Estimation of a Semiparametric Recursive Bivariate Probit Model with Nonparametric Mixing

We consider an extension of the recursive bivariate probit model for estimating the effect of a binary variable on a binary outcome in the presence of unobserved confounders, nonlinear covariate effects and overdispersion. Specifically, the model consists of a system of two binary outcomes with a binary endogenous regressor which includes smooth functions of covariates, hence allowing for flexible functional dependence of the responses on the continuous regressors, and arbitrary random intercepts to deal with overdispersion arising from correlated observations on clusters or from the omission of non‐confounding covariates. We fit the model by maximizing a penalized likelihood using an Expectation‐Maximisation algorithm. The issues of automatic multiple smoothing parameter selection and inference are also addressed. The empirical properties of the proposed algorithm are examined in a simulation study. The method is then illustrated using data from a survey on health, aging and wealth.     

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.     

A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

Fast Bayesian variable screenings for binary response regressions with small sample size

Screening procedures play an important role in data analysis, especially in high-throughput biological studies where the datasets consist of more covariates than independent subjects. In this article, a Bayesian screening procedure is introduced for the binary response models with logit and probit links. In contrast to many screening rules based on marginal information involving one or a few covariates, the proposed Bayesian procedure simultaneously models all covariates and uses closed-form screening statistics. Specifically, we use the posterior means of the regression coefficients as screening statistics; by imposing a generalized g-prior on the regression coefficients, we derive the analytical form of their posterior means and compute the screening statistics without Markov chain Monte Carlo implementation. We evaluate the utility of the proposed Bayesian screening method using simulations and real data analysis. When the sample size is small, the simulation results suggest improved performance with comparable computational cost.     

Markov-switching generalized additive models

We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.     

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

Binary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.     

Using a correlated probit model approximation to estimate the variance for binary matched pairs

A correlated probit model approximation for conditional probabilities (Mendell and Elston 1974) is used to estimate the variance for binary matched pairs data by maximum likelihood. Using asymptotic data, the bias of the estimates is shown to be small for a wide range of intra-class correlations and incidences. This approximation is also compared with other recently published, or implemented, improved approximations. For the small sample examples presented, it shows a substantial advantage over other approximations. The method is extended to allow covariates for each observation, and fitting by iteratively reweighted least squares.     

Markov Poisson regression models for discrete time series. Part 1: Methodology

This paper proposes and investigates a class of Markov Poisson regression models in which Poisson rate functions of covariates are conditional on unobserved states which follow a finite-state Markov chain. Features of the proposed model, estimation, inference, bootstrap confidence intervals, model selection and other implementation issues are discussed. Monte Carlo studies suggest that the proposed estimation method is accurate and reliable for single- and multiple-subject time series data; the choice of starting probabilities for the Markov process has little eff ect on the parameter estimates; and penalized likelihood criteria are reliable for determining the number of states. Part 2 provides applications of the proposed model.     

Semiparametric models of longitudinal and time-to-event data with applications to HIV viral dynamics and CD4 counts

We propose a semiparametric approach based on proportional hazards and copula method to jointly model longitudinal outcomes and the time-to-event. The dependence between the longitudinal outcomes on the covariates is modeled by a copula-based times series, which allows non-Gaussian random effects and overcomes the limitation of the parametric assumptions in existing linear and nonlinear random effects models. A modified partial likelihood method using estimated covariates at failure times is employed to draw statistical inference. The proposed model and method are applied to analyze a set of progression to AIDS data in a study of the association between the human immunodeficiency virus viral dynamics and the time trend in the CD4/CD8 ratio with measurement errors. Simulations are also reported to evaluate the proposed model and method.     

Spatial modelling for binary data using␣a␣hidden conditional autoregressive Gaussian process: a multivariate extension of the probit model

A Bayesian approach to modelling binary data on a regular lattice is introduced. The method uses a hierarchical model where the observed data is the sign of a hidden conditional autoregressive Gaussian process. This approach essentially extends the familiar probit model to dependent data. Markov chain Monte Carlo simulations are used on real and simulated data to estimate the posterior distribution of the spatial dependency parameters and the method is shown to work well. The method can be straightforwardly extended to regression models.     

Estimating models based on Markov jump processes given fragmented observation series

We consider the problem of estimating the rate matrix governing a finite-state Markov jump process given a number of fragmented time series. We propose to concatenate the observed series and to employ the emerging non-Markov process for estimation. We describe the bias arising if standard methods for Markov processes are used for the concatenated process, and provide a post-processing method to correct for this bias. This method applies to discrete-time Markov chains and to more general models based on Markov jump processes where the underlying state process is not observed directly. This is demonstrated in detail for a Markov switching model. We provide applications to simulated time series and to financial market data, where estimators resulting from maximum likelihood methods and Markov chain Monte Carlo sampling are improved using the presented correction.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Latent variable model for mixed correlated power series and ordinal longitudinal responses with non ignorable missing values

We propose a joint model based on a latent variable for analyzing mixed power series and ordinal longitudinal data with and without missing values. A bivariate probit regression model is used for the missing mechanisms. Random effects are used to take into account the correlation between longitudinal responses. A full likelihood-based approach is used to yield maximum-likelihood estimates of the model parameters. Our model is applied to a medical data set, obtained from an observational study on women where the correlated responses are the ordinal response of osteoporosis of the spine and the power series response of the number of joint damages. Sensitivity analysis is also performed to study the influence of small perturbations of the parameters of the missing mechanisms and overdispersion of the model on likelihood displacement.     

Prediction of transplant-free survival in idiopathic pulmonary fibrosis patients using joint models for event times and mixed multivariate longitudinal data

We implement a joint model for mixed multivariate longitudinal measurements, applied to the prediction of time until lung transplant or death in idiopathic pulmonary fibrosis. Specifically, we formulate a unified Bayesian joint model for the mixed longitudinal responses and time-to-event outcomes. For the longitudinal model of continuous and binary responses, we investigate multivariate generalized linear mixed models using shared random effects. Longitudinal and time-to-event data are assumed to be independent conditional on available covariates and shared parameters. A Markov chain Monte Carlo algorithm, implemented in OpenBUGS, is used for parameter estimation. To illustrate practical considerations in choosing a final model, we fit 37 different candidate models using all possible combinations of random effects and employ a deviance information criterion to select a best-fitting model. We demonstrate the prediction of future event probabilities within a fixed time interval for patients utilizing baseline data, post-baseline longitudinal responses, and the time-to-event outcome. The performance of our joint model is also evaluated in simulation studies.     

Evaluating the performance of simple estimators for probit models with two dummy endogenous regressors

This study considers the small sample performance of approximate but simple two-stage estimators for probit models with two endogenous binary covariates. Monte Carlo simulations show that all the considered estimators, including the simulated maximum-likelihood (SML) estimation, of the trivariate probit model are biased in very small samples (N=100). With moderately small samples (N=500), some of the approximations perform as well as the SML estimator when the degree of endogeneity is not very large. Some of the approximations seem robust with higher correlations and are also promising for testing the exogeneity of binary covariates. The methods are used to estimate the impact of employment-based health insurance and health care (HC) on HC use, where the approximations seem to work at least as well as the SML and in some cases better.     

Modelling a non-stationary BINAR(1) Poisson process

ABSTRACT

Non-stationarity in bivariate time series of counts may be induced by a number of time-varying covariates affecting the bivariate responses due to which the innovation terms of the individual series as well as the bivariate dependence structure becomes non-stationary. So far, in the existing models, the innovation terms of individual INAR(1) series and the dependence structure are assumed to be constant even though the individual time series are non-stationary. Under this assumption, the reliability of the regression and correlation estimates is questionable. Besides, the existing estimation methodologies such as the conditional maximum likelihood (CMLE) and the composite likelihood estimation are computationally intensive. To address these issues, this paper proposes a BINAR(1) model where the innovation series follow a bivariate Poisson distribution under some non-stationary distributional assumptions. The method of generalized quasi-likelihood (GQL) is used to estimate the regression effects while the serial and bivariate correlations are estimated using a robust moment estimation technique. The application of model and estimation method is made in the simulated data. The GQL method is also compared with the CMLE, generalized method of moments (GMM) and generalized estimating equation (GEE) approaches where through simulation studies, it is shown that GQL yields more efficient estimates than GMM and equally or slightly more efficient estimates than CMLE and GEE.     

Robust methods for generalized linear models with nonignorable missing covariates

The EM algorithm is often used for finding the maximum likelihood estimates in generalized linear models with incomplete data. In this article, the author presents a robust method in the framework of the maximum likelihood estimation for fitting generalized linear models when nonignorable covariates are missing. His robust approach is useful for downweighting any influential observations when estimating the model parameters. To avoid computational problems involving irreducibly high‐dimensional integrals, he adopts a Metropolis‐Hastings algorithm based on a Markov chain sampling method. He carries out simulations to investigate the behaviour of the robust estimates in the presence of outliers and missing covariates; furthermore, he compares these estimates to the classical maximum likelihood estimates. Finally, he illustrates his approach using data on the occurrence of delirium in patients operated on for abdominal aortic aneurysm.     

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.     

Conditional Heteroscedasticity in Qualitative Response Models of Time Series: A Gibbs-Sampling Approach to the Bank Prime Rate

Previous time series applications of qualitative response models have ignored features of the data, such as conditional heteroscedasticity, that are routinely addressed in time series econometrics of financial data. This article addresses this issue by adding Markov-switching heteroscedasticity to a dynamic ordered probit model of discrete changes in the bank prime lending rate and estimating via the Gibbs sampler. The dynamic ordered probit model of Eichengreen, Watson, and Grossman allows for serial autocorrelation in probit analysis of a time series, and this article demonstrates the relative simplicity of estimating a dynamic ordered probit using the Gibbs sampler instead of the Eichengreen et al. maximum likelihood procedure. In addition, the extension to regime-switching parameters and conditional heteroscedasticity is easy to implement under Gibbs sampling. The article compares tests of goodness of fit between dynamic ordered probit models of the prime rate that have constant variance and conditional heteroscedasticity.