Maximum penalized likelihood estimation of mixed proportional hazard models

Unobservable individual effects in models of duration will cause estimation bias that include the structural parameters as well as the duration dependence. The maximum penalized likelihood estimator is examined as an estimator for the survivor model with heterogeneity. Proofs of the existence and uniqueness of the maximum penalized likelihood estimator in duration model with general forms of unobserved heterogeneity are provided. Some small sample evidence on the behavior of the maximum penalized likelihood estimator is given. The maximum penalized likelihood estimator is shown to be computationally feasible and to provide reasonable estimates in most cases.     

Mixture modelling of recurrent event times with long-term survivors: Analysis of Hutterite birth intervals

We propose a mixture model that combines a discrete-time survival model for analyzing the correlated times between recurrent events, e.g. births, with a logistic regression model for the probability of never experiencing the event of interest, i.e., being a long-term survivor. The proposed survival model incorporates both observed and unobserved heterogeneity in the probability of experiencing the event of interest. We use Gibbs sampling for the fitting of such mixture models, which leads to a computationally intensive solution to the problem of fitting survival models for multiple event time data with long-term survivors. We illustrate our Bayesian approach through an analysis of Hutterite birth histories.     

Nonparametric Panel Estimation of Labor Supply

In this article, we estimate structural labor supply with piecewise-linear budgets and nonseparable endogenous unobserved heterogeneity. We propose a two-stage method to address the endogeneity issue that comes from the correlation between the covariates and unobserved heterogeneity. In the first stage, Evdokimov’s nonparametric de-convolution method serves to identify the conditional distribution of unobserved heterogeneity from the quasi-reduced model that uses panel data. In the second stage, the conditional distribution is plugged into the original structural model to estimate labor supply. We apply this methodology to estimate the labor supply of U.S. married men in 2004 and 2005. Our empirical work demonstrates that ignoring the correlation between the covariates and unobserved heterogeneity will bias the estimates of wage elasticities upward. The labor elasticity estimated from a fixed effects model is less than half of that obtained from a random effects model.     

Specification Tests for Random-effects Transition Models: An Application to a Model of the British Youth Training Scheme

We develop diagnostic tests for random-effects multi-spell multi-state models focusing on: independence between the unobserved heterogeneity and observed covariates; mutual independence of heterogeneity terms; and distributional form. They are applied to a transition model of the British youth labor market, revealing significant misspecifications in our initial model, and allowing us to develop a considerably better-fitting specification that would have been difficult to reach by other means. The improved specification implies reduced estimates of the effectiveness of the youth training scheme (YTS), but we nevertheless retain the conclusion of significant positive effects of YTS on employment prospects.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Keep it simple: estimation strategies for ordered response models with fixed effects

By running Monte Carlo simulations, we compare different estimation strategies of ordered response models in the presence of non-random unobserved heterogeneity. We find that very simple binary recoding schemes deliver parameter estimates with very low bias and high efficiency. Furthermore, if the researcher is interested in the relative size of parameters the simple linear fixed effects model is the method of choice.     

Multilevel regression mixture analysis

Summary.  A two-level regression mixture model is discussed and contrasted with the conventional two-level regression model. Simulated and real data shed light on the modelling alternatives. The real data analyses investigate gender differences in mathematics achievement from the US National Education Longitudinal Survey. The two-level regression mixture analyses show that unobserved heterogeneity should not be presupposed to exist only at level 2 at the expense of level 1. Both the simulated and the real data analyses show that level 1 heterogeneity in the form of latent classes can be mistaken for level 2 heterogeneity in the form of the random effects that are used in conventional two-level regression analysis. Because of this, mixture models have an important role to play in multilevel regression analyses. Mixture models allow heterogeneity to be investigated more fully, more correctly attributing different portions of the heterogeneity to the different levels.     

Parametric Proportional Odds Frailty Models

We define a parametric proportional odds frailty model to describe lifetime data incorporating heterogeneity between individuals. An unobserved individual random effect, called frailty, acts multiplicatively on the odds of failure by time t. We investigate fitting by maximum likelihood and by least squares. For the latter, the parametric survivor function is fitted to the nonparametric Kaplan–Meier estimate at the observed failure times. Bootstrap standard errors and confidence intervals are obtained for the least squares estimates. The models are applied successfully to simulated data and to two real data sets. Least squares estimates appear to have smaller bias than maximum likelihood.     

A mixture latent variable model for modeling mixed data in heterogeneous populations and its applications

Latent variable models are widely used for jointly modeling of mixed data including nominal, ordinal, count and continuous data. In this paper, we consider a latent variable model for jointly modeling relationships between mixed binary, count and continuous variables with some observed covariates. We assume that, given a latent variable, mixed variables of interest are independent and count and continuous variables have Poisson distribution and normal distribution, respectively. As such data may be extracted from different subpopulations, consideration of an unobserved heterogeneity has to be taken into account. A mixture distribution is considered (for the distribution of the latent variable) which accounts the heterogeneity. The generalized EM algorithm which uses the Newton–Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. The standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. Analysis of the primary biliary cirrhosis data is presented as an application of the proposed model.     

Mixture Multi-state Markov Regression Model

Although heterogeneity across individuals may be reduced when a two-state process is extended into a multi-state process, the discrepancy between the observed and the predicted for some states may still exist owing to two possibilities, unobserved mixture distribution in the initial state and the effect of measured covariates on subsequent multi-state disease progression. In the present study, we developed a mixture Markov exponential regression model to take account of the above-mentioned heterogeneity across individuals (subject-to-subject variability) with a systematic model selection based on the likelihood ratio test. The model was successfully demonstrated by an empirical example on surveillance of patients with small hepatocellular carcinoma treated by non-surgical methods. The estimated results suggested that the model with the incorporation of unobserved mixture distribution behaves better than the one without. Complete and partial effects regarding risk factors on different subsequent multi-state transitions were identified using a homogeneous Markov model. The combination of both initial mixture distribution and homogeneous Markov exponential regression model makes a significant contribution to reducing heterogeneity across individuals and over time for disease progression.     

An Alternative Estimation Approach for the Heterogeneity Linear Mixed Model

In this article, an alternative estimation approach is proposed to fit linear mixed effects models where the random effects follow a finite mixture of normal distributions. This heterogeneity linear mixed model is an interesting tool since it relaxes the classical normality assumption and is also perfectly suitable for classification purposes, based on longitudinal profiles. Instead of fitting directly the heterogeneity linear mixed model, we propose to fit an equivalent mixture of linear mixed models under some restrictions which is computationally simpler. Unlike the former model, the latter can be maximized analytically using an EM-algorithm and the obtained parameter estimates can be easily used to compute the parameter estimates of interest.     

Modelling survival data using mixtures of frailties

Frailty models are often used to describe the extra heterogeneity in survival data by introducing an individual random, unobserved effect. The frailty term is usually assumed to act multiplicatively on a baseline hazard function common to all individuals. In order to apply the frailty model, a specific frailty distribution has to be assumed. If at least one of the latent variables is continuous, the frailty must follow a continuous distribution. In this paper, a finite mixture of continuous frailty distributions is used in order to describe situations in which one (or more) of the latent variables separates the population in study into two (or more) subpopulations. Closure properties of the unobserved quantity are given along with the maximum-likelihood estimates under the most common choices of frailty distributions. The model is illustrated on a set of lifetime data.     

A Discrete-Time Multilevel Mixture Model for Event History Data with Long-Term Survivors,with an Application to an Analysis of Contraceptive Sterilization in Bangladesh

Event history models typically assume that the entire population is at risk of experiencing the event of interest throughout the observation period. However, there will often be individuals, referred to as long-term survivors, who may be considered a priori to have a zero hazard throughout the study period. In this paper, a discrete-time mixture model is proposed in which the probability of long-term survivorship and the timing of event occurrence are modelled jointly. Another feature of event history data that often needs to be considered is that they may come from a population with a hierarchical structure. For example, individuals may be nested within geographical regions and individuals in the same region may have similar risks of experiencing the event of interest due to unobserved regional characteristics. Thus, the discrete-time mixture model is extended to allow for clustering in the likelihood and timing of an event within regions. The model is further extended to allow for unobserved individual heterogeneity in the hazard of event occurrence. The proposed model is applied in an analysis of contraceptive sterilization in Bangladesh. The results show that a woman's religion and education level affect her probability of choosing sterilization, but not when she gets sterilized. There is also evidence of community-level variation in sterilization timing, but not in the probability of sterilization.     

Interest-Rate Arbitrage in Currency Baskets: Forecasting Weights and Measuring Risk

This article addresses the problem of the bias of income and expenditure elasticities estimated on pseudopanel data caused by measurement error and unobserved heterogeneity. We gauge these biases empirically by comparing cross-sectional, pseudo-panel, and true panel data from both Polish and U.S. expenditure surveys. Our results suggest that unobserved heterogeneity imparts a downward bias to cross-section estimates of income elasticities of at-home food expenditures and an upward bias to estimates of income elasticities of away-from-home food expenditures. “Within” and first-difference estimators suffer less bias, but only if the effects of measurement error are accounted for with instrumental variables.     

Job Durations With Worker- and Firm-Specific Effects: MCMC Estimation With Longitudinal Employer–Employee Data

We study job durations using a multivariate hazard model allowing for worker-specific and firm-specific unobserved determinants. The latter are captured by unobserved heterogeneity terms or random effects, one at the firm level and another at the worker level. This enables us to decompose the variation in job durations into the relative contribution of the worker and the firm. We also allow the unobserved terms to be correlated in a model that is primarily relevant for markets with small firms. For the empirical analysis, we use a Portuguese longitudinal matched employer–employee dataset. The model is estimated with a Bayesian Markov chain Monte Carlo (MCMC) estimation method. The results imply that unobserved firm characteristics explain almost 40% of the systematic variation in log job durations. In addition, we find a positive correlation between unobserved worker and firm characteristics.     

The probabilistic reduction approach to specifying multinomial logistic regression models in health outcomes research

The paper provides a novel application of the probabilistic reduction (PR) approach to the analysis of multi-categorical outcomes. The PR approach, which systematically takes account of heterogeneity and functional form concerns, can improve the specification of binary regression models. However, its utility for systematically enriching the specification of and inference from models of multi-categorical outcomes has not been examined, while multinomial logistic regression models are commonly used for inference and, increasingly, prediction. Following a theoretical derivation of the PR-based multinomial logistic model (MLM), we compare functional specification and marginal effects from a traditional specification and a PR-based specification in a model of post-stroke hospital discharge disposition and find that the traditional MLM is misspecified. Results suggest that the impact on the reliability of substantive inferences from a misspecified model may be significant, even when model fit statistics do not suggest a strong lack of fit compared with a properly specified model using the PR approach. We identify situations under which a PR-based MLM specification can be advantageous to the applied researcher.     

Effects of unmeasured heterogeneity in the linear transformation model for censored data

We investigate the effect of unobserved heterogeneity in the context of the linear transformation model for censored survival data in the clinical trials setting. The unobserved heterogeneity is represented by a frailty term, with unknown distribution, in the linear transformation model. The bias of the estimate under the assumption of no unobserved heterogeneity when it truly is present is obtained. We also derive the asymptotic relative efficiency of the estimate of treatment effect under the incorrect assumption of no unobserved heterogeneity. Additionally we investigate the loss of power for clinical trials that are designed assuming the model without frailty when, in fact, the model with frailty is true. Numerical studies under a proportional odds model show that the loss of efficiency and the loss of power can be substantial when the heterogeneity, as embodied by a frailty, is ignored. An erratum to this article can be found at     

Accuracy of variational estimates for random graph mixture models

Variational and variational Bayes techniques are popular approaches for statistical inference of complex models but their theoretical properties are still not well known. Because of both unobserved variables and intricate dependency structures, mixture models for random graphs constitute a good case study. We first present four different variational estimates for the parameters of these models. We then compare their accuracy through simulation studies and show that the variational Bayes estimates seem the most accurate for moderate graph size. We finally re-analyse the regulatory network of Escherichia coli with this approach.     

Specification analysis in mixed hazard models and a test of crossing survival functions

The paper investigates diagnostic procedures for the specification of common hazard models in duration analysis. It is shown that under mixed hazard specifications the survival functions of different subgroups cannot cross. A nonparametric test for the crossing of two survival functions is provided and its applications in duration analysis are discussed. In particular, the proportional hazard model with unobserved heterogeneity (PHU) is investigated, and procedures are developed to test whether given data are consistent with the PHU model and whether they contain unobserved heterogeneity within the PHU specification. Examples in which crossing survivals are of substantive concern are discussed, including the dynamics of infectious diseases and the demand for vaccination.