Discrete Response Data Research Articles

Mapping socio-economic status using mixed data: a hierarchical Bayesian approach

Abstract We propose a Bayesian hierarchical model to estimate a socio-economic status (SES) index based on mixed dichotomous and continuous variables. In particular, we extend Quinn’s ([2004]. Bayesian factor analysis for mixed ordinal and continuous responses. Political Analysis, 12(4), 338–353. https://doi.org/10.1093/pan/mph022) and Schliep and Hoeting’s ([2013]. Multilevel latent Gaussian process model for mixed discrete and continuous multivariate response data. Journal of Agricultural, Biological, and Environmental Statistics, 18(4), 492–513. https://doi.org/10.1007/s13253-013-0136-z) factor analysis models for mixed dichotomous and continuous variables by allowing a spatial hierarchical structure of key parameters of the model. Unlike most SES assessment models proposed in the literature, the hierarchical nature of this model enables the use of census observations at the household level without needing to aggregate any information a priori. Therefore, it better accommodates the variability of the SES between census tracts and the number of households per area. The proposed model is used in the estimation of a socio-economic index using 10% of the 2010 Ghana census in the Greater Accra Metropolitan area. Out of the 20 observed variables, the number of people per room, access to water piping and flushable toilets differentiated high and low SES areas the best.

Robust discrete choice models with t-distributed kernel errors

Outliers in discrete choice response data may result from misclassification and misreporting of the response variable and from choice behaviour that is inconsistent with modelling assumptions (e.g. random utility maximisation). In the presence of outliers, standard discrete choice models produce biased estimates and suffer from compromised predictive accuracy. Robust statistical models are less sensitive to outliers than standard non-robust models. This paper analyses two robust alternatives to the multinomial probit (MNP) model. The two models are robit models whose kernel error distributions are heavy-tailed t-distributions to moderate the influence of outliers. The first model is the multinomial robit (MNR) model, in which a generic degrees of freedom parameter controls the heavy-tailedness of the kernel error distribution. The second model, the generalised multinomial robit (Gen-MNR) model, is more flexible than MNR, as it allows for distinct heavy-tailedness in each dimension of the kernel error distribution. For both models, we derive Gibbs samplers for posterior inference. In a simulation study, we illustrate the excellent finite sample properties of the proposed Bayes estimators and show that MNR and Gen-MNR produce more accurate estimates if the choice data contain outliers through the lens of the non-robust MNP model. In a case study on transport mode choice behaviour, MNR and Gen-MNR outperform MNP by substantial margins in terms of in-sample fit and out-of-sample predictive accuracy. The case study also highlights differences in elasticity estimates across models.

Discrete Choice Data with Unobserved Heterogeneity: A Conditional Binary Quantile Model

In political science, data with heterogeneous units are used in many studies, such as those involving legislative proposals in different policy areas, electoral choices by different types of voters, and government formation in varying party systems. To disentangle decision-making mechanisms by units, traditional discrete choice models focus exclusively on the conditional mean and ignore the heterogeneous effects within a population. This paper proposes a conditional binary quantile model that goes beyond this limitation to analyze discrete response data with varying alternative-specific features. This model offers an in-depth understanding of the relationship between the explanatory and response variables. Compared to conditional mean-based models, the conditional binary quantile model relies on weak distributional assumptions and is more robust to distributional misspecification. The model also relaxes the assumption of the independence of irrelevant alternatives, which is often violated in practice. The method is applied to a range of political studies to show the heterogeneous effects of explanatory variables across the conditional distribution. Substantive interpretations from counterfactual scenarios are used to illustrate how the conditional binary quantile model captures unobserved heterogeneity, which extant models fail to do. The results point to the risk of averaging out the heterogeneous effects across units by conditional mean-based models.

Uncertainty quantification for nonlinear difference equations with dependent random inputs via a stochastic Galerkin projection technique

Discrete stochastic systems model discrete response data on some phenomenon with inherent uncertainty. The main goal of uncertainty quantification is to derive the probabilistic features of the stochastic system. This paper deals with theoretical and computational aspects of uncertainty quantification for nonlinear difference equations with dependent random inputs. When the random inputs are independent random variables, a generalized Polynomial Chaos (gPC) approach has been usually used to computationally quantify the uncertainty of stochastic systems. In the gPC technique, the stochastic Galerkin projections are done onto linear spans of orthogonal polynomials from the Askey–Wiener scheme or from Gram–Schmidt orthonormalization procedures. In this regard, recent results have established the algebraic or exponential convergence of these Galerkin projections to the solution process. In this paper, as the random inputs of the difference equation may be dependent, we perform Galerkin projections directly onto linear spans of canonical polynomials. The main contribution of this paper is to study the spectral convergence of these Galerkin projections for the solution process of general random difference equations. Spectral convergence is important to derive the main statistics of the response process at a cheap computational expense. In this regard, the numerical experiments bring to light the theoretical discussion of the paper.

Multilevel quantile function modeling with application to birth outcomes.

Infants born preterm or small for gestational age have elevated rates of morbidity and mortality. Using birth certificate records in Texas from 2002 to 2004 and Environmental Protection Agency air pollution estimates, we relate the quantile functions of birth weight and gestational age to ozone exposure and multiple predictors, including parental age, race, and education level. We introduce a semi-parametric Bayesian quantile approach that models the full quantile function rather than just a few quantile levels. Our multilevel quantile function model establishes relationships between birth weight and the predictors separately for each week of gestational age and between gestational age and the predictors separately across Texas Public Health Regions. We permit these relationships to vary nonlinearly across gestational age, spatial domain and quantile level and we unite them in a hierarchical model via a basis expansion on the regression coefficients that preserves interpretability. Very low birth weight is a primary concern, so we leverage extreme value theory to supplement our model in the tail of the distribution. Gestational ages are recorded in completed weeks of gestation (integer-valued), so we present methodology for modeling quantile functions of discrete response data. In a simulation study we show that pooling information across gestational age and quantile level substantially reduces MSE of predictor effects. We find that ozone is negatively associated with the lower tail of gestational age in south Texas and across the distribution of birth weight for high gestational ages. Our methods are available in the R package BSquare.

Correction of Retransformation Bias in Nonlinear Predictions on Longitudinal Data with Generalized Linear Mixed Models

Researchers often encounter discrete response data in longitudinal analysis. Generalized linear mixed models are generally applied to account for potential lack of independence inherent in longitudinally data. When parameter estimates are used to describe longitudinal processes, random effects, both between and within subjects, need to be retransformed in nonlinear predictions on the response data; otherwise, serious retransformation bias can arise to an unanticipated extent. This study attempts to go beyond existing work by developing a retransformation method deriving statistically robust longitudinal trajectory of nonlinear predictions. Variances of population-averaged nonlinear predictions are approximated by the delta method. The empirical illustration uses longitudinal data from the Asset and Health Dynamics among the Oldest Old study. Our analysis compares three sets of nonlinear predictions of death rate at six time points, from the retransformation method, the best linear unbiased predictor, and the fixed-effects approach, respectively. The results demonstrate that failure to retransform the random components in generalized linear mixed models results in severely biased nonlinear predictions, as well as much reduced standard error approximates.

Understanding spatial filtering for analysis of land use-transport data

This paper summarizes the literature on spatial filtering (SF) for analysis of spatial data. Given the scarcity of its application in transportation and its fledgling nature, preliminary case studies were conducted using continuous and discrete response data sets, for land values and land use, in comparison with results from spatial autoregressive (SAR) models with distance decay parameters estimated using Bayesian techniques. For both the continuous land value and binary land use cases, the SF approach demonstrates great potential as a worthy competitor to more conventional SAR-based models. In addition to offering high fit statistics, somewhat shorter computing times, and more straightforward computations, the SF approach makes explicit the patterns of spatial dependency in the land value and land use data. By controlling for these spatial relationships, the SF approach yields more reliable marginal effects of policy variables of interest. Model results confirm the important role of transportation access (as quantified using distances to a region’s central business district, and various roadway types).

Adaptively combined forecasting for discrete response time series

Adaptive combining is generally a desirable approach for forecasting, which, however, has rarely been explored for discrete response time series. In this paper, we propose an adaptively combined forecasting method for such discrete response data. We demonstrate in theory that the proposed forecast is of the desired adaptation with respect to the widely used squared risk and other significant risk functions under mild conditions. Furthermore, we study the issue of adaptation for the proposed forecasting method in the presence of model screening that is often useful in applications. Our simulation study and two real-world data examples show promise for the proposed approach.

Multilevel Latent Gaussian Process Model for Mixed Discrete and Continuous Multivariate Response Data

We propose a Bayesian model for mixed ordinal and continuous multivariate data to evaluate a latent spatial Gaussian process. Our proposed model can be used in many contexts where mixed continuous and discrete multivariate responses are observed in an effort to quantify an unobservable continuous measurement. In our example, the latent, or unobservable measurement is wetland condition. While predicted values of the latent wetland condition variable produced by the model at each location do not hold any intrinsic value, the relative magnitudes of the wetland condition values are of interest. In addition, by including point-referenced covariates in the model, we are able to make predictions at new locations for both the latent random variable and the multivariate response. Lastly, the model produces ranks of the multivariate responses in relation to the unobserved latent random field. This is an important result as it allows us to determine which response variables are most closely correlated with the latent variable. Our approach offers an alternative to traditional indices based on best professional judgment that are frequently used in ecology. We apply our model to assess wetland condition in the North Platte and Rio Grande River Basins in Colorado. The model facilitates a comparison of wetland condition at multiple locations and ranks the importance of in-field measurements.

A Response Surface Methodology for Modeling Time Series Response Data

This article introduces extensions to classical response surface methods specifically for modeling and exploiting time series response data such as found in aircraft flight test. Classical response surface methods focus on the analysis of discrete response data. In some complex, nonlinear systems, analysts are concerned with the behavior of time series response data. To date, there does not appear to be any formal methods for examining and exploiting these data within a response surface framework. Thus, a mathematical framework is defined and then demonstrated through a sample analysis of empirical flight test data. Copyright © 2012 John Wiley & Sons, Ltd.

Pair Copula Constructions for Multivariate Discrete Data

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.

Kernel-smoothed conditional quantiles of correlated bivariate discrete data

Socio-economic variables are often measured on a discrete scale or rounded to protect confidentiality. Nevertheless, when exploring the effect of a relevant covariate on the outcome distribution of a discrete response variable, virtually all common quantile regression methods require the distribution of the covariate to be continuous. This paper departs from this basic requirement by presenting an algorithm for nonparametric estimation of conditional quantiles when both the response variable and the covariate are discrete. Moreover, we allow the variables of interest to be pairwise correlated. For computational efficiency, we aggregate the data into smaller subsets by a binning operation, and make inference on the resulting prebinned data. Specifically, we propose two kernel-based binned conditional quantile estimators, one for untransformed discrete response data and one for rank-transformed response data. We establish asymptotic properties of both estimators. A practical procedure for jointly selecting band- and binwidth parameters is also presented. Simulation results show excellent estimation accuracy in terms of bias, mean squared error, and confidence interval coverage. Typically prebinning the data leads to considerable computational savings when large datasets are under study, as compared to direct (un)conditional quantile kernel estimation of multivariate data. With this in mind, we illustrate the proposed methodology with an application to a large dataset concerning US hospital patients with congestive heart failure.

A mixture model for the joint analysis of latent developmental trajectories and survival

A general joint modeling framework is proposed that includes a parametric stratified survival component for continuous time survival data, and a mixture multilevel item response component to model latent developmental trajectories given mixed discrete response data. The joint model is illustrated in a real data setting, where the utility of longitudinally measured cognitive function as a predictor for survival is investigated in a group of elderly persons. The object is partly to determine whether cognitive impairment is accompanied by a higher mortality rate. Time-dependent cognitive function is measured using the generalized partial credit model given occasion-specific mini-mental state examination response data. A parametric survival model is applied for the survival information, and cognitive function as a continuous latent variable is included as a time-dependent explanatory variable along with other explanatory information. A mixture model is defined, which incorporates the latent developmental trajectory and the survival component. The mixture model captures the heterogeneity in the developmental trajectories that could not be fully explained by the multilevel item response model and other explanatory variables. A Bayesian modeling approach is pursued, where a Markov chain Monte Carlo algorithm is developed for simultaneous estimation of the joint model parameters. Practical issues as model building and assessment are addressed using the DIC and various posterior predictive tests.

Kernel-Smoothed Conditional Quantiles of Correlated Bivariate Discrete Data

Often socio-economic variables are measured on a discrete scale or rounded to protect confidentiality. Nevertheless, when exploring the effect of a relevant covariate on the whole outcome distribution of a discrete response variable, virtually all common quantile regression methods require the distribution of the covariate to be continuous. This paper departs from this basic requirement by presenting an algorithm for nonparametric estimation of conditional quantiles when both the response variable and the covariate are discretely distributed. Moreover, we allow the variables of interest to be pairwise correlated. For computational efficiency, we aggregate the data into smaller subsets by a binning operation, and make inference on the resulting prebinned data. Specifically, we propose two kernel-based binned conditional quantile estimators, one for untransformed discrete response data and one for rank-transformed response data. We establish asymptotic properties of both estimators. A practical procedure for jointly selecting band- and binwidth parameters is also presented. Simulation results show excellent estimation accuracy in terms of bias, mean squared error, and confidence interval coverage. Typically prebinning the data leads to considerable computational savings when large datasets are under study, as compared to direct (un)conditional quantile kernel estimation of multivariate data. With this in mind, we illustrate the proposed methodology with an application to a large real dataset concerning US hospital patients with congestive heart failure.

BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA WITH A DYNAMIC SPATIAL‐ORDERED PROBIT MODEL

ABSTRACT Many databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This study develops a dynamic spatial‐ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. It incorporates spatial effects in an ordered probit model by allowing for interregional spatial interactions and heteroskedasticity, along with random effects across regions or any clusters of observational units. The model assumes an autoregressive, AR(1), process across latent response values, thereby recognizing time‐series dynamics in panel data sets. The model code and estimation approach is tested on simulated data sets, in order to reproduce known parameter values and provide insights into estimation performance, yielding much more accurate estimates than standard, nonspatial techniques. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of two‐dimensional autocorrelation.

State space mixed models for longitudinal observations with binary and binomial responses

We propose a new class of state space models for longitudinal discrete response data where the observation equation is specified in an additive form involving both deterministic and random linear predictors. These models allow us to explicitly address the effects of trend, seasonal or other time-varying covariates while preserving the power of state space models in modeling serial dependence in the data. We develop a Markov chain Monte Carlo algorithm to carry out statistical inference for models with binary and binomial responses, in which we invoke de Jong and Shephard’s (Biometrika 82(2):339–350, 1995) simulation smoother to establish an efficient sampling procedure for the state variables. To quantify and control the sensitivity of posteriors on the priors of variance parameters, we add a signal-to-noise ratio type parameter in the specification of these priors. Finally, we illustrate the applicability of the proposed state space mixed models for longitudinal binomial response data in both simulation studies and data examples.

A general multilevel multistate competing risks model for event history data, with an application to a study of contraceptive use dynamics

We propose a general discrete time model for multilevel event history data. The model is developed for the analysis of longitudinal repeated episodes within individuals where there are multiple states and multiple types of event (competing risks) which may vary across states. The different transitions are modelled jointly to allow for correlation across transitions in unobserved individual risk factors. Implementation of the methodology using existing multilevel models for discrete response data is described. The model is applied in an analysis of contraceptive use dynamics in Indonesia where transitions from two states, contraceptive use and nonuse, are of interest. A distinction is made between two ways in which an episode of contraceptive use may end: a transition to nonuse or a switch to another method. Before adjusting for covariate effects, there is a strong negative residual correlation between the hazards of a transition from use to nonuse and from nonuse to use; this correlation is due to a tendency for short periods of nonuse after a birth to be followed by long periods of using the same contraceptive method.

Application of conditional moment tests to model checking for generalized linear models.

Generalized linear models (GLMs) are increasingly being used in daily data analysis. However, model checking for GLMs with correlated discrete response data remains difficult. In this paper, through a case study on marginal logistic regression using a real data set, we illustrate the flexibility and effectiveness of using conditional moment tests (CMTs), along with other graphical methods, to do model checking for generalized estimation equation (GEE) analyses. Although CMTs provide an array of powerful diagnostic tests for model checking, they were originally proposed in the econometrics literature and, to our knowledge, have never been applied to GEE analyses. CMTs cover many existing tests, including the (generalized) score test for an omitted covariate, as special cases. In summary, we believe that CMTs provide a class of useful model checking tools.

Semi-nonparametric Distribution-Free Dichotomous Choice Contingent Valuation

We apply a semi-nonparametric distribution-free estimator for binary discrete response data to the estimation of a dichotomous choice contingent valuation model. Using this estimator, mean and median compensating and equivalent variation can be consistently estimated without making nontheoretically motivated assumptions on consumer' preferences. The approach is illustrated using a contingent valuation survey of willingness to pay for reduction of risk of premature death due to exposure to hazardous waste. We find that a conventional parametric estimator and the proposed estimator give similar estimates of unconditional WTP, but that conditional on explanatory variables the estimates are quite different.

Discussion of the Paper by Lee and Nelder

Discussion of the Paper by Lee and Nelder