Estimating probit models with self-selected treatments

Outcomes research often requires estimating the impact of a binary treatment on a binary outcome in a non-randomized setting, such as the effect of taking a drug on mortality. The data often come from self-selected samples, leading to a spurious correlation between the treatment and outcome when standard binary dependent variable techniques, like logit or probit, are used. Intuition suggests that a two-step procedure (analogous to two-stage least squares) might be sufficient to deal with this problem if variables are available that are correlated with the treatment choice but not the outcome.This paper demonstrates the limitations of such a two-step procedure. We show that such estimators will not generally be consistent. We conduct a Monte Carlo exercise to compare the performance of the two-step probit estimator, the two-stage least squares linear probability model estimator, and the multivariate probit. The results from this exercise argue in favour of using the multivariate probit rather than the two-step or linear probability model estimators, especially when there is more than one treatment, when the average probability of the dependent variable is close to 0 or 1, or when the data generating process is not normal. We demonstrate how these different methods perform in an empirical example examining the effect of private and public insurance coverage on the mortality of HIV+ patients.     

A discrete random effects probit model with application to the demand for preventive care

I have developed a random effects probit model in which the distribution of the random intercept is approximated by a discrete density. Monte Carlo results show that only three to four points of support are required for the discrete density to closely mimic normal and chi-squared densities and provide unbiased estimates of the structural parameters and the variance of the random intercept. The empirical application shows that both observed family characteristics and unobserved family-level heterogeneity are important determinants of the demand for preventive care.     

Bayesian neural networks for bivariate binary data: an application to prostate cancer study

Prostate cancer is one of the most common cancers in American men. The cancer could either be locally confined, or it could spread outside the organ. When locally confined, there are several options for treating and curing this disease. Otherwise, surgery is the only option, and in extreme cases of outside spread, it could very easily recur within a short time even after surgery and subsequent radiation therapy. Hence, it is important to know, based on pre-surgery biopsy results how likely the cancer is organ-confined or not.The paper considers a hierarchical Bayesian neural network approach for posterior prediction probabilities of certain features indicative of non-organ confined prostate cancer. In particular, we find such probabilities for margin positivity (MP) and seminal vesicle (SV) positivity jointly. The available training set consists of bivariate binary outcomes indicating the presence or absence of the two. In addition, we have certain covariates such as prostate specific antigen (PSA), gleason score and the indicator for the cancer to be unilateral or bilateral (i.e. spread on one or both sides) in one data set and gene expression microarrays in another data set. We take a hierarchical Bayesian neural network approach to find the posterior prediction probabilities for a test and validation set, and compare these with the actual outcomes for the first data set. In case of the microarray data we use leave one out cross-validation to access the accuracy of our method. We also demonstrate the superiority of our method to the other competing methods through a simulation study. The Bayesian procedure is implemented by an application of the Markov chain Monte Carlo numerical integration technique. For the problem at hand, our Bayesian bivariate neural network procedure is shown to be superior to the classical neural network, Radford Neal's Bayesian neural network as well as bivariate logistic models to predict jointly the MP and SV in a patient in both the data sets as well as in the simulation study.     

Correlated probit analysis of repeatedly measured ordinal and continuous outcomes with application to the Health and Retirement Study

The Health and Retirement Study was designed to evaluate changes in health and labor force participation during and after the transition from working to retirement. Every 2 years, participants provided information about their self‐rated health (SRH), body mass index (BMI), smoking status, and other characteristics. Our goal was to assess the effects of smoking and gender on trajectories of change in BMI and SRH over time. Joint longitudinal analysis of outcome measures is preferable to separate analyses because it allows to account for the correlation between the measures, to test the effects of predictors while controlling type I error, and potentially to improve efficiency. However, because SRH is an ordinal measure while BMI is continuous, formulating a joint model and parameter estimation is challenging. A joint correlated probit model allowed us to seamlessly account for the correlations between the measures over time. Established estimating procedures for such models are based on quasi‐likelihood or numerical approximations that may be biased or fail to converge. Therefore, we proposed a novel expectation–maximization algorithm for parameter estimation and a Monte Carlo bootstrap approach for standard errors approximation. Expectation–maximization algorithms have been previously considered for combinations of binary and/or continuous repeated measures; however, modifications were needed to handle combinations of ordinal and continuous responses. A simulation study demonstrated that the algorithm converged and provided approximately unbiased estimates with sufficiently large sample sizes. In the Health and Retirement Study, male gender and smoking were independently associated with steeper deterioration in self‐rated health and with lower average BMI. Copyright © 2016 John Wiley & Sons, Ltd.     

Choice of dialysis treatment and type of medical unit (private vs public): application of a recursive bivariate probit

ESRD patients have to deal with two choices: the first is related to the dialysis modality; the second concerns the type of dialysis unit (public vs private) where to undertake the treatment. Such a choice is related to unobservable factors, among which there might be patients' clinical factors as well as factors related to the characteristics of each unit. We employ a recursive bivariate probit estimation on a sample of ESRD Sicilian patients in order to evaluate the impact of these factors. Results can have important implications for Sicily in order to organize dialysis services: here, in fact, the number of private centres is higher than in other Italian Regions.     

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.     

Linear mixed models for skew‐normal/independent bivariate responses with an application to periodontal disease

Bivariate clustered (correlated) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed model (LMM) framework with underlying normality assumptions of the random effects and within‐subject errors. However, such normality assumptions might be questionable if the data set particularly exhibits skewness and heavy tails. Using a Bayesian paradigm, we use the skew‐normal/independent (SNI) distribution as a tool for modeling clustered data with bivariate non‐normal responses in an LMM framework. The SNI distribution is an attractive class of asymmetric thick‐tailed parametric structure which includes the skew‐normal distribution as a special case. We assume that the random effects follow multivariate SNI distributions and the random errors follow SNI distributions which provides substantial robustness over the symmetric normal process in an LMM framework. Specific distributions obtained as special cases, viz. the skew‐t, the skew‐slash and the skew‐contaminated normal distributions are compared, along with the default skew‐normal density. The methodology is illustrated through an application to a real data which records the periodontal health status of an interesting population using periodontal pocket depth (PPD) and clinical attachment level (CAL). Copyright © 2010 John Wiley & Sons, Ltd.     

Mixed binary-continuous copula regression models with application to adverse birth outcomes

Bivariate copula regression allows for the flexible combination of two arbitrary, continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution. Motivated by the risk factors for adverse birth outcomes, many of which are dichotomous, we consider mixed binary-continuous responses that extend the bivariate continuous framework to the situation where one response variable is discrete (more precisely, binary) whereas the other response remains continuous. Utilizing the latent continuous representation of binary regression models, we implement a penalized likelihood–based approach for the resulting class of copula regression models and employ it in the context of modeling gestational age and the presence/absence of low birth weight. The analysis demonstrates the advantage of the flexible specification of regression impacts including nonlinear effects of continuous covariates and spatial effects. Our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested.     

Long-term survivor model with bivariate random effects: applications to bone marrow transplant and carcinoma study data

Cured patients (or the so-called long-term survivors) are increasingly being observed in clinical trial studies. As exemplified in two data sets, the bone marrow transplantation study for leukaemia patients and the multi-centre study for patients with carcinoma in the oropharynx, a considerable portion of the patients in these studies are deemed to be cured. With the presence of random hospital/centre effects, a long-term survivor model with bivariate random effects is proposed to analyse clustered survival data with a possible portion of cured patients. This model extends earlier work by allowing random effects in both the cured fraction and the hazard function parts to follow a bivariate normal distribution, which gives a generalized model with an additional correlation parameter governing the relationship between the recovery probability and the instantaneous failure rate due to the hospital/centre effects. By adopting the GLMM formulation, random effects are incorporated in the model via the linear predictor terms. REML estimation of parameters is achieved via the EM algorithm. Application to the two sets of data illustrates the usefulness of the proposed model. A simulation study is conducted to assess the performance of the estimators, under the proposed numerical estimation scheme.     

Joint estimation of multiple disease‐specific sensitivities and specificities via crossed random effects models for correlated reader‐based diagnostic data: application of data cloning

We present a model for describing correlated binocular data from reader‐based diagnostic studies, where the same group of readers evaluates the presence or absence of certain diseases on binocular organs (e.g., fellow eyes) of patients. Multiple random effects are incorporated to meaningfully delineate various associations in the data including crossed random effects to account for reader‐specific variability and to incorporate cross correlations. To overcome the computational complexity involved in the evaluation and maximization of the marginal likelihood, we adopt the data cloning approach, which calculates maximum likelihood estimates under the Bayesian paradigm. The bias and efficiency of the estimates are assessed in two simulation studies. We apply our model to data from a diabetic retinopathy study. Copyright © 2015 John Wiley & Sons, Ltd.     

The effect of information in the utilization of preventive health-care strategies: an application to breast cancer

This paper investigates the net benefit of mammography. A theoretical expected utility (EU) model shows that increases in breast cancer risk, decreases in false-negative and false-positive rates, decreases in cost and increases/decreases in quality of life with early/late-stage breast cancer increase the net benefit of mammography. The theoretical findings are tested in an empirical analysis using Canadian data. The empirical results are broadly consistent with the EU hypothesis. Results suggest that women at higher risk are more likely to obtain a mammogram. In particular, individuals are significantly more likely to have had a time-appropriate mammogram if the mother's cause of death was breast cancer, and if the sister had breast cancer. The results also show that older age (related to higher risk and more accurate mammograms) increases mammography use, and that decreases in time and opportunity costs, and better health behaviours generally have the same effect.     

Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection

Several methodological issues occur in the context of the longitudinal study of HIV markers evolution. Three of them are of particular importance: (i) correlation between CD4+ T lymphocytes (CD4+) and plasma HIV RNA; (ii) left-censoring of HIV RNA due to a lower quantification limit; (iii) and potential informative dropout. We propose a likelihood inference for a parametric joint model including a bivariate linear mixed model for the two markers and a lognormal survival model for the time to drop out. We apply the model to data from patients starting antiretroviral treatment in the CASCADE collaboration where all of the three issues needed to be addressed.     

Multiple indicator hidden Markov model with an application to medical utilization data

Monthly counts of medical visits across several years for persons identified to have alcoholism problems are modeled using two-state hidden Markov models (HMM) in order to describe the effect of alcoholism treatment on the likelihood of persons to be in a 'healthy' or 'unhealthy' state. The medical visits can be classified into different types leading to multivariate counts of medical visits each month. A multiple indicator HMM is introduced, which simultaneously fits the multivariate Poisson counts by assuming a shared hidden state underlying all of them. The multiple indicator HMM borrows information across different types of medical encounters. A univariate HMM based on the total count across types of medical visits each month is also considered. Comparisons between the multiple indicator HMM and the total count HMM are made, as well as comparisons with more traditional longitudinal models that directly model the counts. A Bayesian framework is used for the estimation of the HMM and implementation is in Winbugs.     

Hidden Markov models for zero-inflated Poisson counts with an application to substance use

Paradigms for substance abuse cue-reactivity research involve pharmacological or stressful stimulation designed to elicit stress and craving responses in cocaine-dependent subjects. It is unclear as to whether stress induced from participation in such studies increases drug-seeking behavior. We propose a 2-state Hidden Markov model to model the number of cocaine abuses per week before and after participation in a stress-and cue-reactivity study. The hypothesized latent state corresponds to 'high' or 'low' use. To account for a preponderance of zeros, we assume a zero-inflated Poisson model for the count data. Transition probabilities depend on the prior week's state, fixed demographic variables, and time-varying covariates. We adopt a Bayesian approach to model fitting, and use the conditional predictive ordinate statistic to demonstrate that the zero-inflated Poisson hidden Markov model outperforms other models for longitudinal count data.     

Adjustment for selection bias in cohort studies: An application of a probit model with selectivity to life course epidemiology

Sample attrition is potentially a source of bias in cohort studies. The outcome may not be observed in a considerable proportion of the subjects. This article proposes the application of a probit model with sample selection to handle the problem. Two equations are simultaneously estimated and their error terms allowed to correlate: one regressing an observed outcome on a set of baseline variables, another regressing the probability of the outcome being observed upon a set of (perhaps the same) baseline variables. The method was applied to a study of a birth cohort, half of whose members were interviewed again at age 26. Baseline variables were observed for all the subjects included. The focus was on the association between birth weight and mental health in adults. The probit model with sample selection revealed a stronger and more significant (P = 0.037) relation between birth weight and mental health than an ordinary probit regression model (P = 0.170). Interpretation and practical considerations are discussed.     

Relational event models for longitudinal network data with an application to interhospital patient transfers

The main objective of this paper is to introduce and illustrate relational event models, a new class of statistical models for the analysis of time‐stamped data with complex temporal and relational dependencies. We outline the main differences between recently proposed relational event models and more conventional network models based on the graph‐theoretic formalism typically adopted in empirical studies of social networks. Our main contribution involves the definition and implementation of a marked point process extension of currently available models. According to this approach, the sequence of events of interest is decomposed into two components: (a) event time and (b) event destination. This decomposition transforms the problem of selection of event destination in relational event models into a conditional multinomial logistic regression problem. The main advantages of this formulation are the possibility of controlling for the effect of event‐specific data and a significant reduction in the estimation time of currently available relational event models. We demonstrate the empirical value of the model in an analysis of interhospital patient transfers within a regional community of health care organizations. We conclude with a discussion of how the models we presented help to overcome some the limitations of statistical models for networks that are currently available. Copyright © 2017 John Wiley & Sons, Ltd.     

Robust analysis in joint models: an application to a study on muscular dystrophy

Generalized partial ordinal models occur frequently in biomedical investigations where, along with ordinal longitudinal outcomes, there are time‐dependent covariates that act nonparametrically. In these studies, an association between such outcomes and time to an event is of considerable interest to medical practitioners. The primary objective in the present article is to study the robustness of estimators of the parameters of interest in a joint generalized partial ordinal models and a time‐to‐event model, because in many situations, the estimators in such joint models are sensitive to outliers. A Monte Carlo Metropolis–Hastings Newton Raphson algorithm is proposed for robust estimation. A detailed simulation study was performed to justify the behavior of the proposed estimators. By way of motivation, we consider a data set concerning longitudinal outcomes of children involved in a study on muscular dystrophy. Our analysis revealed some interesting findings that may be useful to medical practitioners. Copyright © 2012 John Wiley & Sons, Ltd.     

Parametric accelerated failure time models with random effects and an application to kidney transplant survival

Accelerated failure time models with a shared random component are described, and are used to evaluate the effect of explanatory factors and different transplant centres on survival times following kidney transplantation. Different combinations of the distribution of the random effects and baseline hazard function are considered and the fit of such models to the transplant data is critically assessed. A mixture model that combines short- and long-term components of a hazard function is then developed, which provides a more flexible model for the hazard function. The model can incorporate different explanatory variables and random effects in each component. The model is straightforward to fit using standard statistical software, and is shown to be a good fit to the transplant data.     

Variable selection in covariate dependent random partition models: an application to urinary tract infection

Lower urinary tract symptoms can indicate the presence of urinary tract infection (UTI), a condition that if it becomes chronic requires expensive and time consuming care as well as leading to reduced quality of life. Detecting the presence and gravity of an infection from the earliest symptoms is then highly valuable. Typically, white blood cell (WBC) count measured in a sample of urine is used to assess UTI. We consider clinical data from 1341 patients in their first visit in which UTI (i.e. WBC ) is diagnosed. In addition, for each patient, a clinical profile of 34 symptoms was recorded. In this paper, we propose a Bayesian nonparametric regression model based on the Dirichlet process prior aimed at providing the clinicians with a meaningful clustering of the patients based on both the WBC (response variable) and possible patterns within the symptoms profiles (covariates). This is achieved by assuming a probability model for the symptoms as well as for the response variable. To identify the symptoms most associated to UTI, we specify a spike and slab base measure for the regression coefficients: this induces dependence of symptoms selection on cluster assignment. Posterior inference is performed through Markov Chain Monte Carlo methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Competing‐risks duration models with correlated random effects: an application to dementia patients’ transition histories

Multi‐state transition models are widely applied tools to analyze individual event histories in the medical or social sciences. In this paper, we propose the use of (discrete‐time) competing‐risks duration models to analyze multi‐transition data. Unlike conventional Markov transition models, these models allow the estimated transition probabilities to depend on the time spent in the current state. Moreover, the models can be readily extended to allow for correlated transition probabilities. A further virtue of these models is that they can be estimated using conventional regression tools for discrete‐response data, such as the multinomial logit model. The latter is implemented in many statistical software packages and can be readily applied by empirical researchers. Moreover, model estimation is feasible, even when dealing with very large data sets, and simultaneously allowing for a flexible form of duration dependence and correlation between transition probabilities. We derive the likelihood function for a model with three competing target states and discuss a feasible and readily applicable estimation method. We also present the results from a simulation study, which indicate adequate performance of the proposed approach. In an empirical application, we analyze dementia patients’ transition probabilities from the domestic setting, taking into account several, partly duration‐dependent covariates. Copyright © 2014 John Wiley & Sons, Ltd.