Marginalized transition random effect models for multivariate longitudinal binary data

Generalized linear models with random effects and/or serial dependence are commonly used to analyze longitudinal data. However, the computation and interpretation of marginal covariate effects can be difficult. This led Heagerty (1999, 2002) to propose models for longitudinal binary data in which a logistic regression is first used to explain the average marginal response. The model is then completed by introducing a conditional regression that allows for the longitudinal, within‐subject, dependence, either via random effects or regressing on previous responses. In this paper, the authors extend the work of Heagerty to handle multivariate longitudinal binary response data using a triple of regression models that directly model the marginal mean response while taking into account dependence across time and across responses. Markov Chain Monte Carlo methods are used for inference. Data from the Iowa Youth and Families Project are used to illustrate the methods.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

Bayesian analysis of correlated mixed categorical data by incorporating historical prior information

In this article, we develop statistical models for analysis of correlated mixed categorical (binary and ordinal) response data arising in medical and epidemi-ologic studies. There is evidence in the literature to suggest that models including correlation structure can lead to substantial improvement in precision of estimation or are more appropriate (accurate). We use a very rich class of scale mixture of multivariate normal (SMMVN) iink functions to accommodate heavy tailed distributions. In order to incorporate available historical information, we propose a unified prior elicitation scheme based on SMMVN-link models. Further, simulation-based techniques are developed to assess model adequacy. Finally, a real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

Addressing misclassification for binary data: probit and t-link regressions

Generalized linear models are addressed to describe the dependence of data on explanatory variables when the binary outcome is subject to misclassification. Both probit and t-link regressions for misclassified binary data under Bayesian methodology are proposed. The computational difficulties have been avoided by using data augmentation. The idea of using a data augmentation framework (with two types of latent variables) is exploited to derive efficient Gibbs sampling and expectation–maximization algorithms. Besides, this formulation has allowed to obtain the probit model as a particular case of the t-link model. Simulation examples are presented to illustrate the model performance when comparing with standard methods that do not consider misclassification. In order to show the potential of the proposed approaches, a real data problem arising when studying hearing loss caused by exposure to occupational noise is analysed.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

Multilevel models for repeated binary outcomes: attitudes and voting over the electoral cycle

Models for fitting longitudinal binary responses are explored by using a panel study of voting intentions. A standard multilevel repeated measures logistic model is shown to be inadequate owing to a substantial proportion of respondents who maintain a constant response over time. A multivariate binary response model is shown to be a better fit to the data.     

二元选择面板模型的设定检验

 本文将RESET检验扩展到二元选择面板数据模型的设定,考察了固定效应Probit模型和Logit模型的设定检验,包括异方差、遗漏变量和分布误设的检验。模拟结果表明Logit模型的RESET设定检验显示良好的水平和功效,而Probit模型的RESET检验可能由于估计方法的选择导致在某些方面的功效表现不好。但总体说来,在二元选择面板数据模型的设定检验上,RESET检验仍然是一个较好的选择。     

Approximate estimation in nonlinear panel data models

This paper is concerned with the estimation of a general class of nonlinear panel data models in which the conditional distribution of the dependent variable and the distribution of the heterogeneity factors are arbitrary. In general, exact analytical results for this problem do not exist. Here, Laplace and small-sigma appriximations for the marginal likelihood are presented. The computation of the MLE from both approximations is straightforward. It is shown that the accuracy of the Laplace approximation depends on both the sample size and the variance of the individual effects, whereas the accuracy of the small-sigma approximation is 0(1) with respect to the sample size. The results are applied to count, duration and probit panel data models. The accuracy of the approximations is evaluated through a Monte Carlo simulation experiment. The approximations are also applied in an analysis of youth unemployment in Australia.     

Patterns of consent: evidence from a general household survey

Summary.  We analyse patterns of consent and consent bias in the context of a large general household survey, the 'Improving survey measurement of income and employment' survey, also addressing issues that arise when there are multiple consent questions. A multivariate probit regression model for four binary outcomes with two incidental truncations is used. We show that there are biases in consent to data linkage with benefit and tax credit administrative records that are held by the Department for Work and Pensions, and with wage and employment data held by employers. There are also biases in respondents' willingness and ability to supply their national insurance number. The biases differ according to the question that is considered. We also show that modelling questions on consent independently rather than jointly may lead to misleading inferences about consent bias. A positive correlation between unobservable individual factors affecting consent to Department for Work and Pensions record linkage and consent to employer record linkage is suggestive of a latent individual consent propensity.     

Secure Bayesian model averaging for horizontally partitioned data

When multiple data owners possess records on different subjects with the same set of attributes—known as horizontally partitioned data—the data owners can improve analyses by concatenating their databases. However, concatenation of data may be infeasible because of confidentiality concerns. In such settings, the data owners can use secure computation techniques to obtain the results of certain analyses on the integrated database without sharing individual records. We present secure computation protocols for Bayesian model averaging and model selection for both linear regression and probit regression. Using simulations based on genuine data, we illustrate the approach for probit regression, and show that it can provide reasonable model selection outputs.     

Fixed Effects and Bias Due to the Incidental Parameters Problem in the Tobit Model

The maximum likelihood estimator (MLE) in nonlinear panel data models with fixed effects is widely understood (with a few exceptions) to be biased and inconsistent when T, the length of the panel, is small and fixed. However, there is surprisingly little theoretical or empirical evidence on the behavior of the estimator on which to base this conclusion. The received studies have focused almost exclusively on coefficient estimation in two binary choice models, the probit and logit models. In this note, we use Monte Carlo methods to examine the behavior of the MLE of the fixed effects tobit model. We find that the estimator's behavior is quite unlike that of the estimators of the binary choice models. Among our findings are that the location coefficients in the tobit model, unlike those in the probit and logit models, are unaffected by the “incidental parameters problem.” But, a surprising result related to the disturbance variance emerges instead - the finite sample bias appears here rather than in the slopes. This has implications for estimation of marginal effects and asymptotic standard errors, which are also examined in this paper. The effects are also examined for the probit and truncated regression models, extending the range of received results in the first of these beyond the widely cited biases in the coefficient estimators.     

Modeling individual migraine severity with autoregressive ordered probit models

This paper considers the problem of modeling migraine severity assessments and their dependence on weather and time characteristics. We take on the viewpoint of a patient who is interested in an individual migraine management strategy. Since factors influencing migraine can differ between patients in number and magnitude, we show how a patient’s headache calendar reporting the severity measurements on an ordinal scale can be used to determine the dominating factors for this special patient. One also has to account for dependencies among the measurements. For this the autoregressive ordinal probit (AOP) model of Müller and Czado (J Comput Graph Stat 14: 320–338, 2005) is utilized and fitted to a single patient’s migraine data by a grouped move multigrid Monte Carlo (GM-MGMC) Gibbs sampler. Initially, covariates are selected using proportional odds models. Model fit and model comparison are discussed. A comparison with proportional odds specifications shows that the AOP models are preferred.     

A comparison of methods for simulating correlated binary variables with specified marginal means and correlations

Simulation studies employed to study properties of estimators for parameters in population-average models for clustered or longitudinal data require suitable algorithms for data generation. Methods for generating correlated binary data that allow general specifications of the marginal mean and correlation structures are particularly useful. We compare an algorithm based on dichotomizing multi-normal variates to one based on a conditional linear family (CLF) of distributions [Qaqish BF. A family of multivariate binary distributions for simulating correlated binary variables with specified marginal means and correlations. Biometrika. 2003;90:455–463] with respect to range restrictions induced on correlations. Examples include generating longitudinal binary data and generating correlated binary data compatible with specified marginal means and covariance structures for bivariate, overdispersed binomial outcomes. Results show the CLF method gives a wider range of correlations for longitudinal data having autocorrelated within-subject associations, while the multivariate probit method gives a wider range of correlations for clustered data having exchangeable-type correlations. In the case of a decaying-product correlation structure, it is shown that the CLF method achieves the nonparametric limits on the range of correlations, which cannot be surpassed by any method.     

A latent variable regression model for asymmetric bivariate ordered categorical data

In many areas of medical research, especially in studies that involve paired organs, a bivariate ordered categorical response should be analyzed. Using a bivariate continuous distribution as the latent variable is an interesting strategy for analyzing these data sets. In this context, the bivariate standard normal distribution, which leads to the bivariate cumulative probit regression model, is the most common choice. In this paper, we introduce another latent variable regression model for modeling bivariate ordered categorical responses. This model may be an appropriate alternative for the bivariate cumulative probit regression model, when postulating a symmetric form for marginal or joint distribution of response data does not appear to be a valid assumption. We also develop the necessary numerical procedure to obtain the maximum likelihood estimates of the model parameters. To illustrate the proposed model, we analyze data from an epidemiologic study to identify some of the most important risk indicators of periodontal disease among students 15-19 years in Tehran, Iran.     

Parametric link modification of both tails in binary regression

Common binary regression models such as logistic or probit regression have been extended to include parametric link transformation families. These binary regression models with parametric link are designed to avoid possible link misspecification and improve fit in some data sets. One and two parameter link families have been proposed in the literature (for a review see Stukel (1988)). However in real data examples published so far only one parameter link families have found to improve the fit significantly. This paper introduces a two parameter link family involving the modification of both tails of the link. An analysis based on computationally tractable Bayesian inference involving Monte Carlo sampling algorithms is presented extending earlier work of Czado (1992, 1993b). Finally, the usefulness of the two tailed link modification will be demonstrated in an example where single tail modification can be significantly improved upon by using a two tailed modification.     

Joint analysis of nonlinear heterogeneous longitudinal data and binary outcome: an application to AIDS clinical studies

Finite mixture models are currently used to analyze heterogeneous longitudinal data. By releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, finite mixture models not only can estimate model parameters but also cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, which might be associated with a clinically important binary outcome. This article develops a joint modeling of a finite mixture of NLME models for longitudinal data in the presence of covariate measurement errors and a logistic regression for a binary outcome, linked by individual latent class indicators, under a Bayesian framework. Simulation studies are conducted to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and logistic regression are fitted separately, followed by an application to a real data set from an AIDS clinical trial, in which the viral dynamics and dichotomized time to the first decline of CD4/CD8 ratio are analyzed jointly.     

A Bayesian approach for misclassified ordinal response data

ABSTRACT

Motivated by a longitudinal oral health study, the Signal-Tandmobiel^® study, a Bayesian approach has been developed to model misclassified ordinal response data. Two regression models have been considered to incorporate misclassification in the categorical response. Specifically, probit and logit models have been developed. The computational difficulties have been avoided by using data augmentation. This idea is exploited to derive efficient Markov chain Monte Carlo methods. Although the method is proposed for ordered categories, it can also be implemented for unordered ones in a simple way. The model performance is shown through a simulation-based example and the analysis of the motivating study.     

Markov structure based logistic regression for repeated measures: An application to diabetes mellitus data

In longitudinal studies, as repeated observations are made on the same individual the response variables will usually be correlated. In analyzing such data, this dependence must be taken into account to avoid misleading inferences. The focus of this paper is to apply a logistic marginal model with Markovian dependence proposed by Azzalini [A. Azzalini, Logistic regression for autocorrelated data with application to repeated measures, Biometrika 81 (1994) 767–775] to the study of the influence of time-dependent covariates on the marginal distribution of the binary response in serially correlated binary data. We have shown how to construct the model so that the covariates relate only to the mean value of the process, independent of the association parameters. After formulating the proposed model for repeated measures data, the same approach is applied to missing data. An application is provided to the diabetes mellitus data of registered patients at the Bangladesh Institute of Research and Rehabilitation in Diabetes, Endocrine and Metabolic Disorders (BIRDEM) in 1984, using both time stationary and time varying covariates.     

Logistic regression analysis of non‐randomized response data collected by the parallel model in sensitive surveys

To study the relationship between a sensitive binary response variable and a set of non‐sensitive covariates, this paper develops a hidden logistic regression to analyse non‐randomized response data collected via the parallel model originally proposed by Tian (2014). This is the first paper to employ the logistic regression analysis in the field of non‐randomized response techniques. Both the Newton–Raphson algorithm and a monotone quadratic lower bound algorithm are developed to derive the maximum likelihood estimates of the parameters of interest. In particular, the proposed logistic parallel model can be used to study the association between a sensitive binary variable and another non‐sensitive binary variable via the measure of odds ratio. Simulations are performed and a study on people's sexual practice data in the United States is used to illustrate the proposed methods.