Statistical models for longitudinal zero‐inflated count data with applications to the substance abuse field

This study fills in the current knowledge gaps in statistical analysis of longitudinal zero‐inflated count data by providing a comprehensive review and comparison of the hurdle and zero‐inflated Poisson models in terms of the conceptual framework, computational advantage, and performance under different real data situations. The design of simulations represents the special features of a well‐known longitudinal study of alcoholism so that the results can be generalizable to the substance abuse field. When the hurdle model is more natural under the conceptual framework of the data, the zero‐inflated Poisson model tends to produce inaccurate estimates. Model performance improves with larger sample sizes, lower proportions of missing data, and lower correlations between covariates. The simulation also shows that the computational strength of the hurdle model disappears when random effects are included. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian joint ordinal and survival modeling for breast cancer risk assessment

We propose a joint model to analyze the structure and intensity of the association between longitudinal measurements of an ordinal marker and time to a relevant event. The longitudinal process is defined in terms of a proportional‐odds cumulative logit model. Time‐to‐event is modeled through a left‐truncated proportional‐hazards model, which incorporates information of the longitudinal marker as well as baseline covariates. Both longitudinal and survival processes are connected by means of a common vector of random effects. General inferences are discussed under the Bayesian approach and include the posterior distribution of the probabilities associated to each longitudinal category and the assessment of the impact of the baseline covariates and the longitudinal marker on the hazard function. The flexibility provided by the joint model makes possible to dynamically estimate individual event‐free probabilities and predict future longitudinal marker values. The model is applied to the assessment of breast cancer risk in women attending a population‐based screening program. The longitudinal ordinal marker is mammographic breast density measured with the Breast Imaging Reporting and Data System (BI‐RADS) scale in biennial screening exams. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt‐PA stroke trial

Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non‐ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non‐ignorable missing data caused by the failure times. We apply the method to the NINDS rt‐PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales, can be treated in the same way. Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐varying effect moderation using the structural nested mean model: estimation using inverse‐weighted regression with residuals

This article considers the problem of examining time‐varying causal effect moderation using observational, longitudinal data in which treatment, candidate moderators, and possible confounders are time varying. The structural nested mean model (SNMM) is used to specify the moderated time‐varying causal effects of interest in a conditional mean model for a continuous response given time‐varying treatments and moderators. We present an easy‐to‐use estimator of the SNMM that combines an existing regression‐with‐residuals (RR) approach with an inverse‐probability‐of‐treatment weighting (IPTW) strategy. The RR approach has been shown to identify the moderated time‐varying causal effects if the time‐varying moderators are also the sole time‐varying confounders. The proposed IPTW+RR approach provides estimators of the moderated time‐varying causal effects in the SNMM in the presence of an additional, auxiliary set of known and measured time‐varying confounders. We use a small simulation experiment to compare IPTW+RR versus the traditional regression approach and to compare small and large sample properties of asymptotic versus bootstrap estimators of the standard errors for the IPTW+RR approach. This article clarifies the distinction between time‐varying moderators and time‐varying confounders. We illustrate the methodology in a case study to assess if time‐varying substance use moderates treatment effects on future substance use. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint mixed‐effects models for causal inference with longitudinal data

Causal inference with observational longitudinal data and time‐varying exposures is complicated due to the potential for time‐dependent confounding and unmeasured confounding. Most causal inference methods that handle time‐dependent confounding rely on either the assumption of no unmeasured confounders or the availability of an unconfounded variable that is associated with the exposure (eg, an instrumental variable). Furthermore, when data are incomplete, validity of many methods often depends on the assumption of missing at random. We propose an approach that combines a parametric joint mixed‐effects model for the study outcome and the exposure with g‐computation to identify and estimate causal effects in the presence of time‐dependent confounding and unmeasured confounding. G‐computation can estimate participant‐specific or population‐average causal effects using parameters of the joint model. The joint model is a type of shared parameter model where the outcome and exposure‐selection models share common random effect(s). We also extend the joint model to handle missing data and truncation by death when missingness is possibly not at random. We evaluate the performance of the proposed method using simulation studies and compare the method to both linear mixed‐ and fixed‐effects models combined with g‐computation as well as to targeted maximum likelihood estimation. We apply the method to an epidemiologic study of vitamin D and depressive symptoms in older adults and include code using SAS PROC NLMIXED software to enhance the accessibility of the method to applied researchers.     

Two‐stage model for time‐varying effects of discrete longitudinal covariates with applications in analysis of daily process data

This study proposes a generalized time‐varying effect model that can be used to characterize a discrete longitudinal covariate process and its time‐varying effect on a later outcome that may be discrete. The proposed method can be applied to examine two important research questions for daily process data: measurement reactivity and predictive validity. We demonstrate these applications using health risk behavior data collected from alcoholic couples through an interactive voice response system. The statistical analysis results show that the effect of measurement reactivity may only be evident in the first week of interactive voice response assessment. Moreover, the level of urge to drink before measurement reactivity takes effect may be more predictive of a later depression outcome. Our simulation study shows that the performance of the proposed method improves with larger sample sizes, more time points, and smaller proportions of zeros in the binary longitudinal covariate. Copyright © 2014 John Wiley & Sons, Ltd.     

A latent factor linear mixed model for high‐dimensional longitudinal data analysis

High‐dimensional longitudinal data involving latent variables such as depression and anxiety that cannot be quantified directly are often encountered in biomedical and social sciences. Multiple responses are used to characterize these latent quantities, and repeated measures are collected to capture their trends over time. Furthermore, substantive research questions may concern issues such as interrelated trends among latent variables that can only be addressed by modeling them jointly. Although statistical analysis of univariate longitudinal data has been well developed, methods for modeling multivariate high‐dimensional longitudinal data are still under development. In this paper, we propose a latent factor linear mixed model (LFLMM) for analyzing this type of data. This model is a combination of the factor analysis and multivariate linear mixed models. Under this modeling framework, we reduced the high‐dimensional responses to low‐dimensional latent factors by the factor analysis model, and then we used the multivariate linear mixed model to study the longitudinal trends of these latent factors. We developed an expectation–maximization algorithm to estimate the model. We used simulation studies to investigate the computational properties of the expectation–maximization algorithm and compare the LFLMM model with other approaches for high‐dimensional longitudinal data analysis. We used a real data example to illustrate the practical usefulness of the model. Copyright © 2013 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.     

A generalized partially linear mean‐covariance regression model for longitudinal proportional data,with applications to the analysis of quality of life data from cancer clinical trials

Motivated by the analysis of quality of life data from a clinical trial on early breast cancer, we propose in this paper a generalized partially linear mean‐covariance regression model for longitudinal proportional data, which are bounded in a closed interval. Cholesky decomposition of the covariance matrix for within‐subject responses and generalized estimation equations are used to estimate unknown parameters and the nonlinear function in the model. Simulation studies are performed to evaluate the performance of the proposed estimation procedures. Our new model is also applied to analyze the data from the cancer clinical trial that motivated this research. In comparison with available models in the literature, the proposed model does not require specific parametric assumptions on the density function of the longitudinal responses and the probability function of the boundary values and can capture dynamic changes of time or other interested variables on both mean and covariance of the correlated proportional responses. Copyright © 2017 John Wiley & Sons, Ltd.     

Three‐part joint modeling methods for complex functional data mixed with zero‐and‐one–inflated proportions and zero‐inflated continuous outcomes with skewness

We take a functional data approach to longitudinal studies with complex bivariate outcomes. This work is motivated by data from a physical activity study that measured 2 responses over time in 5‐minute intervals. One response is the proportion of time active in each interval, a continuous proportions with excess zeros and ones. The other response, energy expenditure rate in the interval, is a continuous variable with excess zeros and skewness. This outcome is complex because there are 3 possible activity patterns in each interval (inactive, partially active, and completely active), and those patterns, which are observed, induce both nonrandom and random associations between the responses. More specifically, the inactive pattern requires a zero value in both the proportion for active behavior and the energy expenditure rate; a partially active pattern means that the proportion of activity is strictly between zero and one and that the energy expenditure rate is greater than zero and likely to be moderate, and the completely active pattern means that the proportion of activity is exactly one, and the energy expenditure rate is greater than zero and likely to be higher. To address these challenges, we propose a 3‐part functional data joint modeling approach. The first part is a continuation‐ratio model to reorder the ordinal valued 3 activity patterns. The second part models the proportions when they are in interval (0,1). The last component specifies the skewed continuous energy expenditure rate with Box‐Cox transformations when they are greater than zero. In this 3‐part model, the regression structures are specified as smooth curves measured at various time points with random effects that have a correlation structure. The smoothed random curves for each variable are summarized using a few important principal components, and the association of the 3 longitudinal components is modeled through the association of the principal component scores. The difficulties in handling the ordinal and proportional variables are addressed using a quasi‐likelihood type approximation. We develop an efficient algorithm to fit the model that also involves the selection of the number of principal components. The method is applied to physical activity data and is evaluated empirically by a simulation study.     

Regression analysis using dependent Polya trees

Many commonly used models for linear regression analysis force overly simplistic shape and scale constraints on the residual structure of data. We propose a semiparametric Bayesian model for regression analysis that produces data‐driven inference by using a new type of dependent Polya tree prior to model arbitrary residual distributions that are allowed to evolve across increasing levels of an ordinal covariate (e.g., time, in repeated measurement studies). By modeling residual distributions at consecutive covariate levels or time points using separate, but dependent Polya tree priors, distributional information is pooled while allowing for broad pliability to accommodate many types of changing residual distributions. We can use the proposed dependent residual structure in a wide range of regression settings, including fixed‐effects and mixed‐effects linear and nonlinear models for cross‐sectional, prospective, and repeated measurement data. A simulation study illustrates the flexibility of our novel semiparametric regression model to accurately capture evolving residual distributions. In an application to immune development data on immunoglobulin G antibodies in children, our new model outperforms several contemporary semiparametric regression models based on a predictive model selection criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

Individualizing drug dosage with longitudinal data

We propose a two‐step procedure to personalize drug dosage over time under the framework of a log‐linear mixed‐effect model. We model patients' heterogeneity using subject‐specific random effects, which are treated as the realizations of an unspecified stochastic process. We extend the conditional quadratic inference function to estimate both fixed‐effect coefficients and individual random effects on a longitudinal training data sample in the first step and propose an adaptive procedure to estimate new patients' random effects and provide dosage recommendations for new patients in the second step. An advantage of our approach is that we do not impose any distribution assumption on estimating random effects. Moreover, the new approach can accommodate more general time‐varying covariates corresponding to random effects. We show in theory and numerical studies that the proposed method is more efficient compared with existing approaches, especially when covariates are time varying. In addition, a real data example of a clozapine study confirms that our two‐step procedure leads to more accurate drug dosage recommendations. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A goodness‐of‐fit test for the ordered stereotype model

This paper presents a new goodness‐of‐fit test for an ordered stereotype model used for an ordinal response variable. The proposed test is based on the well‐known Hosmer–Lemeshow test and its version for the proportional odds regression model. The latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The proposed test takes the use of this new adjusted spacing to partition data. A simulation study shows good performance of the proposed test under a variety of scenarios. Finally, the results of the application in two examples are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic predictions using flexible joint models of longitudinal and time‐to‐event data

Joint models for longitudinal and time‐to‐event data are particularly relevant to many clinical studies where longitudinal biomarkers could be highly associated with a time‐to‐event outcome. A cutting‐edge research direction in this area is dynamic predictions of patient prognosis (e.g., survival probabilities) given all available biomarker information, recently boosted by the stratified/personalized medicine initiative. As these dynamic predictions are individualized, flexible models are desirable in order to appropriately characterize each individual longitudinal trajectory. In this paper, we propose a new joint model using individual‐level penalized splines (P‐splines) to flexibly characterize the coevolution of the longitudinal and time‐to‐event processes. An important feature of our approach is that dynamic predictions of the survival probabilities are straightforward as the posterior distribution of the random P‐spline coefficients given the observed data is a multivariate skew‐normal distribution. The proposed methods are illustrated with data from the HIV Epidemiology Research Study. Our simulation results demonstrate that our model has better dynamic prediction performance than other existing approaches. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Bayesian joint modeling of longitudinal and spatial survival AIDS data

Joint analysis of longitudinal and survival data has received increasing attention in the recent years, especially for analyzing cancer and AIDS data. As both repeated measurements (longitudinal) and time‐to‐event (survival) outcomes are observed in an individual, a joint modeling is more appropriate because it takes into account the dependence between the two types of responses, which are often analyzed separately. We propose a Bayesian hierarchical model for jointly modeling longitudinal and survival data considering functional time and spatial frailty effects, respectively. That is, the proposed model deals with non‐linear longitudinal effects and spatial survival effects accounting for the unobserved heterogeneity among individuals living in the same region. This joint approach is applied to a cohort study of patients with HIV/AIDS in Brazil during the years 2002–2006. Our Bayesian joint model presents considerable improvements in the estimation of survival times of the Brazilian HIV/AIDS patients when compared with those obtained through a separate survival model and shows that the spatial risk of death is the same across the different Brazilian states. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison of mixed-effects quantile stratification propensity adjustment strategies for longitudinal treatment effectiveness analyses of continuous outcomes

The propensity adjustment is used to reduce bias in treatment effectiveness estimates from observational data. We show here that a mixed-effects implementation of the propensity adjustment can reduce bias in longitudinal studies of non-equivalent comparison groups. The strategy examined here involves two stages. Initially, a mixed-effects ordinal logistic regression model of propensity for treatment intensity includes variables that differentiate subjects who receive various doses of time-varying treatments. Second, a mixed-effects linear regression model compares the effectiveness of those ordinal doses on a continuous outcome over time. Here, a simulation study compares bias reduction that is achieved by implementing this propensity adjustment through various forms of stratification. The simulations demonstrate that bias decreased monotonically as the number of quantiles used for stratification increased from two to five. This was particularly pronounced with stronger effects of the confounding variables. The quartile and quintile strategies typically removed in excess of 80-90 per cent of the bias detected in unadjusted models; whereas a median-split approach removed from 20 to 45 per cent of bias. The approach is illustrated in an evaluation of the effectiveness of somatic treatments for major depression in a longitudinal, observational study of affective disorders.     

An index of local sensitivity to non‐ignorability for multivariate longitudinal mixed data with potential non‐random dropout

Multivariate longitudinal data with mixed continuous and discrete responses with the possibility of non‐ignorable missingness are often common in follow‐up medical studies and their analysis needs to be developed. Standard methods of analysis based on the strong and the unverifiable assumption of missing at random (MAR) mechanism could be highly misleading. A way out of this problem is to start with methods that simultaneously allow modelling non‐ignorable mechanism, which includes somehow troubling computations that are often time consuming, then we can use a sensitivity analysis, in which one estimates models under a range of assumptions about non‐ignorability parameters to study the impact of these parameters on key inferences. A general index of sensitivity to non‐ignorability (ISNI) to measure sensitivity of key inferences in a neighborhood of MAR model without fitting a complicated not MAR (NMAR) model for univariate generalized linear models and for models used for univariate longitudinal normal and non‐Gaussian data with potentially NMAR dropout are well presented in the literature. In this paper we extend ISNI methodology to analyze multivariate longitudinal mixed data subject to non‐ignorable dropout in which the non‐ignorable dropout model could be dependent on the mixed responses. The approach is illustrated by analyzing a longitudinal data set in which the general substantive goal of the study is to better understand the relations between parental assessment of child's antisocial behavior and child's reading recognition skill. Copyright © 2010 John Wiley & Sons, Ltd.     

Subjective prior distributions for modeling longitudinal continuous outcomes with non‐ignorable dropout

Substance abuse treatment research is complicated by the pervasive problem of non‐ignorable missing data—i.e. the occurrence of the missing data is related to the unobserved outcomes. Missing data frequently arise due to early client departure from treatment. Pattern‐mixture models (PMMs) are often employed in such situations to jointly model the outcome and the missing data mechanism. PMMs require non‐testable assumptions to identify model parameters. Several approaches to parameter identification have therefore been explored for longitudinal modeling of continuous outcomes, and informative priors have been developed in other contexts. In this paper, we describe an expert interview conducted with five substance abuse treatment clinical experts who have familiarity with the therapeutic community modality of substance abuse treatment and with treatment process scores collected using the Dimensions of Change Instrument. The goal of the interviews was to obtain expert opinion about the rate of change in continuous client‐level treatment process scores for clients who leave before completing two assessments and whose rate of change (slope) in treatment process scores is unidentified by the data. We find that the experts' opinions differed dramatically from widely utilized assumptions used to identify parameters in the PMM. Further, subjective prior assessment allows one to properly address the uncertainty inherent in the subjective decisions required to identify parameters in the PMM and to measure their effect on conclusions drawn from the analysis. Copyright © 2008 John Wiley & Sons, Ltd.