The performance of mixture models in heterogeneous closed population capture-recapture.

Dorazio and Royle (2003, Biometrics 59, 351-364) investigated the behavior of three mixture models for closed population capture-recapture analysis in the presence of individual heterogeneity of capture probability. Their simulations were from the beta-binomial distribution, with analyses from the beta-binomial, the logit-normal, and the finite mixture (latent class) models. In this response, simulations from many different distributions give a broader picture of the relative value of the beta-binomial and the finite mixture models, and provide some preliminary insights into the situations in which these models are useful.     

Mixture models for estimating the size of a closed population when capture rates vary among individuals

We develop a parameterization of the beta-binomial mixture that provides sensible inferences about the size of a closed population when probabilities of capture or detection vary among individuals. Three classes of mixture models (beta-binomial, logistic-normal, and latent-class) are fitted to recaptures of snowshoe hares for estimating abundance and to counts of bird species for estimating species richness. In both sets of data, rates of detection appear to vary more among individuals (animals or species) than among sampling occasions or locations. The estimates of population size and species richness are sensitive to model-specific assumptions about the latent distribution of individual rates of detection. We demonstrate using simulation experiments that conventional diagnostics for assessing model adequacy, such as deviance, cannot be relied on for selecting classes of mixture models that produce valid inferences about population size. Prior knowledge about sources of individual heterogeneity in detection rates, if available, should be used to help select among classes of mixture models that are to be used for inference.     

On comparison of mixture models for closed population capture-recapture studies

Summary .  A mixture model is a natural choice to deal with individual heterogeneity in capture–recapture studies. Pledger (2000, Biometrics 56, 434–442; 2005, Biometrics 61, 868–876) advertised the use of the two-point mixture model. Dorazio and Royle (2003, Biometrics 59, 351–364; 2005, Biometrics 61, 874–876) suggested that the beta-binomial model has advantages. The controversy is related to the nonidentifiability of the population size ( Link, 2003 , Biometrics 59, 1123–1130) and certain boundary problems. The total bias is decomposed into an intrinsic bias, an approximation bias, and an estimation bias. We propose to assess the approximation bias, the estimation bias, and the variance, with the intrinsic bias excluded when comparing different estimators. The boundary problems in both models and their impacts are investigated. Real epidemiological and ecological examples are analyzed.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

On Identifiability in Capture–Recapture Models

Summary .   We study the issue of identifiability of mixture models in the context of capture–recapture abundance estimation for closed populations. Such models are used to take account of individual heterogeneity in capture probabilities, but their validity was recently questioned by Link (2003, Biometrics 59, 1123–1130) on the basis of their nonidentifiability. We give a general criterion for identifiability of the mixing distribution, and apply it to establish identifiability within families of mixing distributions that are commonly used in this context, including finite and beta mixtures. Our analysis covers binomial and geometrically distributed outcomes. In an example we highlight the difference between the identifiability issue considered here and that in classical binomial mixture models.     

Random-effects meta-analysis models for the odds ratio in the case of rare events under different data-generating models: A simulation study

Meta-analysis of binary data is challenging when the event under investigation is rare, and standard models for random-effects meta-analysis perform poorly in such settings. In this simulation study, we investigate the performance of different random-effects meta-analysis models in terms of point and interval estimation of the pooled log odds ratio in rare events meta-analysis. First and foremost, we evaluate the performance of a hypergeometric-normal model from the family of generalized linear mixed models (GLMMs), which has been recommended, but has not yet been thoroughly investigated for rare events meta-analysis. Performance of this model is compared to performance of the beta-binomial model, which yielded favorable results in previous simulation studies, and to the performance of models that are frequently used in rare events meta-analysis, such as the inverse variance model and the Mantel–Haenszel method. In addition to considering a large number of simulation parameters inspired by real-world data settings, we study the comparative performance of the meta-analytic models under two different data-generating models (DGMs) that have been used in past simulation studies. The results of this study show that the hypergeometric-normal GLMM is useful for meta-analysis of rare events when moderate to large heterogeneity is present. In addition, our study reveals important insights with regard to the performance of the beta-binomial model under different DGMs from the binomial-normal family. In particular, we demonstrate that although misalignment of the beta-binomial model with the DGM affects its performance, it shows more robustness to the DGM than its competitors.     

On a likelihood-based goodness-of-fit test of the beta-binomial model

When faced with proportion data that exhibit extra-binomial variation, data analysts often consider the beta-binomial distribution as an alternative model to the more common binomial distribution. A typical example occurs in toxicological experiments with laboratory animals, where binary observations on fetuses within a litter are often correlated with each other. In such instances, it may be of interest to test for the goodness of fit of the beta-binomial model; this effort is complicated, however, when there is large variability among the litter sizes. We investigate a recent goodness-of-fit test proposed by Brooks et al. (1997, Biometrics 53, 1097-1115) but find that it lacks the ability to distinguish between the beta-binomial model and some severely non-beta-binomial models. Other tests and models developed in their article are quite useful and interesting but are not examined herein.     

An Evaluation of Beta-Binomial and Dirichlet-Trinomial Models for the Analysis of Reproductive and Developmental Effects

The common endpoints for the evaluation of reproductive and developmental toxic effects are the number of dead/resorbed fetuses, the number of malformed fetuses, and the number of normal fetuses for each litter. The joint distribution of the three endpoints could be modelled by a Dirichlettrinomial distribution or by a product of two-beta-binomial distributions. A simulation experiment is used to investigate the biases of the maximum likelihood estimate (MLE) for the probability of adverse effects under the Dirichlet-trinomial model and the beta-binomial model. Also, the type I errors and powers of the likelihood ratio test for comparing the difference between treatment and control are evaluated for the two underlying models. In estimation, the two MLE's are comparable, the bias estimates are small. In testing, the likelihood ratio test is generally more powerful under the Dirichlet-trinomial model than the beta-binomial model. The type I error rate is greater than the nominal level using the Dirichlet-trinomial model in some cases, when the data are generated from the two-beta-binomial model, and it is less than the nominal level using the beta-binomial model in other cases, when the data are generated from the Dirichlet-trinomial model.     

A latent-class mixture model for incomplete longitudinal Gaussian data

Summary .   In the analyses of incomplete longitudinal clinical trial data, there has been a shift, away from simple methods that are valid only if the data are missing completely at random, to more principled ignorable analyses, which are valid under the less restrictive missing at random assumption. The availability of the necessary standard statistical software nowadays allows for such analyses in practice. While the possibility of data missing not at random (MNAR) cannot be ruled out, it is argued that analyses valid under MNAR are not well suited for the primary analysis in clinical trials. Rather than either forgetting about or blindly shifting to an MNAR framework, the optimal place for MNAR analyses is within a sensitivity-analysis context. One such route for sensitivity analysis is to consider, next to selection models, pattern-mixture models or shared-parameter models. The latter can also be extended to a latent-class mixture model, the approach taken in this article. The performance of the so-obtained flexible model is assessed through simulations and the model is applied to data from a depression trial.     

Minimum Hellinger distance estimation for finite mixtures of Poisson regression models and its applications

Minimum Hellinger distance estimation (MHDE) has been shown to discount anomalous data points in a smooth manner with first-order efficiency for a correctly specified model. An estimation approach is proposed for finite mixtures of Poisson regression models based on MHDE. Evidence from Monte Carlo simulations suggests that MHDE is a viable alternative to the maximum likelihood estimator when the mixture components are not well separated or the model parameters are near zero. Biometrical applications also illustrate the practical usefulness of the MHDE method.     

A stabilized moment estimator for the beta-binomial distribution

The beta-binomial distribution has been proposed as a model for the incorporation of historical control data in the analysis of rodent carcinogenesis bioassays. Low spontaneous tumor incidences along with the small number and sizes of historical control groups combine to make the moment and maximum likelihood estimates of the beta-binomial parameters deficient. We therefore propose a stabilized moment estimator for one of the parameters. The stabilized moment estimator is similar to the ridge regression estimator and introduces a shrinkage parameter. Computer simulations were run to examine the behavior of the stabilized moment estimator. The effect of the stabilized moment estimator on the score test for dose-related trend is considered both on simulated data and on an example from the literature.     

Comparison of random-effects meta-analysis models for the relative risk in the case of rare events: A simulation study

Pooling the relative risk (RR) across studies investigating rare events, for example, adverse events, via meta-analytical methods still presents a challenge to researchers. The main reason for this is the high probability of observing no events in treatment or control group or both, resulting in an undefined log RR (the basis of standard meta-analysis). Other technical challenges ensue, for example, the violation of normality assumptions, or bias due to exclusion of studies and application of continuity corrections, leading to poor performance of standard approaches. In the present simulation study, we compared three recently proposed alternative models (random-effects [RE] Poisson regression, RE zero-inflated Poisson [ZIP] regression, binomial regression) to the standard methods in conjunction with different continuity corrections and to different versions of beta-binomial regression. Based on our investigation of the models' performance in 162 different simulation settings informed by meta-analyses from the Cochrane database and distinguished by different underlying true effects, degrees of between-study heterogeneity, numbers of primary studies, group size ratios, and baseline risks, we recommend the use of the RE Poisson regression model. The beta-binomial model recommended by Kuss (2015) also performed well. Decent performance was also exhibited by the ZIP models, but they also had considerable convergence issues. We stress that these recommendations are only valid for meta-analyses with larger numbers of primary studies. All models are applied to data from two Cochrane reviews to illustrate differences between and issues of the models. Limitations as well as practical implications and recommendations are discussed; a flowchart summarizing recommendations is provided.     

Multivariate methods for clustered binary data with multiple subclasses, with application to binary longitudinal data.

Clustered binary data occur frequently in biostatistical work. Several approaches have been proposed for the analysis of clustered binary data. In Rosner (1984, Biometrics 40, 1025-1035), a polychotomous logistic regression model was proposed that is a generalization of the beta-binomial distribution and allows for unit- and subunit-specific covariates, while controlling for clustering effects. One assumption of this model is that all pairs of subunits within a cluster are equally correlated. This is appropriate for ophthalmologic work where clusters are generally of size 2, but may be inappropriate for larger cluster sizes. A beta-binomial mixture model is introduced to allow for multiple subclasses within a cluster and to estimate odds ratios relating outcomes for pairs of subunits within a subclass as well as in different subclasses. To include covariates, an extension of the polychotomous logistic regression model is proposed, which allows one to estimate effects of unit-, class-, and subunit-specific covariates, while controlling for clustering using the beta-binomial mixture model. This model is applied to the analysis of respiratory symptom data in children collected over a 14-year period in East Boston, Massachusetts, in relation to maternal and child smoking, where the unit is the child and symptom history is divided into early-adolescent and late-adolescent symptom experience.     

Minimum Hellinger distance estimation for k-component poisson mixture with random effects

Summary .   The k-component Poisson regression mixture with random effects is an effective model in describing the heterogeneity for clustered count data arising from several latent subpopulations. However, the residual maximum likelihood estimation (REML) of regression coefficients and variance component parameters tend to be unstable and may result in misleading inferences in the presence of outliers or extreme contamination. In the literature, the minimum Hellinger distance (MHD) estimation has been investigated to obtain robust estimation for finite Poisson mixtures. This article aims to develop a robust MHD estimation approach for k-component Poisson mixtures with normally distributed random effects. By applying the Gaussian quadrature technique to approximate the integrals involved in the marginal distribution, the marginal probability function of the k-component Poisson mixture with random effects can be approximated by the summation of a set of finite Poisson mixtures. Simulation study shows that the MHD estimates perform satisfactorily for data without outlying observation(s), and outperform the REML estimates when data are contaminated. Application to a data set of recurrent urinary tract infections (UTI) with random institution effects demonstrates the practical use of the robust MHD estimation method.     

Mixture Models for Distance Sampling Detection Functions

We present a new class of models for the detection function in distance sampling surveys of wildlife populations, based on finite mixtures of simple parametric key functions such as the half-normal. The models share many of the features of the widely-used “key function plus series adjustment” (K+A) formulation: they are flexible, produce plausible shapes with a small number of parameters, allow incorporation of covariates in addition to distance and can be fitted using maximum likelihood. One important advantage over the K+A approach is that the mixtures are automatically monotonic non-increasing and non-negative, so constrained optimization is not required to ensure distance sampling assumptions are honoured. We compare the mixture formulation to the K+A approach using simulations to evaluate its applicability in a wide set of challenging situations. We also re-analyze four previously problematic real-world case studies. We find mixtures outperform K+A methods in many cases, particularly spiked line transect data (i.e., where detectability drops rapidly at small distances) and larger sample sizes. We recommend that current standard model selection methods for distance sampling detection functions are extended to include mixture models in the candidate set.     

Sample size for gene expression microarray experiments

MOTIVATION: Microarray experiments often involve hundreds or thousands of genes. In a typical experiment, only a fraction of genes are expected to be differentially expressed; in addition, the measured intensities among different genes may be correlated. Depending on the experimental objectives, sample size calculations can be based on one of the three specified measures: sensitivity, true discovery and accuracy rates. The sample size problem is formulated as: the number of arrays needed in order to achieve the desired fraction of the specified measure at the desired family-wise power at the given type I error and (standardized) effect size. RESULTS: We present a general approach for estimating sample size under independent and equally correlated models using binomial and beta-binomial models, respectively. The sample sizes needed for a two-sample z-test are computed; the computed theoretical numbers agree well with the Monte Carlo simulation results. But, under more general correlation structures, the beta-binomial model can underestimate the needed samples by about 1-5 arrays. CONTACT: jchen@nctr.fda.gov.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Generalized parametric cure models for relative survival

Cure models are used in time-to-event analysis when not all individuals are expected to experience the event of interest, or when the survival of the considered individuals reaches the same level as the general population. These scenarios correspond to a plateau in the survival and relative survival function, respectively. The main parameters of interest in cure models are the proportion of individuals who are cured, termed the cure proportion, and the survival function of the uncured individuals. Although numerous cure models have been proposed in the statistical literature, there is no consensus on how to formulate these. We introduce a general parametric formulation of mixture cure models and a new class of cure models, termed latent cure models, together with a general estimation framework and software, which enable fitting of a wide range of different models. Through simulations, we assess the statistical properties of the models with respect to the cure proportion and the survival of the uncured individuals. Finally, we illustrate the models using survival data on colon cancer, which typically display a plateau in the relative survival. As demonstrated in the simulations, mixture cure models which are not guaranteed to be constant after a finite time point, tend to produce accurate estimates of the cure proportion and the survival of the uncured. However, these models are very unstable in certain cases due to identifiability issues, whereas LC models generally provide stable results at the price of more biased estimates.     

Finite-size effects of Kirkwood–Buff integrals from molecular simulations

The modelling of thermodynamic properties of liquids from local density fluctuations is relevant to many chemical and biological processes. The Kirkwood–Buff (KB) theory connects the microscopic structure of isotropic liquids with macroscopic properties such as partial derivatives of activity coefficients, partial molar volumes and compressibilities. Originally, KB integrals were formulated for open and infinite systems which are difficult to access with standard Molecular Dynamics (MD) simulations. Recently, KB integrals for finite and open systems were formulated (J Phys Chem Lett. 2013;4:235). From the scaling of KB integrals for finite subvolumes, embedded in larger reservoirs, with the inverse of the size of these subvolumes, estimates for KB integrals in the thermodynamic limit are obtained. Two system size effects are observed in MD simulations: (1) effects due to the size of the simulation box and the size of the finite subvolume embedded in the simulation box, and (2) effects due to computing radial distribution functions (RDF) from a closed and finite system. In this study, we investigate the two effects in detail by computing KB integrals using the following methods: (1) Monte Carlo simulations of finite subvolumes of a liquid with an analytic RDF and (2) MD simulations of a WCA mixture for various simulation box sizes, but at the same thermodynamic state. We investigate the effect of the size of the simulation box and quantify the differences compared to KB integrals computed in the thermodynamic limit. We demonstrate that calculations of KB integrals should not be extended beyond half the size of the simulation box. For finite-size effects related to the RDF, we find that the Van der Vegt correction (J Chem Theory Comput. 2013;9:1347) yields the most accurate results.     

Parametric Models for Estimating the Number of Classes

We consider parametric distributions intended to model heterogeneity in population size estimation, especially parametric stochastic abundance models for species richness estimation. We briefly review (conditional) maximum likelihood estimation of the number of species, and summarize the results of fitting 7 candidate models to frequency‐count data, from a database of >40000 such instances, mostly arising from microbial ecology. We consider error estimation, goodness‐of‐fit assessment, data subsetting, and other practical matters. We find that, although the array of candidate models can be improved, finite mixtures of a small number of components (point masses or simple diffuse distributions) represent a promising direction. Finally we consider the connections between parametric models for abundance and incidence data, again noting the usefulness of finite mixture models. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)