Finite mixture distributions, sequential likelihood and the EM algorithm

A popular way to account for unobserved heterogeneity is to assume that the data are drawn from a finite mixture distribution. A barrier to using finite mixture models is that parameters that could previously be estimated in stages must now be estimated jointly: using mixture distributions destroys any additive separability of the log-likelihood function. We show, however, that an extension of the EM algorithm reintroduces additive separability, thus allowing one to estimate parameters sequentially during each maximization step. In establishing this result, we develop a broad class of estimators for mixture models. Returning to the likelihood problem, we show that, relative to full information maximum likelihood, our sequential estimator can generate large computational savings with little loss of efficiency.

Probabilistic subgroup identification using Bayesian finite mixture modelling : a case study in Parkinson’s disease phenotype identification

This article explores the use of probabilistic classification, namely finite mixture modelling, for identification of complex disease phenotypes, given cross-sectional data. In particular, if focuses on posterior probabilities of subgroup membership, a standard output of finite mixture modelling, and how the quantification of uncertainty in these probabilities can lead to more detailed analyses. Using a Bayesian approach, we describe two practical uses of this uncertainty: (i) as a means of describing a person’s membership to a single or multiple latent subgroups and (ii) as a means of describing identified subgroups by patient-centred covariates not included in model estimation. These proposed uses are demonstrated on a case study in Parkinson’s disease (PD), where latent subgroups are identified using multiple symptoms from the Unified Parkinson’s Disease Rating Scale (UPDRS).

Examining the cost efficiency of Chinese hydroelectric companies using a finite mixture model

This paper evaluates the operational activities of Chinese hydroelectric power companies over the period 2000-2010 using a finite mixture model that controls for unobserved heterogeneity. In so doing, a stochastic frontier latent class model, which allows for the existence of different technologies, is adopted to estimate cost frontiers. This procedure not only enables us to identify different groups among the hydro-power companies analysed, but also permits the analysis of their cost efficiency. The main result is that three groups are identified in the sample, each equipped with different technologies, suggesting that distinct business strategies need to be adapted to the characteristics of China's hydro-power companies. Some managerial implications are developed. © 2012 Elsevier B.V.

On Finite Mixture of Pareto and Beta Distnbutions

Department of Statistics, Cochin University of Science and Technology

The effect of (mis-specified) GARCH filters on the finite sample distribution of the BDS test

This paper considers the effect of using a GARCH filter on the properties of the BDS test statistic as well as a number of other issues relating to the application of the test. It is found that, for certain values of the user-adjustable parameters, the finite sample distribution of the test is far-removed from asymptotic normality. In particular, when data generated from some completely different model class are filtered through a GARCH model, the frequency of rejection of iid falls, often substantially. The implication of this result is that it might be inappropriate to use non-rejection of iid of the standardised residuals of a GARCH model as evidence that the GARCH model ‘fits’ the data.

Finite mixture regression model with random effects: application to neonatal hospital length of stay

A two-component mixture regression model that allows simultaneously for heterogeneity and dependency among observations is proposed. By specifying random effects explicitly in the linear predictor of the mixture probability and the mixture components, parameter estimation is achieved by maximising the corresponding best linear unbiased prediction type log-likelihood. Approximate residual maximum likelihood estimates are obtained via an EM algorithm in the manner of generalised linear mixed model (GLMM). The method can be extended to a g-component mixture regression model with the component density from the exponential family, leading to the development of the class of finite mixture GLMM. For illustration, the method is applied to analyse neonatal length of stay (LOS). It is shown that identification of pertinent factors that influence hospital LOS can provide important information for health care planning and resource allocation. (C) 2002 Elsevier Science B.V. All rights reserved.

Response-based segmentation using Finite Mixture Partial Least Squares: theoretical foundations and an application to American Customer Satisfaction Index Data

When applying multivariate analysis techniques in information systems and social science disciplines, such as management information systems (MIS) and marketing, the assumption that the empirical data originate from a single homogeneous population is often unrealistic. When applying a causal modeling approach, such as partial least squares (PLS) path modeling, segmentation is a key issue in coping with the problem of heterogeneity in estimated cause-and-effect relationships. This chapter presents a new PLS path modeling approach which classifies units on the basis of the heterogeneity of the estimates in the inner model. If unobserved heterogeneity significantly affects the estimated path model relationships on the aggregate data level, the methodology will allow homogenous groups of observations to be created that exhibit distinctive path model estimates. The approach will, thus, provide differentiated analytical outcomes that permit more precise interpretations of each segment formed. An application on a large data set in an example of the American customer satisfaction index (ACSI) substantiates the methodology’s effectiveness in evaluating PLS path modeling results.

Overfitting Bayesian mixture models with an unknown number of components

This paper proposes solutions to three issues pertaining to the estimation of finite mixture models with an unknown number of components: the non-identifiability induced by overfitting the number of components, the mixing limitations of standard Markov Chain Monte Carlo (MCMC) sampling techniques, and the related label switching problem. An overfitting approach is used to estimate the number of components in a finite mixture model via a Zmix algorithm. Zmix provides a bridge between multidimensional samplers and test based estimation methods, whereby priors are chosen to encourage extra groups to have weights approaching zero. MCMC sampling is made possible by the implementation of prior parallel tempering, an extension of parallel tempering. Zmix can accurately estimate the number of components, posterior parameter estimates and allocation probabilities given a sufficiently large sample size. The results will reflect uncertainty in the final model and will report the range of possible candidate models and their respective estimated probabilities from a single run. Label switching is resolved with a computationally light-weight method, Zswitch, developed for overfitted mixtures by exploiting the intuitiveness of allocation-based relabelling algorithms and the precision of label-invariant loss functions. Four simulation studies are included to illustrate Zmix and Zswitch, as well as three case studies from the literature. All methods are available as part of the R package Zmix, which can currently be applied to univariate Gaussian mixture models.

Probe-level linear model fitting and mixture modeling results in high accuracy detection of differential gene expression

Affiliation: Institut de recherche en immunologie et en cancérologie, Université de Montréal

Mixture modeling for joint analysis of survival, discrete, and continuous data

Mixture modeling is commonly used to model categorical latent variables that represent subpopulations in which population membership is unknown but can be inferred from the data. In relatively recent years, the potential of finite mixture models has been applied in time-to-event data. However, the commonly used survival mixture model assumes that the effects of the covariates involved in failure times differ across latent classes, but the covariate distribution is homogeneous. The aim of this dissertation is to develop a method to examine time-to-event data in the presence of unobserved heterogeneity under a framework of mixture modeling. A joint model is developed to incorporate the latent survival trajectory along with the observed information for the joint analysis of a time-to-event variable, its discrete and continuous covariates, and a latent class variable. It is assumed that the effects of covariates on survival times and the distribution of covariates vary across different latent classes. The unobservable survival trajectories are identified through estimating the probability that a subject belongs to a particular class based on observed information. We applied this method to a Hodgkin lymphoma study with long-term follow-up and observed four distinct latent classes in terms of long-term survival and distributions of prognostic factors. Our results from simulation studies and from the Hodgkin lymphoma study demonstrated the superiority of our joint model compared with the conventional survival model. This flexible inference method provides more accurate estimation and accommodates unobservable heterogeneity among individuals while taking involved interactions between covariates into consideration.^