Joint regression and association modeling of longitudinal ordinal data

We propose models for longitudinal, or otherwise clustered, ordinal data. The association between subunit responses is characterized by dependence ratios (Ekholm, Smith, and McDonald, 1995, Biometrika 82, 847-854), which are extended from the binary to the multicategory case. The joint probabilities of the subunit responses are expressed as explicit functions of the marginal means and the dependence ratios of all orders, obtaining a computational advantage for likelihood-based inference. Equal emphasis is put on finding regression models for the univariate cumulative probabilities, and on deriving the dependence ratios from meaningful association-generating mechanisms. A data set on the effects of treatment with Fluvoxamine, which has been analyzed in parts before (Molenberghs, Kenward, and Lesaffre, 1997, Biometrika 84, 33-44), is analyzed in its entirety. Selection models are used for studying the sensitivity of the results to drop-out.     

A general class of pattern mixture models for nonignorable dropout with many possible dropout times

Summary .   In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.     

Regressive models for risk prediction of repeated multinomial outcomes: An illustration using Health and Retirement Study data

Life expectancy is increasing in many countries and this may lead to a higher frequency of adverse health outcomes. Therefore, there is a growing demand for predicting the risk of a sequence of events based on specified factors from repeated outcomes. We proposed regressive models and a framework to predict the joint probabilities of a sequence of events for multinomial outcomes from longitudinal studies. The Markov chain is used to link marginal and sequence of conditional probabilities to predict the joint probability. Marginal and sequence of conditional probabilities are estimated using marginal and regressive models. An application is shown using the Health and Retirement Study data. The bias of parameter estimates for all models from all bootstrap simulation is less than 1% in most of the cases. The estimated mean squared error is also very low. Results from the simulation study show negligible bias and the usefulness of the proposed model. The proposed model and framework would be useful to solve real-life problems from various fields and big data analysis.     

Marginal regression analysis of a multivariate binary response

We propose the use of the mean parameter for regression analysisof a multivariate binary response. We model the associationusing dependence ratios defined in terms of the mean parameter,the components of which are the joint success probabilitiesof all orders. This permits flexible modelling of higher-orderassociations, using maximum likelihood estimation. We reanalysetwo data sets, one with variable cluster size and the othera longitudinal data set with constant cluster size.     

Integrating sample similarities into latent class analysis: a tree-structured shrinkage approach

This paper is concerned with using multivariate binary observations to estimate the probabilities of unobserved classes with scientific meanings. We focus on the setting where additional information about sample similarities is available and represented by a rooted weighted tree. Every leaf in the given tree contains multiple samples. Shorter distances over the tree between the leaves indicate a priori higher similarity in class probability vectors. We propose a novel data integrative extension to classical latent class models with tree-structured shrinkage. The proposed approach enables (1) borrowing of information across leaves, (2) estimating data-driven leaf groups with distinct vectors of class probabilities, and (3) individual-level probabilistic class assignment given the observed multivariate binary measurements. We derive and implement a scalable posterior inference algorithm in a variational Bayes framework. Extensive simulations show more accurate estimation of class probabilities than alternatives that suboptimally use the additional sample similarity information. A zoonotic infectious disease application is used to illustrate the proposed approach. The paper concludes by a brief discussion on model limitations and extensions.     

Estimation of operating characteristics for dependent diagnostic tests based on latent Markov models

We describe a method for making inferences about the joint operating characteristics of multiple diagnostic tests applied longitudinally and in the absence of a definitive reference test. Log-linear models are adopted for the classification distributions conditional on the latent state, where inclusion of appropriate interaction terms accommodates conditional dependencies among the tests. A marginal likelihood is constructed by marginalizing over a latent two-state Markov process. Specific latent processes we consider include a first-order Markov model, a second-order Markov model, and a time-nonhomogeneous Markov model, although the method is described in full generality. Adaptations to handle missing data are described. Model diagnostics are considered based on the bootstrap distribution of conditional residuals. The methods are illustrated by application to a study of diffuse bilateral infiltrates among patients in intensive care wards in which the objective was to assess aspects of validity and clinical agreement.     

The Analysis of Correlated Binary Outcomes Using Multivariate Logistic Regression

A method for analyzing correlated binary outcomes when the responses are distinct measurements made simultaneously on a single individual is presented. This extension of univariate logistic regression allows us to model the dependence of the responses on a set of covariates while estimating the degree of association among them. For the case of two dichotomous outcomes, a form of the cumulative bivariate logistic distribution proposed by Gumbel is used to characterize their joint probabilities in terms of logistic marginal probabilities and the correlation coefficient of the responses. The model is then extended to accommodate three or more dichotomous outcomes. A two-step approximation to fitting the multivariate logistic model is also described.     

Generalized hierarchical multivariate CAR models for areal data

In the fields of medicine and public health, a common application of areal data models is the study of geographical patterns of disease. When we have several measurements recorded at each spatial location (for example, information on p>/= 2 diseases from the same population groups or regions), we need to consider multivariate areal data models in order to handle the dependence among the multivariate components as well as the spatial dependence between sites. In this article, we propose a flexible new class of generalized multivariate conditionally autoregressive (GMCAR) models for areal data, and show how it enriches the MCAR class. Our approach differs from earlier ones in that it directly specifies the joint distribution for a multivariate Markov random field (MRF) through the specification of simpler conditional and marginal models. This in turn leads to a significant reduction in the computational burden in hierarchical spatial random effect modeling, where posterior summaries are computed using Markov chain Monte Carlo (MCMC). We compare our approach with existing MCAR models in the literature via simulation, using average mean square error (AMSE) and a convenient hierarchical model selection criterion, the deviance information criterion (DIC; Spiegelhalter et al., 2002, Journal of the Royal Statistical Society, Series B64, 583-639). Finally, we offer a real-data application of our proposed GMCAR approach that models lung and esophagus cancer death rates during 1991-1998 in Minnesota counties.     

A parametric copula model for analysis of familial binary data

Modeling the joint distribution of a binary trait (disease) within families is a tedious challenge, owing to the lack of a general statistical model with desirable properties such as the multivariate Gaussian model for a quantitative trait. Models have been proposed that either assume the existence of an underlying liability variable, the reality of which cannot be checked, or provide estimates of aggregation parameters that are dependent on the ordering of family members and on family size. We describe how a class of copula models for the analysis of exchangeable categorical data can be incorporated into a familial framework. In this class of models, the joint distribution of binary outcomes is characterized by a function of the given marginals. This function, referred to as a "copula," depends on an aggregation parameter that is weakly dependent on the marginal distributions. We propose to decompose a nuclear family into two sets of equicorrelated data (parents and offspring), each of which is characterized by an aggregation parameter (alphaFM and alphaSS, respectively). The marginal probabilities are modeled through a logistic representation. The advantage of this model is that it provides estimates of the aggregation parameters that are independent of family size and does not require any arbitrary ordering of sibs. It can be incorporated easily into segregation or combined segregation-linkage analysis and does not require extensive computer time. As an illustration, we applied this model to a combined segregation-linkage analysis of levels of plasma angiotensin I-converting enzyme (ACE) dichotomized into two classes according to the median. The conclusions of this analysis were very similar to those we had reported in an earlier familial analysis of quantitative ACE levels.     

Exchangeability in multivariate Markov chain models.

Time-homogeneous Markov chain models with state space [0, 1]k are useful in analysis of binary follow-up data on k individuals that interact. The number of parameters increases exponentially with k so more restrictive models are imperative for statistical inference. The hypothesis that the matrix of transition probabilities is invariant under permutation of individuals is discussed. It is shown that if individuals are exchangeable, then the process counting the number of individuals occupying a given state is a Markov chain. This reduction of data is sufficient if either at most a single individual may change state between two consecutive time points or if a state is absorbing. Similar results are obtained for exchangeability within two subgroups. Inference in the multivariate process reduces to a univariate problem if individuals are independent given the group's previous response. It is shown how conditional independence could be tested assuming exchangeability. The different hypotheses re examined in an analysis of the occurrence of bacteria in milk samples of Danish dairy cattle.     

Bayesian multivariate logistic regression

Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models that do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple noninformative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. The method is illustrated with application to a neurotoxicology study.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time

Within the pattern-mixture modeling framework for informative dropout, conditional linear models (CLMs) are a useful approach to deal with dropout that can occur at any point in continuous time (not just at observation times). However, in contrast with selection models, inferences about marginal covariate effects in CLMs are not readily available if nonidentity links are used in the mean structures. In this article, we propose a CLM for long series of longitudinal binary data with marginal covariate effects directly specified. The association between the binary responses and the dropout time is taken into account by modeling the conditional mean of the binary response as well as the dependence between the binary responses given the dropout time. Specifically, parameters in both the conditional mean and dependence models are assumed to be linear or quadratic functions of the dropout time; and the continuous dropout time distribution is left completely unspecified. Inference is fully Bayesian. We illustrate the proposed model using data from a longitudinal study of depression in HIV-infected women, where the strategy of sensitivity analysis based on the extrapolation method is also demonstrated.     

Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

Marginal models for correlated binary responses with multiple classes and multiple levels of nesting.

A model for correlated binary data is presented. Marginal probabilities and odds ratios are allowed to have general regression structures that include multiple classes and multiple levels of nesting. Estimation is done through the generalized estimating equations approach of Liang and Zeger (1986, Biometrika 73, 13-22). They are contrasted with conditional models and recommendations for choosing between the two are given. Examples from genetic epidemiology are presented.     

Marginalized transition models and likelihood inference for longitudinal categorical data

Marginal generalized linear models are now frequently used for the analysis of longitudinal data. Semiparametric inference for marginal models was introduced by Liang and Zeger (1986, Biometrics 73, 13-22). This article develops a general parametric class of serial dependence models that permits likelihood-based marginal regression analysis of binary response data. The methods naturally extend the first-order Markov models of Azzalini (1994, Biometrika 81, 767-775) and prove computationally feasible for long series.     

Tractable Bayes of skew-elliptical link models for correlated binary data

Correlated binary response data with covariates are ubiquitous in longitudinal or spatial studies. Among the existing statistical models, the most well-known one for this type of data is the multivariate probit model, which uses a Gaussian link to model dependence at the latent level. However, a symmetric link may not be appropriate if the data are highly imbalanced. Here, we propose a multivariate skew-elliptical link model for correlated binary responses, which includes the multivariate probit model as a special case. Furthermore, we perform Bayesian inference for this new model and prove that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior. The new methodology is illustrated by an application to COVID-19 data from three different counties of the state of California, USA. By jointly modeling extreme spikes in weekly new cases, our results show that the spatial dependence cannot be neglected. Furthermore, the results also show that the skewed latent structure of our proposed model improves the flexibility of the multivariate probit model and provides a better fit to our highly imbalanced dataset.     

Two‐stage estimation for multivariate recurrent event data with a dependent terminal event

Recurrent event data arise in longitudinal follow‐up studies, where each subject may experience the same type of events repeatedly. The work in this article is motivated by the data from a study of repeated peritonitis for patients on peritoneal dialysis. Due to the aspects of medicine and cost, the peritonitis cases were classified into two types: Gram‐positive and non‐Gram‐positive peritonitis. Further, since the death and hemodialysis therapy preclude the occurrence of recurrent events, we face multivariate recurrent event data with a dependent terminal event. We propose a flexible marginal model, which has three characteristics: first, we assume marginal proportional hazard and proportional rates models for terminal event time and recurrent event processes, respectively; second, the inter‐recurrences dependence and the correlation between the multivariate recurrent event processes and terminal event time are modeled through three multiplicative frailties corresponding to the specified marginal models; third, the rate model with frailties for recurrent events is specified only on the time before the terminal event. We propose a two‐stage estimation procedure for estimating unknown parameters. We also establish the consistency of the two‐stage estimator. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is applied to the peritonitis cohort data that motivated this study.     

Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution

The multivariate binomial logit-normal distribution is a mixture distribution for which, (i) conditional on a set of success probabilities and sample size indices, a vector of counts is independent binomial variates, and (ii) the vector of logits of the parameters has a multivariate normal distribution. We use this distribution to model multivariate binomial-type responses using a vector of random effects. The vector of logits of parameters has a mean that is a linear function of explanatory variables and has an unspecified or partly specified covariance matrix. The model generalizes and provides greater flexibility than the univariate model that uses a normal random effect to account for positive correlations in clustered data. The multivariate model is useful when different elements of the response vector refer to different characteristics, each of which may naturally have its own random effect. It is also useful for repeated binary measurement of a single response when there is a nonexchangeable association structure, such as one often expects with longitudinal data or when negative association exists for at least one pair of responses. We apply the model to an influenza study with repeated responses in which some pairs are negatively associated and to a developmental toxicity study with continuation-ratio logits applied to an ordinal response with clustered observations.     

Bayesian analysis of correlated evolution of discrete characters by reversible-jump Markov chain Monte Carlo

We describe a Bayesian method for investigating correlated evolution of discrete binary traits on phylogenetic trees. The method fits a continuous-time Markov model to a pair of traits, seeking the best fitting models that describe their joint evolution on a phylogeny. We employ the methodology of reversible-jump (RJ) Markov chain Monte Carlo to search among the large number of possible models, some of which conform to independent evolution of the two traits, others to correlated evolution. The RJ Markov chain visits these models in proportion to their posterior probabilities, thereby directly estimating the support for the hypothesis of correlated evolution. In addition, the RJ Markov chain simultaneously estimates the posterior distributions of the rate parameters of the model of trait evolution. These posterior distributions can be used to test among alternative evolutionary scenarios to explain the observed data. All results are integrated over a sample of phylogenetic trees to account for phylogenetic uncertainty. We implement the method in a program called RJ Discrete and illustrate it by analyzing the question of whether mating system and advertisement of estrus by females have coevolved in the Old World monkeys and great apes.