A two‐stage approach for dynamic prediction of time‐to‐event distributions

Dynamic prediction uses longitudinal biomarkers for real‐time prediction of an individual patient's prognosis. This is critical for patients with an incurable disease such as cancer. Biomarker trajectories are usually not linear, nor even monotone, and vary greatly across individuals. Therefore, it is difficult to fit them with parametric models. With this consideration, we propose an approach for dynamic prediction that does not need to model the biomarker trajectories. Instead, as a trade‐off, we assume that the biomarker effects on the risk of disease recurrence are smooth functions over time. This approach turns out to be computationally easier. Simulation studies show that the proposed approach achieves stable estimation of biomarker effects over time, has good predictive performance, and is robust against model misspecification. It is a good compromise between two major approaches, namely, (i) joint modeling of longitudinal and survival data and (ii) landmark analysis. The proposed method is applied to patients with chronic myeloid leukemia. At any time following their treatment with tyrosine kinase inhibitors, longitudinally measured BCR‐ABL gene expression levels are used to predict the risk of disease progression. Copyright © 2016 John Wiley & Sons, Ltd.     

Adverse subpopulation regression for multivariate outcomes with high‐dimensional predictors

Biomedical studies have a common interest in assessing relationships between multiple related health outcomes and high‐dimensional predictors. For example, in reproductive epidemiology, one may collect pregnancy outcomes such as length of gestation and birth weight and predictors such as single nucleotide polymorphisms in multiple candidate genes and environmental exposures. In such settings, there is a need for simple yet flexible methods for selecting true predictors of adverse health responses from a high‐dimensional set of candidate predictors. To address this problem, one may either consider linear regression models for the continuous outcomes or convert these outcomes into binary indicators of adverse responses using predefined cutoffs. The former strategy has the disadvantage of often leading to a poorly fitting model that does not predict risk well, whereas the latter approach can be very sensitive to the cutoff choice. As a simple yet flexible alternative, we propose a method for adverse subpopulation regression, which relies on a two‐component latent class model, with the dominant component corresponding to (presumed) healthy individuals and the risk of falling in the minority component characterized via a logistic regression. The logistic regression model is designed to accommodate high‐dimensional predictors, as occur in studies with a large number of gene by environment interactions, through the use of a flexible nonparametric multiple shrinkage approach. The Gibbs sampler is developed for posterior computation. We evaluate the methods with the use of simulation studies and apply these to a genetic epidemiology study of pregnancy outcomes. Copyright © 2012 John Wiley & Sons, Ltd.     

Using trajectories from a bivariate growth curve as predictors in a Cox regression model

An important research objective in most psychiatric clinical trials of maintenance treatment is to find predictors of recurrence of illness. In those trials, patients are first admitted into an open treatment period also called acute treatment. If they respond to the treatment and are considered to have stable remission from the illness, they enter the second phase of the trial where they are randomized into different arms of the 'maintenance treatments'. Often, more than one response variable is measured longitudinally in the acute treatment phase to monitor treatment responses. Trajectories of these response measures are believed to have predictive ability for recurrences in the maintenance phase of the trial. By using a bivariate growth curve from two such longitudinal measures, we developed a method to use the estimated trajectories of each subject in a Cox regression model to predict recurrence in the maintenance phase. To adjust for the parameter estimation errors, we applied a full likelihood approach based on the conditional expectations of the predictors. Simulation studies indicate that the estimation error corrected estimators for the Cox model parameters are less biased when compared to the naive regression estimators without accounting for these errors. The uniqueness of this method lies in estimating trajectories from bivariate unequally spaced longitudinal response measures. An illustrative example is provided with data from a maintenance treatment trial for major depression in an elderly population. Visual Fortran 90 programs were developed to implement the algorithm.     

Joint modeling of longitudinal data and informative dropout time in the presence of multiple changepoints

In longitudinal studies of patients with the human immunodeficiency virus (HIV), objectives of interest often include modeling of individual-level trajectories of HIV ribonucleic acid (RNA) as a function of time. Such models can be used to predict the effects of different treatment regimens or to classify subjects into subgroups with similar trajectories. Empirical evidence, however, suggests that individual trajectories often possess multiple points of rapid change, which may vary from subject to subject. Additionally, some individuals may end up dropping out of the study and the tendency to drop out may be related to the level of the biomarker. Modeling of individual viral RNA profiles is challenging in the presence of these changes, and currently available methods do not address all the issues such as multiple changes, informative dropout, clustering, etc. in a single model. In this article, we propose a new joint model, where a multiple-changepoint model is proposed for the longitudinal viral RNA response and a proportional hazards model for the time of dropout process. Dirichlet process (DP) priors are used to model the distribution of the individual random effects and error distribution. In addition to robustifying the model against possible misspecifications, the DP leads to a natural clustering of subjects with similar trajectories which can be of importance in itself. Sharing of information among subjects with similar trajectories also results in improved parameter estimation. A fully Bayesian approach for model fitting and prediction is implemented using MCMC procedures on the ACTG 398 clinical trial data. The proposed model is seen to give rise to improved estimates of individual trajectories when compared with a parametric approach.     

Dynamic prediction of Alzheimer's disease progression using features of multiple longitudinal outcomes and time-to-event data

This paper is motivated by combining serial neurocognitive assessments and other clinical variables for monitoring the progression of Alzheimer's disease (AD). We propose a novel framework for the use of multiple longitudinal neurocognitive markers to predict the progression of AD. The conventional joint modeling longitudinal and survival data approach is not applicable when there is a large number of longitudinal outcomes. We introduce various approaches based on functional principal component for dimension reduction and feature extraction from multiple longitudinal outcomes. We use these features to extrapolate the health outcome trajectories and use scores on these features as predictors in a Cox proportional hazards model to conduct predictions over time. We propose a personalized dynamic prediction framework that can be updated as new observations collected to reflect the patient's latest prognosis, and thus intervention could be initiated in a timely manner. Simulation studies and application to the Alzheimer's Disease Neuroimaging Initiative dataset demonstrate the robustness of the method for the prediction of future health outcomes and risks of target events under various scenarios.     

Joint modeling of repeated multivariate cognitive measures and competing risks of dementia and death: a latent process and latent class approach

Joint models initially dedicated to a single longitudinal marker and a single time‐to‐event need to be extended to account for the rich longitudinal data of cohort studies. Multiple causes of clinical progression are indeed usually observed, and multiple longitudinal markers are collected when the true latent trait of interest is hard to capture (e.g., quality of life, functional dependency, and cognitive level). These multivariate and longitudinal data also usually have nonstandard distributions (discrete, asymmetric, bounded, etc.). We propose a joint model based on a latent process and latent classes to analyze simultaneously such multiple longitudinal markers of different natures, and multiple causes of progression. A latent process model describes the latent trait of interest and links it to the observed longitudinal outcomes using flexible measurement models adapted to different types of data, and a latent class structure links the longitudinal and cause‐specific survival models. The joint model is estimated in the maximum likelihood framework. A score test is developed to evaluate the assumption of conditional independence of the longitudinal markers and each cause of progression given the latent classes. In addition, individual dynamic cumulative incidences of each cause of progression based on the repeated marker data are derived. The methodology is validated in a simulation study and applied on real data about cognitive aging obtained from a large population‐based study. The aim is to predict the risk of dementia by accounting for the competing death according to the profiles of semantic memory measured by two asymmetric psychometric tests. Copyright © 2015 John Wiley & Sons, Ltd.     

Imputation of systematically missing predictors in an individual participant data meta‐analysis: a generalized approach using MICE

Individual participant data meta‐analyses (IPD‐MA) are increasingly used for developing and validating multivariable (diagnostic or prognostic) risk prediction models. Unfortunately, some predictors or even outcomes may not have been measured in each study and are thus systematically missing in some individual studies of the IPD‐MA. As a consequence, it is no longer possible to evaluate between‐study heterogeneity and to estimate study‐specific predictor effects, or to include all individual studies, which severely hampers the development and validation of prediction models. Here, we describe a novel approach for imputing systematically missing data and adopt a generalized linear mixed model to allow for between‐study heterogeneity. This approach can be viewed as an extension of Resche‐Rigon's method (Stat Med 2013), relaxing their assumptions regarding variance components and allowing imputation of linear and nonlinear predictors. We illustrate our approach using a case study with IPD‐MA of 13 studies to develop and validate a diagnostic prediction model for the presence of deep venous thrombosis. We compare the results after applying four methods for dealing with systematically missing predictors in one or more individual studies: complete case analysis where studies with systematically missing predictors are removed, traditional multiple imputation ignoring heterogeneity across studies, stratified multiple imputation accounting for heterogeneity in predictor prevalence, and multilevel multiple imputation (MLMI) fully accounting for between‐study heterogeneity. We conclude that MLMI may substantially improve the estimation of between‐study heterogeneity parameters and allow for imputation of systematically missing predictors in IPD‐MA aimed at the development and validation of prediction models. Copyright © 2015 John Wiley & Sons, Ltd.     

Empirical constrained Bayes predictors accounting for non‐detects among repeated measures

When the prediction of subject‐specific random effects is of interest, constrained Bayes predictors (CB) have been shown to reduce the shrinkage of the widely accepted Bayes predictor while still maintaining desirable properties, such as optimizing mean‐square error subsequent to matching the first two moments of the random effects of interest. However, occupational exposure and other epidemiologic (e.g. HIV) studies often present a further challenge because data may fall below the measuring instrument's limit of detection. Although methodology exists in the literature to compute Bayes estimates in the presence of non‐detects (Bayes_ND), CB methodology has not been proposed in this setting. By combining methodologies for computing CBs and Bayes_ND, we introduce two novel CBs that accommodate an arbitrary number of observable and non‐detectable measurements per subject. Based on application to real data sets (e.g. occupational exposure, HIV RNA) and simulation studies, these CB predictors are markedly superior to the Bayes predictor and to alternative predictors computed using ad hoc methods in terms of meeting the goal of matching the first two moments of the true random effects distribution. Copyright © 2010 John Wiley & Sons, Ltd.     

Predicting longitudinal trajectories of health probabilities with random‐effects multinomial logit regression

Researchers often encounter longitudinal health data characterized with three or more ordinal or nominal categories. Random‐effects multinomial logit models are generally applied to account for potential lack of independence inherent in such clustered data. When parameter estimates are used to describe longitudinal processes, however, random effects, both between and within individuals, need to be retransformed for correctly predicting outcome probabilities. This study attempts to go beyond existing work by developing a retransformation method that derives longitudinal growth trajectories of unbiased health probabilities. We estimated variances of the predicted probabilities by using the delta method. Additionally, we transformed the covariates’ regression coefficients on the multinomial logit function, not substantively meaningful, to the conditional effects on the predicted probabilities. The empirical illustration uses the longitudinal data from the Asset and Health Dynamics among the Oldest Old. Our analysis compared three sets of the predicted probabilities of three health states at six time points, obtained from, respectively, the retransformation method, the best linear unbiased prediction, and the fixed‐effects approach. The results demonstrate that neglect of retransforming random errors in the random‐effects multinomial logit model results in severely biased longitudinal trajectories of health probabilities as well as overestimated effects of covariates on the probabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

Mixed‐effects Gaussian process functional regression models with application to dose–response curve prediction

We propose a new semiparametric model for functional regression analysis, combining a parametric mixed‐effects model with a nonparametric Gaussian process regression model, namely a mixed‐effects Gaussian process functional regression model. The parametric component can provide explanatory information between the response and the covariates, whereas the nonparametric component can add nonlinearity. We can model the mean and covariance structures simultaneously, combining the information borrowed from other subjects with the information collected from each individual subject. We apply the model to dose–response curves that describe changes in the responses of subjects for differing levels of the dose of a drug or agent and have a wide application in many areas. We illustrate the method for the management of renal anaemia. An individual dose–response curve is improved when more information is included by this mechanism from the subject/patient over time, enabling a patient‐specific treatment regime. Copyright © 2012 John Wiley & Sons, Ltd.     

A joint model for interval‐censored functional decline trajectories under informative observation

Multi‐state models are useful for modelling disease progression where the state space of the process is used to represent the discrete disease status of subjects. Often, the disease process is only observed at clinical visits, and the schedule of these visits can depend on the disease status of patients. In such situations, the frequency and timing of observations may depend on transition times that are themselves unobserved in an interval‐censored setting. There is a potential for bias if we model a disease process with informative observation times as a non‐informative observation scheme with pre‐specified examination times. In this paper, we develop a joint model for the disease and observation processes to ensure valid inference because the follow‐up process may itself contain information about the disease process. The transitions for each subject are modelled using a Markov process, where bivariate subject‐specific random effects are used to link the disease and observation models. Inference is based on a Bayesian framework, and we apply our joint model to the analysis of a large study examining functional decline trajectories of palliative care patients. Copyright © 2015 John Wiley & Sons, Ltd.     

Functional regression analysis using an F test for longitudinal data with large numbers of repeated measures

Longitudinal data sets from certain fields of biomedical research often consist of several variables repeatedly measured on each subject yielding a large number of observations. This characteristic complicates the use of traditional longitudinal modelling strategies, which were primarily developed for studies with a relatively small number of repeated measures per subject. An innovative way to model such 'wide' data is to apply functional regression analysis, an emerging statistical approach in which observations of the same subject are viewed as a sample from a functional space. Shen and Faraway introduced an F test for linear models with functional responses. This paper illustrates how to apply this F test and functional regression analysis to the setting of longitudinal data. A smoking cessation study for methadone-maintained tobacco smokers is analysed for demonstration. In estimating the treatment effects, the functional regression analysis provides meaningful clinical interpretations, and the functional F test provides consistent results supported by a mixed-effects linear regression model. A simulation study is also conducted under the condition of the smoking data to investigate the statistical power for the F test, Wilks' likelihood ratio test, and the linear mixed-effects model using AIC.     

High‐dimensional variable selection and prediction under competing risks with application to SEER‐Medicare linked data

Competing risk analysis considers event times due to multiple causes or of more than one event types. Commonly used regression models for such data include (1) cause‐specific hazards model, which focuses on modeling one type of event while acknowledging other event types simultaneously, and (2) subdistribution hazards model, which links the covariate effects directly to the cumulative incidence function. Their use in the presence of high‐dimensional predictors are largely unexplored. Motivated by an analysis using the linked SEER‐Medicare database for the purposes of predicting cancer versus noncancer mortality for patients with prostate cancer, we study the accuracy of prediction and variable selection of existing machine learning methods under both models using extensive simulation experiments, including different approaches to choosing penalty parameters in each method. We then apply the optimal approaches to the analysis of the SEER‐Medicare data.     

A joint logistic regression and covariate‐adjusted continuous‐time Markov chain model

The use of longitudinal measurements to predict a categorical outcome is an increasingly common goal in research studies. Joint models are commonly used to describe two or more models simultaneously by considering the correlated nature of their outcomes and the random error present in the longitudinal measurements. However, there is limited research on joint models with longitudinal predictors and categorical cross‐sectional outcomes. Perhaps the most challenging task is how to model the longitudinal predictor process such that it represents the true biological mechanism that dictates the association with the categorical response. We propose a joint logistic regression and Markov chain model to describe a binary cross‐sectional response, where the unobserved transition rates of a two‐state continuous‐time Markov chain are included as covariates. We use the method of maximum likelihood to estimate the parameters of our model. In a simulation study, coverage probabilities of about 95%, standard deviations close to standard errors, and low biases for the parameter values show that our estimation method is adequate. We apply the proposed joint model to a dataset of patients with traumatic brain injury to describe and predict a 6‐month outcome based on physiological data collected post‐injury and admission characteristics. Our analysis indicates that the information provided by physiological changes over time may help improve prediction of long‐term functional status of these severely ill subjects. Copyright © 2017 John Wiley & Sons, Ltd.     

Semiparametric modeling and analysis of longitudinal method comparison data

Studies comparing two or more methods of measuring a continuous variable are routinely conducted in biomedical disciplines with the primary goal of measuring agreement between the methods. Often, the data are collected by following a cohort of subjects over a period of time. This gives rise to longitudinal method comparison data where there is one observation trajectory for each method on every subject. It is not required that observations from all methods be available at each observation time. The multiple trajectories on the same subjects are dependent. We propose modeling the trajectories nonparametrically through penalized regression splines within the framework of mixed‐effects models. The model also uses random effects of subjects and their interactions to capture dependence in observations from the same subjects. It additionally allows the within‐subject errors of each method to be correlated. It is fit using the method of maximum likelihood. Agreement between the methods is evaluated by performing inference on measures of agreement, such as concordance correlation coefficient and total deviation index, which are functions of parameters of the assumed model. Simulations indicate that the proposed methodology performs reasonably well for 30 or more subjects. Its application is illustrated by analyzing a dataset of percentage body fat measurements. Copyright © 2017 John Wiley & Sons, Ltd.     

On the uncertainty of individual prediction because of sampling predictors

Prediction of an outcome for a given unit based on prediction models built on a training sample plays a major role in many research areas. The uncertainty of the prediction is predominantly characterized by the subject sampling variation in current practice, where prediction models built on hypothetically re‐sampled units yield variable predictions for the same unit of interest. It is almost always true that the predictors used to build prediction models are simply a subset of the entirety of factors related to the outcome. Following the frequentist principle, we can account for the variation because of hypothetically re‐sampled predictors used to build the prediction models. This is particularly important in medicine where the prediction has important and sometime life‐death consequences on a patient's health status. In this article, we discuss some rationale along this line in the context of medicine. We propose a simple approach to estimate the standard error of the prediction that accounts for the variation because of sampling both subjects and predictors under logistic and Cox regression models. A simulation study is presented to support our argument and demonstrate the performance of our method. The concept and method are applied to a real data set. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian approach for flexible modeling of semicompeting risks data

Semicompeting risks data arise when two types of events, non‐terminal and terminal, are observed. When the terminal event occurs first, it censors the non‐terminal event, but not vice versa. To account for possible dependent censoring of the non‐terminal event by the terminal event and to improve prediction of the terminal event using the non‐terminal event information, it is crucial to model their association properly. Motivated by a breast cancer clinical trial data analysis, we extend the well‐known illness–death models to allow flexible random effects to capture heterogeneous association structures in the data. Our extension also represents a generalization of the popular shared frailty models that usually assume that the non‐terminal event does not affect the hazards of the terminal event beyond a frailty term. We propose a unified Bayesian modeling approach that can utilize existing software packages for both model fitting and individual‐specific event prediction. The approach is demonstrated via both simulation studies and a breast cancer data set analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating a time‐dependent concordance index for survival prediction models with covariate dependent censoring

Given a predictive marker and a time‐to‐event response variable, the proportion of concordant pairs in a data set is called concordance index. A specifically useful marker is the risk predicted by a survival regression model. This article extends the existing methodology for applications where the length of the follow‐up period depends on the predictor variables. A class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution. The estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive. In this framework, all kinds of prediction models can be assessed, and time trends in the discrimination ability of a model can be captured by varying the truncation time point. For illustration, we re‐analyze a study on risk prediction for prostate cancer patients. The effects of misspecification of the censoring model are studied in simulated data. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian random effects selection in mixed accelerated failure time model for interval‐censored data

In many medical problems that collect multiple observations per subject, the time to an event is often of interest. Sometimes, the occurrence of the event can be recorded at regular intervals leading to interval‐censored data. It is further desirable to obtain the most parsimonious model in order to increase predictive power and to obtain ease of interpretation. Variable selection and often random effects selection in case of clustered data become crucial in such applications. We propose a Bayesian method for random effects selection in mixed effects accelerated failure time (AFT) models. The proposed method relies on the Cholesky decomposition on the random effects covariance matrix and the parameter‐expansion method for the selection of random effects. The Dirichlet prior is used to model the uncertainty in the random effects. The error distribution for the accelerated failure time model has been specified using a Gaussian mixture to allow flexible error density and prediction of the survival and hazard functions. We demonstrate the model using extensive simulations and the Signal Tandmobiel Study^®. Copyright © 2013 John Wiley & Sons, Ltd.     

Statistical models for longitudinal biomarkers of disease onset

We consider the analysis of serial biomarkers to screen and monitor individuals in a given population for onset of a specific disease of interest. The biomarker readings are subject to error. We survey some of the existing literature and concentrate on two recently proposed models. The first is a fully Bayesian hierarchical structure for a mixed effects segmented regression model. Posterior estimates of the changepoint (onset time) distribution are obtained by Gibbs sampling. The second is a hidden changepoint model in which the onset time distribution is estimated by maximum likelihood using the EM algorithm. Both methods lead to a dynamic index that represents a strength of evidence that onset has occurred by the current time in an individual subject. The methods are applied to some large data sets concerning prostate specific antigen (PSA) as a serial marker for prostate cancer. Rules based on the indices are compared to standard diagnostic criteria through the use of ROC curves adapted for longitudinal data.