilevel and latent variable models. Performance of pseudo-maximum-likelihood with different methods for handling level 1 weights is investigated in a Monte Carlo experiment. Pseudo-maximum-likelihood estimators of (conditional) regression coefficients perform well for large cluster sizes but are biased for small cluster sizes. In contrast, estimators of marginal effects perform well in both situations. We conclude that caution must be exercised in pseudo-maximum-likelihood estimation for small cluster sizes when level 1 weights are used.     

Asymptotic results for fitting marginal hazards models from stratified case-cohort studies with multiple disease outcomes

In stratified case-cohort designs, samplings of case-cohort samples are conducted via a stratified random sampling based on covariate information available on the entire cohort members. In this paper, we extended the work of Kang & Cai (2009) to a generalized stratified case-cohort study design for failure time data with multiple disease outcomes. Under this study design, we developed weighted estimating procedures for model parameters in marginal multiplicative intensity models and for the cumulative baseline hazard function. The asymptotic properties of the estimators are studied using martingales, modern empirical process theory, and results for finite population sampling.     

A practical sampling approach for a Bayesian mixture model with unknown number of components

Recently, mixture distribution becomes more and more popular in many scientific fields. Statistical computation and analysis of mixture models, however, are extremely complex due to the large number of parameters involved. Both EM algorithms for likelihood inference and MCMC procedures for Bayesian analysis have various difficulties in dealing with mixtures with unknown number of components. In this paper, we propose a direct sampling approach to the computation of Bayesian finite mixture models with varying number of components. This approach requires only the knowledge of the density function up to a multiplicative constant. It is easy to implement, numerically efficient and very practical in real applications. A simulation study shows that it performs quite satisfactorily on relatively high dimensional distributions. A well-known genetic data set is used to demonstrate the simplicity of this method and its power for the computation of high dimensional Bayesian mixture models.     

Variance estimation in complex survey sampling for generalized linear models

Summary.  Complex survey sampling is often used to sample a fraction of a large finite population. In general, the survey is conducted so that each unit (e.g. subject) in the sample has a different probability of being selected into the sample. For generalizability of the sample to the population, both the design and the probability of being selected into the sample must be incorporated in the analysis. In this paper we focus on non-standard regression models for complex survey data. In our motivating example, which is based on data from the Medical Expenditure Panel Survey, the outcome variable is the subject's 'total health care expenditures in the year 2002'. Previous analyses of medical cost data suggest that the variance is approximately equal to the mean raised to the power of 1.5, which is a non-standard variance function. Currently, the regression parameters for this model cannot be easily estimated in standard statistical software packages. We propose a simple two-step method to obtain consistent regression parameter and variance estimates; the method proposed can be implemented within any standard sample survey package. The approach is applicable to complex sample surveys with any number of stages.     

Bayesian weighted inference from surveys

Data from large surveys are often supplemented with sampling weights that are designed to reflect unequal probabilities of response and selection inherent in complex survey sampling methods. We propose two methods for Bayesian estimation of parametric models in a setting where the survey data and the weights are available, but where information on how the weights were constructed is unavailable. The first approach is to simply replace the likelihood with the pseudo likelihood in the formulation of Bayes theorem. This is proven to lead to a consistent estimator but also leads to credible intervals that suffer from systematic undercoverage. Our second approach involves using the weights to generate a representative sample which is integrated into a Markov chain Monte Carlo (MCMC) or other simulation algorithms designed to estimate the parameters of the model. In the extensive simulation studies, the latter methodology is shown to achieve performance comparable to the standard frequentist solution of pseudo maximum likelihood, with the added advantage of being applicable to models that require inference via MCMC. The methodology is demonstrated further by fitting a mixture of gamma densities to a sample of Australian household income.     

Variable selection in finite mixture of regression models using the skew-normal distribution

Variable selection in finite mixture of regression (FMR) models is frequently used in statistical modeling. The majority of applications of variable selection in FMR models use a normal distribution for regression error. Such assumptions are unsuitable for a set of data containing a group or groups of observations with asymmetric behavior. In this paper, we introduce a variable selection procedure for FMR models using the skew-normal distribution. With appropriate choice of the tuning parameters, we establish the theoretical properties of our procedure, including consistency in variable selection and the oracle property in estimation. To estimate the parameters of the model, a modified EM algorithm for numerical computations is developed. The methodology is illustrated through numerical experiments and a real data example.     

Discrete-time survival analysis under ranked set sampling: an application to Turkish motor insurance data

Survival models with continuous-time data are still superior methods of survival analysis. However when the survival data is discrete, taking it as continuous leads the researchers to incorrect results and interpretations. The discrete-time survival model has some advantages in applications such as it can be used for non-proportional hazards, time-varying covariates and tied observations. However, it has a disadvantage about the reconstruction of the survival data and working with big data sets. Actuaries are often rely on complex and big data whereas they have to be quick and efficient for short period analysis. Using the mass always creates inefficient processes and consumes time. Therefore sampling design becomes more and more important in order to get reliable results. In this study, we take into account sampling methods in discrete-time survival model using a real data set on motor insurance. To see the efficiency of the proposed methodology we conducted a simulation study.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Maximum likelihood methods for complex sample data: logistic regression and discrete proportional hazards models

To estimate model parameters from complex sample data. we apply maximum likelihood techniques to the complex sample data from the finite population, which is treated as a sample from an i nfinite superpopulation. General asymptotic distribution theory is developed and then applied to both logistic regression and discrete proportional hazards models. Data from the Lipid Research Clinics Program areused to illustrate each model, demonstrating the effects on inference of neglecting the sampling design during parameter estimation. These empirical results also shed light on the issue of model-based vs. design-based inferences.     

A Thresholding Algorithm for Order Selection in Finite Mixture Models

Order selection is an important step in the application of finite mixture models. Classical methods such as AIC and BIC discourage complex models with a penalty directly proportional to the number of mixing components. In contrast, Chen and Khalili propose to link the penalty to two types of overfitting. In particular, they introduce a regularization penalty to merge similar subpopulations in a mixture model, where the shrinkage idea of regularized regression is seamlessly employed. However, the new method requires an effective and efficient algorithm. When the popular expectation-maximization (EM)-algorithm is used, we need to maximize a nonsmooth and nonconcave objective function in the M-step, which is computationally challenging. In this article, we show that such an objective function can be transformed into a sum of univariate auxiliary functions. We then design an iterative thresholding descent algorithm (ITD) to efficiently solve the associated optimization problem. Unlike many existing numerical approaches, the new algorithm leads to sparse solutions and thereby avoids undesirable ad hoc steps. We establish the convergence of the ITD and further assess its empirical performance using both simulations and real data examples.     

Investigating impacts of complex sampling on latent growth curve modelling

We investigate the impacts of complex sampling on point and standard error estimates in latent growth curve modelling of survey data. Methodological issues are illustrated with empirical evidence from the analysis of longitudinal data on life satisfaction trajectories using data from the British Household Panel Survey, a national representative survey in Great Britain. A multi-process second-order latent growth curve model with conditional linear growth is used to study variation in the two perceived life satisfaction latent factors considered. The benefits of accounting for the complex survey design are considered, including obtaining unbiased both point and standard error estimates, and therefore correctly specified confidence intervals and statistical tests. We conclude that, even for the rather elaborated longitudinal data models that were considered, estimation procedures are affected by variance-inflating impacts of complex sampling.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

使用普查数据模拟MPPS抽样方法的研究

MPPS抽样即多变量与规模成比例的概率抽样,是20世纪90年代才提出来的一种抽样设计。近年来,中国有关部门与美国农业部国家农业署合作,进行了MPPS抽样设计的试点,来解决多目标调查问题。但是MPPS抽样在中国的应用非常有限。对MPPS抽样进行简单的回顾,介绍了它的基本估计,并对其应用进行了数据模拟研究。模拟中采用了系统抽样和泊松抽样的方法,根据实际调查数据得到了明确的结果。还对泊松抽样的一种变形永久随机数抽样的方法进行了模拟研究,并对它的一种误用情况进行了模拟比较,得到了具有说服力的结果。     

A simple root selection method for univariate finite normal mixture models

It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.     

Finite Mixture of Generalized Semiparametric Models: Variable Selection via Penalized Estimation

Selection of the important variables is one of the most important model selection problems in statistical applications. In this article, we address variable selection in finite mixture of generalized semiparametric models. To overcome computational burden, we introduce a class of variable selection procedures for finite mixture of generalized semiparametric models using penalized approach for variable selection. Estimation of nonparametric component will be done via multivariate kernel regression. It is shown that the new method is consistent for variable selection and the performance of proposed method will be assessed via simulation.     

Multilevel modelling of survey data: impact of the two-level weights used in the pseudolikelihood

Approaches that use the pseudolikelihood to perform multilevel modelling on survey data have been presented in the literature. To avoid biased estimates due to unequal selection probabilities, conditional weights can be introduced at each level. Less-biased estimators can also be obtained in a two-level linear model if the level-1 weights are scaled. In this paper, we studied several level-2 weights that can be introduced into the pseudolikelihood when the sampling design and the hierarchical structure of the multilevel model do not match. Two-level and three-level models were studied. The present work was motivated by a study that aims to estimate the contributions of lead sources to polluting the interior floor dust of the rooms within dwellings. We performed a simulation study using the real data collected from a French survey to achieve our objective. We conclude that it is preferable to use unweighted analyses or, at the most, to use conditional level-2 weights in a two-level or a three-level model. We state some warnings and make some recommendations.     

Bayesian Semiparametric Regression for Median Residual Life

Abstract.  With survival data there is often interest not only in the survival time distribution but also in the residual survival time distribution. In fact, regression models to explain residual survival time might be desired. Building upon recent work of Kottas & Gelfand [ J. Amer. Statist. Assoc. 96 (2001) 1458], we formulate a semiparametric median residual life regression model induced by a semiparametric accelerated failure time regression model. We utilize a Bayesian approach which allows full and exact inference. Classical work essentially ignores covariates and is either based upon parametric assumptions or is limited to asymptotic inference in non-parametric settings. No regression modelling of median residual life appears to exist. The Bayesian modelling is developed through Dirichlet process mixing. The models are fitted using Gibbs sampling. Residual life inference is implemented extending the approach of Gelfand & Kottas [ J. Comput. Graph. Statist. 11 (2002) 289]. Finally, we present a fairly detailed analysis of a set of survival times with moderate censoring for patients with small cell lung cancer.