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Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Joint mean-covariance regression modeling with unconstrained parametrization for continuous longitudinal data has provided statisticians and practitioners with a powerful analytical device. How to develop a delineation of such a regression framework amongst discrete longitudinal responses remains an open and more challenging problem. This paper studies a novel mean-correlation regression for a family of generic discrete responses. Targeting the joint distributions of the discrete longitudinal responses, our regression approach is constructed by using a copula model whose correlation parameters are represented in hyperspherical coordinates with no constraint on their support. To overcome computational intractability in maximizing the full likelihood function of the model, we propose a computationally efficient pairwise likelihood approach. A pairwise likelihood ratio test is then constructed and validated for statistical inferences. We show that the resulting estimators of our approaches are consistent and asymptotically normal. We demonstrate the effectiveness, parsimoniousness and desirable performance of the proposed approach by analyzing three data sets and conducting extensive simulations.

Studies on event occurrence aim to investigate if and when subjects experience a particular event. The timing of events may be measured continuously using thin precise units or discretely using time periods. The latter metric of time is often used in social science research and the generalized linear model (GLM) is an appropriate model for data analysis. While the design of trials with continuous-time survival endpoints has been extensively studied, hardly any guidelines are available for trials with discrete-time survival endpoints. This article studies the relationship between sample size and power to detect a treatment effect in a trial with two treatment conditions. The authors use the exponential and the Weibull survival functions to represent constant and varying hazard rates. Furthermore, logit and complementary log—log link functions are used. For constant hazard rates, the power depends on the event proportions at the end of the trial in both treatment arms and on the number of time periods. For varying hazard rates, the power also depends on the shape of the survival functions and different power levels are observed in each time period in case the logit link function is used. For any survival function, power decreases if attrition is present. The authors provide R code to perform the power calculations.

We study discrete panel data methods where unobserved heterogeneity is revealed in a first step, in environments where population heterogeneity is not discrete. We focus on two-step grouped fixed-effects (GFE) estimators, where individuals are first classified into groups using kmeans clustering, and the model is then estimated allowing for group-specific heterogeneity. Our framework relies on two key properties: heterogeneity is a function—possibly nonlinear and time-varying—of a low-dimensional continuous latent type, and informative moments are available for classification. We illustrate the method in a model of wages and labor market participation, and in a probit model with time-varying heterogeneity. We derive asymptotic expansions of two-step GFE estimators as the number of groups grows with the two dimensions of the panel. We propose a data-driven rule for the number of groups, and discuss bias reduction and inference.

Intensive longitudinal designs are increasingly popular, as are dynamic structural equation models (DSEM) to accommodate unique features of these designs. Many helpful resources on DSEM exist, though they focus on continuous outcomes while categorical outcomes are omitted, briefly mentioned, or considered as a straightforward extension. This viewpoint regarding categorical outcomes is not unwarranted for technical audiences, but there are non-trivial nuances in model building and interpretation with categorical outcomes that are not necessarily straightforward for empirical researchers. Furthermore, categorical outcomes are common given that binary behavioral indicators or Likert responses are frequently solicited as low-burden variables to discourage participant non-response. This tutorial paper is therefore dedicated to providing an accessible treatment of DSEM in M plus exclusively for categorical outcomes. We cover the general probit model whereby the raw categorical responses are assumed to come from an underlying normal process. We cover probit DSEM and expound why existing treatments have considered categorical outcomes as a straightforward extension of the continuous case. Data from a motivating ecological momentary assessment study with a binary outcome are used to demonstrate an unconditional model, a model with disaggregated covariates, and a model for data with a time trend. We provide annotated M plus code for these models and discuss interpretation of the results. We then discuss model specification and interpretation in the case of an ordinal outcome and provide an example to highlight differences between ordinal and binary outcomes. We conclude with a discussion of caveats and extensions.

Multivariate discrete response data can be found in diverse fields, including econometrics, finance, biometrics, and psychometrics. Our contribution, through this study, is to introduce a new class of models for multivariate discrete data based on pair copula constructions (PCCs) that has two major advantages. First, by deriving the conditions under which any multivariate discrete distribution can be decomposed as a PCC, we show that discrete PCCs attain highly flexible dependence structures. Second, the computational burden of evaluating the likelihood for an m -dimensional discrete PCC only grows quadratically with m . This compares favorably to existing models for which computing the likelihood either requires the evaluation of 2 ᵐ terms or slow numerical integration methods. We demonstrate the high quality of inference function for margins and maximum likelihood estimates, both under a simulated setting and for an application to a longitudinal discrete dataset on headache severity. This article has online supplementary material.

Objectives. We estimated the incidence of homelessness during the transition to adulthood and identified the risk and protective factors that predict homelessness during this transition. Methods. Using data from the Midwest Evaluation of the Adult Functioning of Former Foster Youth, a longitudinal study of youths aging out of foster care in 3 Midwestern states, and a bounds approach, we estimated the cumulative percentage of youths who become homeless during the transition to adulthood. We also estimated a discrete time hazard model that predicted first reported episode of homelessness. Results. Youths aging out of foster care are at high risk for becoming homeless during the transition to adulthood. Between 31% and 46% of our study participants had been homeless at least once by age 26 years. Running away while in foster care, greater placement instability, being male, having a history of physical abuse, engaging in more delinquent behaviors, and having symptoms of a mental health disorder were associated with an increase in the relative risk of becoming homeless. Conclusions. Policy and practice changes are needed to reduce the risk that youths in foster care will become homeless after aging out.

Nonseparable panel models are important in a variety of economic settings, including discrete choice. This paper gives identification and estimation results for nonseparable models under time-homogeneity conditions that are like \"time is randomly assigned\" or \"time is an instrument.\" Partial-identification results for average and quantile effects are given for discrete regressors, under static or dynamic conditions, in fully nonparametric and in semiparametric models, with time effects. It is shown that the usual, linear, fixed-effects estimator is not a consistent estimator of the identified average effect, and a consistent estimator is given. A simple estimator of identified quantile treatment effects is given, providing a solution to the important problem of estimating quantile treatment effects from panel data. Bounds for overall effects in static and dynamic models are given. The dynamic bounds provide a partial-identification solution to the important problem of estimating the effect of state dependence in the presence of unobserved heterogeneity. The impact of T, the number of time periods, is shown by deriving shrinkage rates for the identified set as T grows. We also consider semiparametric, discrete-choice models and find that semiparametric panel bounds can be much tighter than nonparametric bounds. Computationally convenient methods for semiparametric models are presented. We propose a novel inference method that applies in panel data and other settings and show that it produces uniformly valid confidence regions in large samples. We give empirical illustrations.

Incentive-based public transport demand management (PTDM) can effectively mitigate overcrowding issues in crowded urban rail systems. Analyzing passengers’ behavioral responses to the incentive can guide the design, implementation, and update of PTDM strategies. Though several studies reported passengers’ responses to fare incentives, they focused on passengers’ short-term behavioral responses. Limited studies explore passengers’ longitudinal behavioral responses for different types of adopters, which is important for policy assessment and adjustment. This paper explores and models passengers’ longitudinal behavior response to a pre-peak fare discount incentive using 18 months of smartcard data in public transport in Hong Kong. We classified adopters into six types based on their temporal travel pattern changes before and after the promotion. The longitudinal analysis reveals that among all adopters, 19% of users change their departure times to take advantage of fare discounts but do not contribute to the goal of reducing peak-hour travel. However, these adopters are more likely to sustain their changed behavior in a long term which is not desired by the incentive program. The spatial analysis shows that the origin station distribution of late adopters is relatively more diverse than the early adopters with more trips starting from distant areas. The diffusion modeling shows that the majority adopters are innovators and the word-of-mouth diffusion effect (imitators) is marginal. The discrete choice model results highlight the heterogeneous impact of factors on different types of adopters and their values of time changes. The significant factors common to adopters are: departure time flexibility, the expected money savings, the required departure time changes, and work locations. The findings are useful for public transport planners and policymakers for informed incentive design and management.