Mixed Correlated Bivariate Ordinal and Negative Binomial Longitudinal Responses with Nonignorable Missing Values

Regression models with random effects are proposed for joint analysis of negative binomial and ordinal longitudinal data with nonignorable missing values under fully parametric framework. The presented model simultaneously considers a multivariate probit regression model for the missing mechanisms, which provides the ability of examining the missing data assumptions and a multivariate mixed model for the responses. Random effects are used to take into account the correlation between longitudinal responses of the same individual. A full likelihood-based approach that allows yielding maximum likelihood estimates of the model parameters is used. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the ordinal response of osteoporosis of the spine and negative binomial response is the number of joint damage. A sensitivity of the results to the assumptions is also investigated. The effect of some covariates on all responses are investigated simultaneously.     

Asymptotic inference in discrete response models

Regression models for discrete responses have found numerous applications. We consider logit, probit and cumulative logit models for qualitative data, and the loglinear and linear Poisson model for counted data. Statistical analysis of these models relies heavily on asymptotic likelihood theory, i.e. asymptotic properties of the maximum likelihood estimator and the likelihood ratio as well as related test statistics. In practical situations, previously published conditions assuring these properties may be too strong, or it is difficult to see whether they apply. This paper contributes to a clarification of this point and characterizes to some extent situations where asymptotic theory is applicable and where it is not. In particular, sharp upper bounds on the admissible growth of regressors are given.     

A generalized ordered Probit model

Abstract

The ordered probit and logit models, based on the normal and logistic distributions, can yield biased and inconsistent estimators when the distributions are misspecified. A generalized ordered response model is introduced which can reduce the impact of distributional misspecification. An empirical exploration of various determinants of life satisfaction suggests possible benefits of allowing for diverse distributional characteristics. These improvements are confirmed using a Monte Carlo study to contrast the performance of the flexible parametric specifications to the probit and logit specifications.     

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.     

Regression Analysis of Multivariate Fractional Data

The present article discusses alternative regression models and estimation methods for dealing with multivariate fractional response variables. Both conditional mean models, estimable by quasi-maximum likelihood, and fully parametric models (Dirichlet and Dirichlet-multinomial), estimable by maximum likelihood, are considered. A new parameterization is proposed for the parametric models, which accommodates the most common specifications for the conditional mean (e.g., multinomial logit, nested logit, random parameters logit, dogit). The text also discusses at some length the specification analysis of fractional regression models, proposing several tests that can be performed through artificial regressions. Finally, an extensive Monte Carlo study evaluates the finite sample properties of most of the estimators and tests considered.     

Probit and Logit Model Selection

Monte Carlo experiments are conducted to compare the Bayesian and sample theory model selection criteria in choosing the univariate probit and logit models. We use five criteria: the deviance information criterion (DIC), predictive deviance information criterion (PDIC), Akaike information criterion (AIC), weighted, and unweighted sums of squared errors. The first two criteria are Bayesian while the others are sample theory criteria. The results show that if data are balanced none of the model selection criteria considered in this article can distinguish the probit and logit models. If data are unbalanced and the sample size is large the DIC and AIC choose the correct models better than the other criteria. We show that if unbalanced binary data are generated by a leptokurtic distribution the logit model is preferred over the probit model. The probit model is preferred if unbalanced data are generated by a platykurtic distribution. We apply the model selection criteria to the probit and logit models that link the ups and downs of the returns on S&P500 to the crude oil price.     

A non-normal class of distribution function for dose-binary response curve*

A non-normal class of distribution (Edgeworth Series distribution) function in three and four parameters has been considered for dose-binary response relationship. This class accounts for the non-normality (expressed in terms of skewness and kurtosis) present in the relationship in addition to the usual location and scale parameters (generally considered by two parameter models). We present the maximum likelihood method of estimation of the parameters and test of probit (normal distribution) hypothesis. Edgeworth Series distribution when fitted to the data of Milicer & Szczotka (1966) showed an excellent closeness to the observed values, significant improvement over probit and logit fit (Aranda-Ordaz, 1981), and better fit compared to Prentice (1976) model.     

Confirmatory Factor Analysis of Ordinal Data Using Full‐Information Adaptive Quadrature

We conducted confirmatory factor analysis (CFA) of responses (N=803) to a self‐reported measure of optimism, using full‐information estimation via adaptive quadrature (AQ), an alternative estimation method for ordinal data. We evaluated AQ results in terms of the number of iterations required to achieve convergence, model fit, parameter estimates, standard errors (SE), and statistical significance, across four link‐functions (logit, probit, log‐log, complimentary log‐log) using 3–10 and 20 quadrature points. We compared AQ results with those obtained using maximum likelihood, robust maximum likelihood, and robust diagonally weighted least‐squares estimation. Compared to the other two link‐functions, logit and probit not only produced fit statistics, parameters estimates, SEs, and levels of significance that varied less across numbers of quadrature points, but also fitted the data better and provided larger completely standardised loadings than did maximum likelihood and diagonally weighted least‐squares. Our findings demonstrate the viability of using full‐information AQ to estimate CFA models with real‐world ordinal data.     

A Monte Carlo comparison of alternative methods of maximum likelihood ranking in racing sports

Applications of maximum likelihood techniques to rank competitors in sports are commonly based on the assumption that each competitor's performance is a function of a deterministic component that represents inherent ability and a stochastic component that the competitor has limited control over. Perhaps based on an appeal to the central limit theorem, the stochastic component of performance has often been assumed to be a normal random variable. However, in the context of a racing sport, this assumption is problematic because the resulting model is the computationally difficult rank-ordered probit. Although a rank-ordered logit is a viable alternative, a Thurstonian paired-comparison model could also be applied. The purpose of this analysis was to compare the performance of the rank-ordered logit and Thurstonian paired-comparison models given the objective of ranking competitors based on ability. Monte Carlo simulations were used to generate race results based on a known ranking of competitors, assign rankings from the results of the two models, and judge performance based on Spearman's rank correlation coefficient. Results suggest that in many applications, a Thurstonian model can outperform a rank-ordered logit if each competitor's performance is normally distributed.     

On probit versus logit dynamic mixed models for binary panel data

In this paper, we consider inferences in a binary dynamic mixed model. The existing estimation approaches mainly estimate the regression effects and the dynamic dependence parameters either through the estimation of the random effects or by avoiding the random effects technically. Under the assumption that the random effects follow a Gaussian distribution, we propose a generalized quasilikelihood (GQL) approach for the estimation of the parameters of the dynamic mixed models. The proposed approach is computationally less cumbersome than the exact maximum likelihood (ML) approach. We also carry out the GQL estimation under two competitive, namely, probit and logit mixed models, and discuss both the asymptotic and small-sample behaviour of their estimators.     

Two-Part Models for Analysis of Agatston Scores with Possible Proportionality Constraints

ABSTRACT

Logit-linear and probit-linear two-part models can be used to analyze data that are a mixture of zeros and positive continuous responses. The slopes in the linear part of a model can be constrained to be proportional to the slopes in the logit or probit part. In this article, it is shown that implementing such a constraint will decrease (in Loewner ordering) the asymptotic covariance matrix of the maximum likelihood estimates. A case study is provided using coronary artery calcification data from the Multi-Ethnic Study of Atherosclerosis.     

Consistency of the estimator of binary response models based on AUC maximization

This paper examines the asymptotic properties of a binary response model estimator based on maximization of the Area Under receiver operating characteristic Curve (AUC). Given certain assumptions, AUC maximization is a consistent method of binary response model estimation up to normalizations. As AUC is equivalent to Mann-Whitney U statistics and Wilcoxon test of ranks, maximization of area under ROC curve is equivalent to the maximization of corresponding statistics. Compared to parametric methods, such as logit and probit, AUC maximization relaxes assumptions about error distribution, but imposes some restrictions on the distribution of explanatory variables, which can be easily checked, since this information is observable.     

Decriminalization and Marijuana Smoking Prevalence: Evidence From Australia

This article uses the 2001 National Drug Strategy Household Survey to assess the impact of marijuana decriminalization policy on marijuana smoking prevalence in Australia. Both parametric and nonparametric methods are used. The parametric approach includes endogenous probit switching, two-part, sample selection, and standard dummy variable models, while the nonparametric approach uses propensity score stratification matching. Specification analyses are also conducted. A nonparametric kernel-based test is constructed to select between parametric and nonparametric models, and the likelihood ratio test is used to choose among parametric models. Our analyses favor the endogenous switching model where decriminalization increases the probability of smoking by 16.2%.     

Specification tests in ordered logit and probit models

In this paper, I study the application of various specification tests to ordered logit and probit models with heteroskedastic errors, with the primary focus on the ordered probit model. The tests are Lagrange multiplier tests, information matrix tests, and chi-squared goodness of fit tests. The alternatives are omitted variables in the regression equation, omitted varaibles in the equation describing the heteroskedasticity, and non-logistic/non-normal errors. The alternative error distributions include a generalized logistic distribution in the ordered logit model and the Pearson family in the ordered.     

Robustness of Approaches to ROC Curve Modeling under Misspecification of the Underlying Probability Model

A variety of statistical regression models have been proposed for the comparison of ROC curves for different markers across covariate groups. Pepe developed parametric models for the ROC curve that induce a semiparametric model for the market distributions to relax the strong assumptions in fully parametric models. We investigate the analysis of the power ROC curve using these ROC-GLM models compared to the parametric exponential model and the estimating equations derived from the usual partial likelihood methods in time-to-event analyses. In exploring the robustness to violations of distributional assumptions, we find that the ROC-GLM provides an extra measure of robustness.     

A new algorithm to estimate monotone nonparametric link functions and a comparison with parametric approach

The generalized linear model (GLM) is a class of regression models where the means of the response variables and the linear predictors are joined through a link function. Standard GLM assumes the link function is fixed, and one can form more flexible GLM by either estimating the flexible link function from a parametric family of link functions or estimating it nonparametically. In this paper, we propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be monotone. It is equivalent to fit the generalized single index model with monotonicity constraint. We also conduct extensive simulation studies to compare our nonparametric approach for estimating link function with various parametric approaches, including traditional logit, probit and robit link functions, and two recently developed link functions, the generalized extreme value link and the symmetric power logit link. The simulation study shows that the link function estimated nonparametrically by our proposed algorithm performs well under a wide range of different true link functions and outperforms parametric approaches when they are misspecified. A real data example is used to illustrate the results.     

Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models

This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained. The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers. However, because the proposal distributions are available in closed form they can be implemented without numerical methods for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor, and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress. Its application to hierarchical models and Poisson regression are briefly discussed.     

Optimal Binary Prediction for Group Decision Making

We address the problem of optimally forecasting a binary variable for a heterogeneous group of decision makers facing various (binary) decision problems that are tied together only by the unknown outcome. A typical example is a weather forecaster who needs to estimate the probability of rain tomorrow and then report it to the public. Given a conditional probability model for the outcome of interest (e.g., logit or probit), we introduce the idea of maximum welfare estimation and derive conditions under which traditional estimators, such as maximum likelihood or (nonlinear) least squares, are asymptotically socially optimal even when the underlying model is misspecified.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Modelling forces of infection by using monotone local polynomials

Summary. On the basis of serological data from prevalence studies of rubella, mumps and hepatitis A, the paper describes a flexible local maximum likelihood method for the estimation of the rate at which susceptible individuals acquire infection at different ages. In contrast with parametric models that have been used before in the literature, the local polynomial likelihood method allows this age-dependent force of infection to be modelled without making any assumptions about the parametric structure. Moreover, this method allows for simultaneous nonparametric estimation of age-specific incidence and prevalence. Unconstrained models may lead to negative estimates for the force of infection at certain ages. To overcome this problem and to guarantee maximal flexibility, the local smoother can be constrained to be monotone. It turns out that different parametric and nonparametric estimates of the force of infection can exhibit considerably different qualitative features like location and the number of maxima, emphasizing the importance of a well-chosen flexible statistical model.