Joint modeling of missing data due to non‐participation and death in longitudinal aging studies

Specific age‐related hypotheses are tested in population‐based longitudinal studies. At specific time intervals, both the outcomes of interest and the time‐varying covariates are measured. When participants are approached for follow‐up, some participants do not provide data. Investigations may show that many have died before the time of follow‐up whereas others refused to participate. Some of these non‐participants do not provide data at later follow‐ups. Few statistical methods for missing data distinguish between ‘non‐participation’ and ‘death’ among study participants. The augmented inverse probability‐weighted estimators are most commonly used in marginal structure models when data are missing at random. Treating non‐participation and death as the same, however, may lead to biased estimates and invalid inferences. To overcome this limitation, a multiple inverse probability‐weighted approach is presented to account for two types of missing data, non‐participation and death, when using a marginal mean model. Under certain conditions, the multiple weighted estimators are consistent and asymptotically normal. Simulation studies will be used to study the finite sample efficiency of the multiple weighted estimators. The proposed method will be applied to study the risk factors associated with the cognitive decline among the aging adults, using data from the Chicago Health and Aging Project (CHAP). Copyright © 2010 John Wiley & Sons, Ltd.     

Functional linear models for zero‐inflated count data with application to modeling hospitalizations in patients on dialysis

We propose functional linear models for zero‐inflated count data with a focus on the functional hurdle and functional zero‐inflated Poisson (ZIP) models. Although the hurdle model assumes the counts come from a mixture of a degenerate distribution at zero and a zero‐truncated Poisson distribution, the ZIP model considers a mixture of a degenerate distribution at zero and a standard Poisson distribution. We extend the generalized functional linear model framework with a functional predictor and multiple cross‐sectional predictors to model counts generated by a mixture distribution. We propose an estimation procedure for functional hurdle and ZIP models, called penalized reconstruction, geared towards error‐prone and sparsely observed longitudinal functional predictors. The approach relies on dimension reduction and pooling of information across subjects involving basis expansions and penalized maximum likelihood techniques. The developed functional hurdle model is applied to modeling hospitalizations within the first 2 years from initiation of dialysis, with a high percentage of zeros, in the Comprehensive Dialysis Study participants. Hospitalization counts are modeled as a function of sparse longitudinal measurements of serum albumin concentrations, patient demographics, and comorbidities. Simulation studies are used to study finite sample properties of the proposed method and include comparisons with an adaptation of standard principal components regression. Copyright © 2014 John Wiley & Sons, Ltd.     

Practical recommendations for reporting Fine‐Gray model analyses for competing risk data

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause‐specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event‐free. Subdistribution hazard ratios obtained from the Fine‐Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine‐Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients.     

Time‐varying effect modeling with longitudinal data truncated by death: conditional models,interpretations, and inference

Recent studies found that infection‐related hospitalization was associated with increased risk of cardiovascular (CV) events, such as myocardial infarction and stroke in the dialysis population. In this work, we develop time‐varying effects modeling tools in order to examine the CV outcome risk trajectories during the time periods before and after an initial infection‐related hospitalization. For this, we propose partly conditional and fully conditional partially linear generalized varying coefficient models (PL‐GVCMs) for modeling time‐varying effects in longitudinal data with substantial follow‐up truncation by death. Unconditional models that implicitly target an immortal population is not a relevant target of inference in applications involving a population with high mortality, like the dialysis population. A partly conditional model characterizes the outcome trajectory for the dynamic cohort of survivors, where each point in the longitudinal trajectory represents a snapshot of the population relationships among subjects who are alive at that time point. In contrast, a fully conditional approach models the time‐varying effects of the population stratified by the actual time of death, where the mean response characterizes individual trends in each cohort stratum. We compare and contrast partly and fully conditional PL‐GVCMs in our aforementioned application using hospitalization data from the United States Renal Data System. For inference, we develop generalized likelihood ratio tests. Simulation studies examine the efficacy of estimation and inference procedures. Copyright © 2015 John Wiley & Sons, Ltd.     

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo‐likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite‐sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non‐proportionality. The relative merits of these methods are discussed in concluding remarks. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian adjustment for covariate measurement errors: A flexible parametric approach

In most epidemiological investigations, the study units are people, the outcome variable (or the response) is a health‐related event, and the explanatory variables are usually environmental and/or socio‐demographic factors. The fundamental task in such investigations is to quantify the association between the explanatory variables (covariates/exposures) and the outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely the relevant covariates are measured. In many instances, we cannot measure some of the covariates accurately. Rather, we can measure noisy (mismeasured) versions of them. In statistical terminology, mismeasurement in continuous covariates is known as measurement errors or errors‐in‐variables. Regression analyses based on mismeasured covariates lead to biased inference about the true underlying response–covariate associations. In this paper, we suggest a flexible parametric approach for avoiding this bias when estimating the response–covariate relationship through a logistic regression model. More specifically, we consider the flexible generalized skew‐normal and the flexible generalized skew‐t distributions for modeling the unobserved true exposure. For inference and computational purposes, we use Bayesian Markov chain Monte Carlo techniques. We investigate the performance of the proposed flexible parametric approach in comparison with a common flexible parametric approach through extensive simulation studies. We also compare the proposed method with the competing flexible parametric method on a real‐life data set. Though emphasis is put on the logistic regression model, the proposed method is unified and is applicable to the other generalized linear models, and to other types of non‐linear regression models as well. Copyright © 2009 John Wiley & Sons, Ltd.     

Regression modeling of longitudinal data with outcome‐dependent observation times: extensions and comparative evaluation

Conventional longitudinal data analysis methods assume that outcomes are independent of the data‐collection schedule. However, the independence assumption may be violated, for example, when a specific treatment necessitates a different follow‐up schedule than the control arm or when adverse events trigger additional physician visits in between prescheduled follow‐ups. Dependence between outcomes and observation times may introduce bias when estimating the marginal association of covariates on outcomes using a standard longitudinal regression model. We formulate a framework of outcome‐observation dependence mechanisms to describe conditional independence given observed observation‐time process covariates or shared latent variables. We compare four recently developed semi‐parametric methods that accommodate one of these mechanisms. To allow greater flexibility, we extend these methods to accommodate a combination of mechanisms. In simulation studies, we show how incorrectly specifying the outcome‐observation dependence may yield biased estimates of covariate‐outcome associations and how our proposed extensions can accommodate a greater number of dependence mechanisms. We illustrate the implications of different modeling strategies in an application to bladder cancer data. In longitudinal studies with potentially outcome‐dependent observation times, we recommend that analysts carefully explore the conditional independence mechanism between the outcome and observation‐time processes to ensure valid inference regarding covariate‐outcome associations. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian proportional hazards model with time-varying regression coefficients: a penalized Poisson regression approach

One can fruitfully approach survival problems without covariates in an actuarial way. In narrow time bins, the number of people at risk is counted together with the number of events. The relationship between time and probability of an event can then be estimated with a parametric or semi-parametric model. The number of events observed in each bin is described using a Poisson distribution with the log mean specified using a flexible penalized B-splines model with a large number of equidistant knots. Regression on pertinent covariates can easily be performed using the same log-linear model, leading to the classical proportional hazard model. We propose to extend that model by allowing the regression coefficients to vary in a smooth way with time. Penalized B-splines models will be proposed for each of these coefficients. We show how the regression parameters and the penalty weights can be estimated efficiently using Bayesian inference tools based on the Metropolis-adjusted Langevin algorithm.     

Estimating the effect of a rare time‐dependent treatment on the recurrent event rate

In many observational studies, the objective is to estimate the effect of treatment or state‐change on the recurrent event rate. If treatment is assigned after the start of follow‐up, traditional methods (eg, adjustment for baseline‐only covariates or fully conditional adjustment for time‐dependent covariates) may give biased results. We propose a two‐stage modeling approach using the method of sequential stratification to accurately estimate the effect of a time‐dependent treatment on the recurrent event rate. At the first stage, we estimate the pretreatment recurrent event trajectory using a proportional rates model censored at the time of treatment. Prognostic scores are estimated from the linear predictor of this model and used to match treated patients to as yet untreated controls based on prognostic score at the time of treatment for the index patient. The final model is stratified on matched sets and compares the posttreatment recurrent event rate to the recurrent event rate of the matched controls. We demonstrate through simulation that bias due to dependent censoring is negligible, provided the treatment frequency is low, and we investigate a threshold at which correction for dependent censoring is needed. The method is applied to liver transplant (LT), where we estimate the effect of development of post‐LT End Stage Renal Disease (ESRD) on rate of days hospitalized.     

A maximum likelihood method for secondary analysis of nested case‐control data

Many epidemiological studies use a nested case‐control (NCC) design to reduce cost while maintaining study power. Because NCC sampling is conditional on the primary outcome, routine application of logistic regression to analyze a secondary outcome will generally be biased. Recently, many studies have proposed several methods to obtain unbiased estimates of risk for a secondary outcome from NCC data. Two common features of all current methods requires that the times of onset of the secondary outcome are known for cohort members not selected into the NCC study and the hazards of the two outcomes are conditionally independent given the available covariates. This last assumption will not be plausible when the individual frailty of study subjects is not captured by the measured covariates. We provide a maximum‐likelihood method that explicitly models the individual frailties and also avoids the need to have access to the full cohort data. We derive the likelihood contribution by respecting the original sampling procedure with respect to the primary outcome. We use proportional hazard models for the individual hazards, and Clayton's copula is used to model additional dependence between primary and secondary outcomes beyond that explained by the measured risk factors. We show that the proposed method is more efficient than weighted likelihood and is unbiased in the presence of shared frailty for the primary and secondary outcome. We illustrate the method with an application to a study of risk factors for diabetes in a Swedish cohort. Copyright © 2014 John Wiley & Sons, Ltd.     

Flexible modeling of the cumulative effects of time‐dependent exposures on the hazard

Many epidemiological studies assess the effects of time‐dependent exposures, where both the exposure status and its intensity vary over time. One example that attracts public attention concerns pharmacoepidemiological studies of the adverse effects of medications. The analysis of such studies poses challenges for modeling the impact of complex time‐dependent drug exposure, especially given the uncertainty about the way effects cumulate over time and about the etiological relevance of doses taken in different time periods. We present a flexible method for modeling cumulative effects of time‐varying exposures, weighted by recency, represented by time‐dependent covariates in the Cox proportional hazards model. The function that assigns weights to doses taken in the past is estimated using cubic regression splines. We validated the method in simulations and applied it to re‐assess the association between exposure to a psychotropic drug and fall‐related injuries in the elderly. Copyright © 2009 John Wiley & Sons, Ltd.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐ignorable missingness in logistic regression

Nonresponses and missing data are common in observational studies. Ignoring or inadequately handling missing data may lead to biased parameter estimation, incorrect standard errors and, as a consequence, incorrect statistical inference and conclusions. We present a strategy for modelling non‐ignorable missingness where the probability of nonresponse depends on the outcome. Using a simple case of logistic regression, we quantify the bias in regression estimates and show the observed likelihood is non‐identifiable under non‐ignorable missing data mechanism. We then adopt a selection model factorisation of the joint distribution as the basis for a sensitivity analysis to study changes in estimated parameters and the robustness of study conclusions against different assumptions. A Bayesian framework for model estimation is used as it provides a flexible approach for incorporating different missing data assumptions and conducting sensitivity analysis. Using simulated data, we explore the performance of the Bayesian selection model in correcting for bias in a logistic regression. We then implement our strategy using survey data from the 45 and Up Study to investigate factors associated with worsening health from the baseline to follow‐up survey. Our findings have practical implications for the use of the 45 and Up Study data to answer important research questions relating to health and quality‐of‐life. Copyright © 2017 John Wiley & Sons, Ltd.     

Exploratory assessment of treatment‐dependent random‐effects distribution using gradient functions

In analyzing repeated measurements from randomized controlled trials with mixed‐effects models, it is important to carefully examine the conventional normality assumption regarding the random‐effects distribution and its dependence on treatment allocation in order to avoid biased estimation and correctly interpret the estimated random‐effects distribution. In this article, we propose the use of a gradient function method in modeling with the different random‐effects distributions depending on the treatment allocation. This method can be effective for considering in advance whether a proper fit requires a model that allows dependence of the random‐effects distribution on covariates, or for finding the subpopulations in the random effects.     

Using the jackknife for estimation in log link Bernoulli regression models

Bernoulli (or binomial) regression using a generalized linear model with a log link function, where the exponentiated regression parameters have interpretation as relative risks, is often more appropriate than logistic regression for prospective studies with common outcomes. In particular, many researchers regard relative risks to be more intuitively interpretable than odds ratios. However, for the log link, when the outcome is very prevalent, the likelihood may not have a unique maximum. To circumvent this problem, a ‘COPY method’ has been proposed, which is equivalent to creating for each subject an additional observation with the same covariates except the response variable has the outcome values interchanged (1's changed to 0's and 0's changed to 1's). The original response is given weight close to 1, while the new observation is given a positive weight close to 0; this approach always leads to convergence of the maximum likelihood algorithm, except for problems with convergence due to multicollinearity among covariates. Even though this method produces a unique maximum, when the outcome is very prevalent, and/or the sample size is relatively small, the COPY method can yield biased estimates. Here, we propose using the jackknife as a bias‐reduction approach for the COPY method. The proposed method is motivated by a study of patients undergoing colorectal cancer surgery. Copyright © 2014 John Wiley & Sons, Ltd.     

Ensemble learning of inverse probability weights for marginal structural modeling in large observational datasets

Inverse probability weights used to fit marginal structural models are typically estimated using logistic regression. However, a data‐adaptive procedure may be able to better exploit information available in measured covariates. By combining predictions from multiple algorithms, ensemble learning offers an alternative to logistic regression modeling to further reduce bias in estimated marginal structural model parameters. We describe the application of two ensemble learning approaches to estimating stabilized weights: super learning (SL), an ensemble machine learning approach that relies on V‐fold cross validation, and an ensemble learner (EL) that creates a single partition of the data into training and validation sets. Longitudinal data from two multicenter cohort studies in Spain (CoRIS and CoRIS‐MD) were analyzed to estimate the mortality hazard ratio for initiation versus no initiation of combined antiretroviral therapy among HIV positive subjects. Both ensemble approaches produced hazard ratio estimates further away from the null, and with tighter confidence intervals, than logistic regression modeling. Computation time for EL was less than half that of SL. We conclude that ensemble learning using a library of diverse candidate algorithms offers an alternative to parametric modeling of inverse probability weights when fitting marginal structural models. With large datasets, EL provides a rich search over the solution space in less time than SL with comparable results. Copyright © 2014 John Wiley & Sons, Ltd.     

Longitudinal latent variable models given incompletely observed biomarkers and covariates

In this paper, we analyze a two‐level latent variable model for longitudinal data from the National Growth and Health Study where surrogate outcomes or biomarkers and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to re‐express the desired model as a joint distribution of variables, including the biomarkers, that are subject to missingness conditional on all of the covariates that are completely observed, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. We show that the over‐identified joint model produces biased estimation of the latent variable model and describe how to impose constraints on the joint model so that it has a one‐to‐one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation‐maximization algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

log-binomial模型估计的患病比及其应用

[导读]探讨有和无连续协变量时的log-binomial模型估计患病比的统计方法及其应用.文中使用禁烟立法态度与吸烟关联的研究实例,比较log-binomial模型估计的现患比(PR)与logistic回归模型估计的优势比(OR).当模型中无连续协变量时,采用最大似然估计拟合log-binomial模型;当因含有连续协变量导致模型不收敛时,则采用COPY方法估计PR.分别估计男、女禁烟立法态度与吸烟的关联.由于女性吸烟率低,用PR与OR所估计的关联结果相似.而男性吸烟率较高,OR值明显大于PR.当年龄作为连续协变量纳入模型时,导致log-binomial模型不收敛,采用COPY方法解决此问题.所有分析均在SAS软件中实现.结论:当患病率较高时,PR比OR更好地测量了暴露与疾病的关联.文中给出log-binomial回归模型和COPY方法估计PR的SAS程序.     

Bivariate logistic regression: modelling the association of small for gestational age births in twin gestations.

Clustered binary responses, such as disease status in twins, frequently arise in perinatal health and other epidemiologic applications. The scientific objective involves modelling both the marginal mean responses, such as the probability of disease, and the within-cluster association of the multivariate responses. In this regard, bivariate logistic regression is a useful procedure with advantages that include (i) a single maximization of the joint probability distribution of the bivariate binary responses, and (ii) modelling the odds ratio describing the pairwise association between the two binary responses in relation to several covariates. In addition, since the form of the joint distribution of the bivariate binary responses is assumed known, parameters for the regression model can be estimated by the method of maximum likelihood. Hence, statistical inferences may be based on likelihood ratio tests and profile likelihood confidence intervals. We apply bivariate logistic regression to a perinatal database comprising 924 twin foetuses resulting from 462 pregnancies to model obstetric and clinical risk factors for the association of small for gestational age births in twin gestations.