Joint analysis of left‐censored longitudinal biomarker and binary outcome via latent class modeling

Joint latent class modeling is an appealing approach for evaluating the association between a longitudinal biomarker and clinical outcome when the study population is heterogeneous. The link between the biomarker trajectory and the risk of event is reflected by the latent classes, which accommodate the underlying population heterogeneity. The estimation of joint latent class models may be complicated by the censored data in the biomarker measurements due to detection limits. We propose a modified likelihood function under the parametric assumption of biomarker distribution and develop a Monte Carlo expectation‐maximization algorithm for joint analysis of a biomarker and a binary outcome. We conduct simulation studies to demonstrate the satisfactory performance of our Monte Carlo expectation‐maximization algorithm and the superiority of our method to the naive imputation method for handling censored biomarker data. In addition, we apply our method to the Genetic and Inflammatory Markers of Sepsis study to investigate the role of inflammatory biomarker profile in predicting 90‐day mortality for patients hospitalized with community‐acquired pneumonia.     

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.     

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.     

Multivariate longitudinal data analysis with censored and intermittent missing responses

The multivariate linear mixed model (MLMM) has emerged as an important analytical tool for longitudinal data with multiple outcomes. However, the analysis of multivariate longitudinal data could be complicated by the presence of censored measurements because of a detection limit of the assay in combination with unavoidable missing values arising when subjects miss some of their scheduled visits intermittently. This paper presents a generalization of the MLMM approach, called the MLMM‐CM, for a joint analysis of the multivariate longitudinal data with censored and intermittent missing responses. A computationally feasible expectation maximization–based procedure is developed to carry out maximum likelihood estimation within the MLMM‐CM framework. Moreover, the asymptotic standard errors of fixed effects are explicitly obtained via the information‐based method. We illustrate our methodology by using simulated data and a case study from an AIDS clinical trial. Experimental results reveal that the proposed method is able to provide more satisfactory performance as compared with the traditional MLMM approach.     

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.     

Joint multiple imputation for longitudinal outcomes and clinical events that truncate longitudinal follow‐up

Longitudinal cohort studies often collect both repeated measurements of longitudinal outcomes and times to clinical events whose occurrence precludes further longitudinal measurements. Although joint modeling of the clinical events and the longitudinal data can be used to provide valid statistical inference for target estimands in certain contexts, the application of joint models in medical literature is currently rather restricted because of the complexity of the joint models and the intensive computation involved. We propose a multiple imputation approach to jointly impute missing data of both the longitudinal and clinical event outcomes. With complete imputed datasets, analysts are then able to use simple and transparent statistical methods and standard statistical software to perform various analyses without dealing with the complications of missing data and joint modeling. We show that the proposed multiple imputation approach is flexible and easy to implement in practice. Numerical results are also provided to demonstrate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal survival time‐related cut‐point with censored data

In biomedical research and practice, continuous biomarkers are often used for diagnosis and prognosis, with a cut‐point being established on the measurement to aid binary classification. When survival time is examined for the purposes of disease prognostication and is found to be related to the baseline measure of a biomarker, employing a single cut‐point on the biomarker may not be very informative. Using survival time‐dependent sensitivity and specificity, we extend a concordance probability‐based objective function to select survival time‐related cut‐points. To estimate the objective function with censored survival data, we adopt a non‐parametric procedure for time‐dependent receiver operational characteristics curves, which uses nearest neighbor estimation techniques. In a simulation study, the proposed method, when used to select a cut‐point to optimally predict survival at a given time within a specified range, yields satisfactory results. We apply the procedure to estimate survival time‐dependent cut‐point on the prognostic biomarker of serum bilirubin among patients with primary biliary cirrhosis. Copyright © 2014 John Wiley & Sons, Ltd.     

Joint modeling of quantitative longitudinal data and censored survival time

BACKGROUND: In epidemiology, we are often interested in the association between the evolution of a quantitative variable and the onset of an event. The aim of this paper is to present a joint model for the analysis of Gaussian repeated data and survival time. Such models allow, for example, to perform survival analysis when a time-dependent explanatory variable is measured intermittently, or to study the evolution of a quantitative marker conditionally to an event. METHODS: They are constructed by combining a mixed model for repeated Gaussian variables and a survival model which can be parametric or semi-parametric (Cox model). RESULTS: We discuss the hypotheses underlying the different joint models proposed in the literature and the necessary assumptions for maximum likelihood estimation. The interest of these methods is illustrated with a study of the natural history of dementia in a cohort of elderly persons.     

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.     

Two-part hidden Markov models for semicontinuous longitudinal data with nonignorable missing covariates

This study develops a two-part hidden Markov model (HMM) for analyzing semicontinuous longitudinal data in the presence of missing covariates. The proposed model manages a semicontinuous variable by splitting it into two random variables: a binary indicator for determining the occurrence of excess zeros at all occasions and a continuous random variable for examining its actual level. For the continuous longitudinal response, an HMM is proposed to describe the relationship between the observation and unobservable finite-state transition processes. The HMM consists of two major components. The first component is a transition model for investigating how potential covariates influence the probabilities of transitioning from one hidden state to another. The second component is a conditional regression model for examining the state-specific effects of covariates on the response. A shared random effect is introduced to each part of the model to accommodate possible unobservable heterogeneity among observation processes and the nonignorability of missing covariates. A Bayesian adaptive least absolute shrinkage and selection operator (lasso) procedure is developed to conduct simultaneous variable selection and estimation. The proposed methodology is applied to a study on the Alzheimer's Disease Neuroimaging Initiative dataset. New insights into the pathology of Alzheimer's disease and its potential risk factors are obtained.     

Regression with incomplete covariates and left‐truncated time‐to‐event data

Studies of chronic diseases routinely sample individuals subject to conditions on an event time of interest. In epidemiology, for example, prevalent cohort studies aiming to evaluate risk factors for survival following onset of dementia require subjects to have survived to the point of screening. In clinical trials designed to assess the effect of experimental cancer treatments on survival, patients are required to survive from the time of cancer diagnosis to recruitment. Such conditions yield samples featuring left‐truncated event time distributions. Incomplete covariate data often arise in such settings, but standard methods do not deal with the fact that individuals’ covariate distributions are also affected by left truncation. We describe an expectation–maximization algorithm for dealing with incomplete covariate data in such settings, which uses the covariate distribution conditional on the selection criterion. We describe an extension to deal with subgroup analyses in clinical trials for the case in which the stratification variable is incompletely observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Correction of bias from non‐random missing longitudinal data using auxiliary information

Missing data are common in longitudinal studies due to drop‐out, loss to follow‐up, and death. Likelihood‐based mixed effects models for longitudinal data give valid estimates when the data are missing at random (MAR). These assumptions, however, are not testable without further information. In some studies, there is additional information available in the form of an auxiliary variable known to be correlated with the missing outcome of interest. Availability of such auxiliary information provides us with an opportunity to test the MAR assumption. If the MAR assumption is violated, such information can be utilized to reduce or eliminate bias when the missing data process depends on the unobserved outcome through the auxiliary information. We compare two methods of utilizing the auxiliary information: joint modeling of the outcome of interest and the auxiliary variable, and multiple imputation (MI). Simulation studies are performed to examine the two methods. The likelihood‐based joint modeling approach is consistent and most efficient when correctly specified. However, mis‐specification of the joint distribution can lead to biased results. MI is slightly less efficient than a correct joint modeling approach and can also be biased when the imputation model is mis‐specified, though it is more robust to mis‐specification of the imputation distribution when all the variables affecting the missing data mechanism and the missing outcome are included in the imputation model. An example is presented from a dementia screening study. Copyright © 2009 John Wiley & Sons, Ltd.     

Pseudo maximum likelihood approach for the analysis of multivariate left‐censored longitudinal data

The linear mixed effects model based on a full likelihood is one of the few methods available to model longitudinal data subject to left censoring. However, a full likelihood approach is complicated algebraically because of the large dimension of the numeric computations, and maximum likelihood estimation can be computationally prohibitive when the data are heavily censored. Moreover, for mixed models, the complexity of the computation increases as the dimension of the random effects in the model increases. We propose a method based on pseudo likelihood that simplifies the computational complexities, allows a wide class of multivariate models, and that can be used for many different data structures including settings where the level of censoring is high. The motivation for this work comes from the need for a joint model to assess the joint effect of pro‐inflammatory and anti‐inflammatory biomarker data on 30‐day mortality status while simultaneously accounting for longitudinal left censoring and correlation between markers in the analysis of Genetic and Inflammatory Markers for Sepsis study conducted at the University of Pittsburgh. Two markers, interleukin‐6 and interleukin‐10, which naturally are correlated because of a shared similar biological pathways and are left‐censored because of the limited sensitivity of the assays, are considered to determine if higher levels of these markers is associated with an increased risk of death after accounting for the left censoring and their assumed correlation. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling time‐varying effects with generalized and unsynchronized longitudinal data

We propose novel estimation approaches for generalized varying coefficient models that are tailored for unsynchronized, irregular and infrequent longitudinal designs/data. Unsynchronized longitudinal data refer to the time‐dependent response and covariate measurements for each individual measured at distinct time points. Data from the Comprehensive Dialysis Study motivate the proposed methods. We model the potential age‐varying association between infection‐related hospitalization status and the inflammatory marker, C‐reactive protein, within the first 2 years from initiation of dialysis. We cannot directly apply traditional longitudinal modeling to unsynchronized data, and no method exists to estimate time‐varying or age‐varying effects for generalized outcomes (e.g., binary or count data) to date. In addition, through the analysis of the Comprehensive Dialysis Study data and simulation studies, we show that preprocessing steps, such as binning, needed to synchronize data to apply traditional modeling can lead to significant loss of information in this context. In contrast, the proposed approaches discard no observation; they exploit the fact that although there is little information in a single subject trajectory because of irregularity and infrequency, the moments of the underlying processes can be accurately and efficiently recovered by pooling information from all subjects using functional data analysis. We derive subject‐specific mean response trajectory predictions and study finite sample properties of the estimators. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint modeling of survival time and longitudinal data with subject-specific changepoints in the covariates

Joint models are frequently used in survival analysis to assess the relationship between time-to-event data and time-dependent covariates, which are measured longitudinally but often with errors. Routinely, a linear mixed-effects model is used to describe the longitudinal data process, while the survival times are assumed to follow the proportional hazards model. However, in some practical situations, individual covariate profiles may contain changepoints. In this article, we assume a two-phase polynomial random effects with subject-specific changepoint model for the longitudinal data process and the proportional hazards model for the survival times. Our main interest is in the estimation of the parameter in the hazards model. We incorporate a smooth transition function into the changepoint model for the longitudinal data and develop the corrected score and conditional score estimators, which do not require any assumption regarding the underlying distribution of the random effects or that of the changepoints. The estimators are shown to be asymptotically equivalent and their finite-sample performance is examined via simulations. The methods are applied to AIDS clinical trial data.     

Recurrent event data analysis with intermittently observed time‐varying covariates

Although recurrent event data analysis is a rapidly evolving area of research, rigorous studies on estimation of the effects of intermittently observed time‐varying covariates on the risk of recurrent events have been lacking. Existing methods for analyzing recurrent event data usually require that the covariate processes are observed throughout the entire follow‐up period. However, covariates are often observed periodically rather than continuously. We propose a novel semiparametric estimator for the regression parameters in the popular proportional rate model. The proposed estimator is based on an estimated score function where we kernel smooth the mean covariate process. We show that the proposed semiparametric estimator is asymptotically unbiased, normally distributed, and derives the asymptotic variance. Simulation studies are conducted to compare the performance of the proposed estimator and the simple methods carrying forward the last covariates. The different methods are applied to an observational study designed to assess the effect of group A streptococcus on pharyngitis among school children in India. Copyright © 2016 John Wiley & Sons, Ltd.     

Weighted estimating equations for longitudinal studies with death and non-monotone missing time-dependent covariates and outcomes

We propose a marginal modeling approach to estimate the association between a time-dependent covariate and an outcome in longitudinal studies where some study participants die during follow-up and both variables have non-monotone response patterns. The proposed method is an extension of weighted estimating equations that allows the outcome and covariate to have different missing-data patterns. We present methods for both random and non-random missing-data mechanisms. A study of functional recovery in a cohort of elderly female hip-fracture patients motivates the approach.     

Frailty models for pneumonia to death with a left‐censored covariate

Frailty models are multiplicative hazard models for studying association between survival time and important clinical covariates. When some values of a clinical covariate are unobserved but known to be below a threshold called the limit of detection (LOD), naive approaches ignoring this problem, such as replacing the undetected value by the LOD or half of the LOD, often produce biased parameter estimate with larger mean squared error of the estimate. To address the LOD problem in a frailty model, we propose a flexible smooth nonparametric density estimator along with Simpson's numerical integration technique. This is an extension of an existing method in the likelihood framework for the estimation and inference of the model parameters. The proposed new method shows the estimators are asymptotically unbiased and gives smaller mean squared error of the estimates. Compared with the existing method, the proposed new method does not require distributional assumptions for the underlying covariates. Simulation studies were conducted to evaluate the performance of the new method in realistic scenarios. We illustrate the use of the proposed method with a data set from Genetic and Inflammatory Markers of Sepsis study in which interlekuin‐10 was subject to LOD. Copyright © 2015 John Wiley & Sons, Ltd.     

A comparison of hospital performance with non-ignorable missing covariates: an application to trauma care data

Trauma is a term used in medicine for describing physical injury. The prospective evaluation of the care of injured patients aims to improve the management of a trauma system and acts as an ongoing audit of trauma care. One of the principal techniques used to evaluate the effectiveness of trauma care at different hospitals is through a comparative outcome analysis. In such an analysis, a national 'league table' can be compiled to determine which hospitals are better at managing trauma care. One of the problems with the conventional analysis is that key covariates for measuring physiological injury can often be missing. It is also hypothesized that this missingness is not missing at random (NMAR). We describe the methods used to assess the performance of hospitals in a trauma setting and implement the method of weights for generalized linear models to account for the missing covariate data, when we suspect the missing data mechanism is NMAR using a Monte Carlo EM algorithm. Through simulation work and application to the trauma data we demonstrate the affect the missing covariate data can have on the performance of hospitals and how the conclusions we draw from the analysis can differ. We highlight the differences in hospital performance and the ranking of hospitals.     

Multiple imputation for IPD meta‐analysis: allowing for heterogeneity and studies with missing covariates

Recently, multiple imputation has been proposed as a tool for individual patient data meta‐analysis with sporadically missing observations, and it has been suggested that within‐study imputation is usually preferable. However, such within study imputation cannot handle variables that are completely missing within studies. Further, if some of the contributing studies are relatively small, it may be appropriate to share information across studies when imputing. In this paper, we develop and evaluate a joint modelling approach to multiple imputation of individual patient data in meta‐analysis, with an across‐study probability distribution for the study specific covariance matrices. This retains the flexibility to allow for between‐study heterogeneity when imputing while allowing (i) sharing information on the covariance matrix across studies when this is appropriate, and (ii) imputing variables that are wholly missing from studies. Simulation results show both equivalent performance to the within‐study imputation approach where this is valid, and good results in more general, practically relevant, scenarios with studies of very different sizes, non‐negligible between‐study heterogeneity and wholly missing variables. We illustrate our approach using data from an individual patient data meta‐analysis of hypertension trials. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.