Joint modelling of repeated measurements and time-to-event outcomes: the fourth Armitage lecture

In many longitudinal studies, the outcomes recorded on each subject include both a sequence of repeated measurements at pre-specified times and the time at which an event of particular interest occurs: for example, death, recurrence of symptoms or drop out from the study. The event time for each subject may be recorded exactly, interval censored or right censored. The term joint modelling refers to the statistical analysis of the resulting data while taking account of any association between the repeated measurement and time-to-event outcomes. In this paper, we first discuss different approaches to joint modelling and argue that the analysis strategy should depend on the scientific focus of the study. We then describe in detail a particularly simple, fully parametric approach. Finally, we use this approach to re-analyse data from a clinical trial of drug therapies for schizophrenic patients, in which the event time is an interval-censored or right-censored time to withdrawal from the study due to adverse side effects.     

Estimating the distribution of the window period for recent HIV infections: a comparison of statistical methods

In the past few years a number of antibody biomarkers have been developed to distinguish between recent and established Human Immunodeficiency Virus (HIV) infection. Typically, a specific threshold/cut-off of the biomarker is chosen, values below which are indicative of recent infections. Such biomarkers have attracted considerable interest as the basis for incidence estimation using a cross-sectional sample. An estimate of HIV incidence can be obtained from the prevalence of recent infection, as measured in the sample, and knowledge of the time spent in the recent infection state, known as the window period. However, such calculations are based on a number of assumptions concerning the distribution of the window period. We compare two statistical methods for estimating the mean and distribution of a window period using data on repeated measurements of an antibody biomarker from a cohort of HIV seroconverters. The methods account for the interval-censored nature of both the date of seroconversion and the date of crossing a specific threshold. We illustrate the methods using repeated measurements of the Avidity Index (AI) and make recommendations about the choice of threshold for this biomarker so that the resulting window period satisfies the assumptions for incidence estimation.     

Youden Index and the optimal threshold for markers with mass at zero

The Youden Index is often used as a summary measure of the receiver operating characteristic curve. It measures the effectiveness of a diagnostic marker and permits the selection of an optimal threshold value or cutoff point for the biomarker of interest. Some markers, while basically continuous and positive, have a spike or positive mass of probability at the value zero. We provide a flexible modeling approach for estimating the Youden Index and its associated cutoff point for such spiked data and compare it with the standard empirical approach. We show how this modeling approach can be adjusted to take covariate information into account. This approach is applied to data on the Coronary Calcium Score, a marker for atherosclerosis.     

The analysis of multivariate interval-censored survival data

This paper considers a marginal approach for the analysis of the effect of covariates on multivariate interval-censored survival data.Interval censoring of multivariate events can occur when the events are not directly observable but are detected by periodically performing clinical examinations or laboratory tests. The method assumes the marginal distribution for each event is based on a discrete analogue of the proportional hazards model for interval-censored data. A robust estimator for the covariance matrix is developed that accounts for the correlation between events. A simulation study comparing the performance of this method and a midpoint imputation approach indicates the parameter estimates from the proposed method are less biased. Furthermore, even when the events are only modestly correlated, ignoring the correlation can result in erroneous variance estimators. The method is illustrated using data from an ongoing clinical trial involving subjects with systemic lupus erythematosus.     

Analysis of interval-censored data from circular migrant and non-migrant sexual partnerships using the EM algorithm

In epidemiological studies where subjects are seen periodically on follow-up visits, interval-censored data occur naturally. The exact time the change of state (such as HIV seroconversion) occurs is not known exactly, only that it occurred sometime within a specific time interval. Methods of estimation for interval-censored data are readily available when data are independent. However, methods for correlated interval-censored data are not well developed. This paper considers an approach for estimating the parameters when data are interval-censored and correlated within sexual partnerships. We consider the exact event times for interval-censored observations as unobserved data, only known to be between two time points. Dependency induced by sexual partnerships is modelled as frailties assuming a gamma distribution for frailties and an exponential distribution on the time to infection. This formulation facilitates application of the expectation-maximization (EM) algorithm. Maximization process maximizes the standard survival frailty model. Results show high degree of heterogeneity between sexual partnerships. Intervention strategies aimed at combating the spread of HIV and other sexually transmitted infections (STI)s should treat sexual partnerships as social units and fully incorporate the effects of migration in their strategies.     

Multiple imputation for interval censored data with auxiliary variables

We propose a non-parametric multiple imputation scheme, NPMLE imputation, for the analysis of interval censored survival data. Features of the method are that it converts interval-censored data problems to complete data or right censored data problems to which many standard approaches can be used, and that measures of uncertainty are easily obtained. In addition to the event time of primary interest, there are frequently other auxiliary variables that are associated with the event time. For the goal of estimating the marginal survival distribution, these auxiliary variables may provide some additional information about the event time for the interval censored observations. We extend the imputation methods to incorporate information from auxiliary variables with potentially complex structures. To conduct the imputation, we use a working failure-time proportional hazards model to define an imputing risk set for each censored observation. The imputation schemes consist of using the data in the imputing risk sets to create an exact event time for each interval censored observation. In simulation studies we show that the use of multiple imputation methods can improve the efficiency of estimators and reduce the effect of missing visits when compared to simpler approaches. We apply the approach to cytomegalovirus shedding data from an AIDS clinical trial, in which CD4 count is the auxiliary variable.     

Evaluating factors associated with STD infection in a study with interval-censored event times and an unknown proportion of participants not at risk for disease

Sexually transmitted diseases (STD) are a major cause of morbidity and mortality world-wide. Because of their association with an increased risk of infection with human immunodeficiency virus, the prevention and control of STD are particularly important. Studies designed to evaluate factors associated with the transmission of STD can pose a number of statistical challenges, however. Two such concerns are the interval-censored event times that result from spacing between follow-up test visits, and an unknown proportion of study participants who are not at risk for infection. Researchers in various fields of study have used parametric mixture models to account for individuals not at risk. Owing to non-identifiability concerns within the mixture model framework, however, it is not always possible to distinguish between effects of explanatory variables on the distribution of event times for at-risk individuals and their effects on the probability of being at risk. We address these issues using data from a clinical trial designed to investigate the effectiveness of an intravaginal microbicide in preventing male-to-female transmission of STD. Factors associated with time to infection among at-risk women are initially identified by fitting right-truncated models to the interval-censored event times of participants who tested positive for STD, and hence are known to have been at risk. Subsequently, factors associated with the probability of being at risk are evaluated using mixture models that incorporate information contributed by the right-censored event-free times of uninfected study participants.     

A two-sample test with interval censored data via multiple imputation

Interval censored data arise naturally in large scale panel studies where subjects can only be followed periodically and the event of interest can only be recorded as having occurred between two examination times. In this paper we consider the problem of comparing two interval-censored samples. We propose to impute exact failure times from interval-censored observations to obtain right censored data, then apply existing techniques, such as Harrington and Fleming's G(rho) tests to imputed right censored data. To appropriately account for variability, a multiple imputation algorithm based on the approximate Bayesian bootstrap (ABB) is discussed. Through simulation studies we find that it performs well. The advantage of our proposal is its simplicity to implement and adaptability to incorporate many existing two-sample comparison techniques for right censored data. The method is illustrated by reanalysing the Breast Cosmesis Study data set.     

Extension of the association structure in joint models to include weighted cumulative effects

Motivated by a study measuring diabetes‐related risk factors and complications, we postulate an extension to the standard formulation of joint models for longitudinal and survival outcomes, wherein the longitudinal outcome has a cumulative effect on the hazard of the event, weighted by recency. We focus on the relationship between the biomarker HbA1c and the development of sight threatening retinopathy, since the impact of the HbA1c marker on the risk of sight threatening retinopathy is expected to be cumulative, with the evolution of the HbA1c marker over time contributing to progressively greater damage to the vascular structure of the retina. Opting for a parametric approach, we propose the use of the normal and skewed normal probability density functions as weight functions, estimating the relevant parameters directly from the data. The use of the recency‐weighted cumulative effect specification allows us to incorporate differences in the development of the longitudinal profile over time in the calculation of hazard ratios between subjects. The proposed functions provide us with parameters with clinically relevant interpretations while retaining a degree of flexibility. In addition, they also allow answering of important clinical questions regarding the relative importance of various segments of the biomarkers history in the estimation of the risk of the event. Copyright © 2017 John Wiley & Sons, Ltd.     

A Bayesian analysis of doubly censored data using a hierarchical Cox model

Two common statistical problems in pooling survival data from several studies are addressed. The first problem is that the data are doubly censored in that the origin is interval censored and the endpoint event may be right censored. Two approaches to incorporate the uncertainty of interval-censored origins are developed, and then compared with more usual analyses using imputation of a single fixed value for each origin. The second problem is that the data are collected from multiple studies and it is likely that heterogeneity exists among the study populations. A random-effects hierarchical Cox proportional hazards model is therefore used.The scientific problem motivating this work is a pooled survival analysis of data sets from three studies to examine the effect of GB virus type C (GBV-C) coinfection on survival of HIV-infected individuals. The time of HIV infection is the origin and for each subject this time is unknown, but is known to lie later than the last time at which the subject was known to be HIV negative, and earlier than the first time the subject was known to be HIV positive. The use of an approximate Bayesian approach using the partial likelihood as the likelihood is recommended because it more appropriately incorporates the uncertainty of interval-censored HIV infection times.     

Analysis of recurrent event data with incomplete observation gaps

In analysis of recurrent event data, recurrent events are not completely experienced when the terminating event occurs before the end of a study. To make valid inference of recurrent events, several methods have been suggested for accommodating the terminating event (Statist. Med. 1997; 16:911-924; Biometrics 2000; 56:554-562). In this paper, our interest is to consider a particular situation, where intermittent dropouts result in observation gaps during which no recurrent events are observed. In this situation, risk status varies over time and the usual definition of risk variable is not applicable. In particular, we consider the case when information on the observation gap is incomplete, that is, the starting time of intermittent dropout is known but the terminating time is not available. This incomplete information is modeled in terms of an interval-censored mechanism. Our proposed method is applied to the study of the Young Traffic Offenders Program on conviction rates, wherein a certain proportion of subjects experienced suspensions with intermittent dropouts during the study.     

SARS incubation and quarantine times: when is an exposed individual known to be disease free?

The setting of a quarantine time for an emerging infectious disease will depend on current knowledge concerning incubation times. Methods for the analysis of information on incubation times are investigated with a particular focus on inference regarding a possible maximum incubation time, after which an exposed individual would be known to be disease free. Data from the Hong Kong SARS epidemic are used for illustration. The incorporation of interval-censored data is considered and comparison is made with percentile estimation. Results suggest that a wide class of models for incubation times should be considered because the apparent informativeness of a likelihood depends on the choice and generalizability of a model. There will usually remain a probability of releasing from quarantine some infected individuals and the impact of early release will depend on the size of the epidemic.     

Multiple augmentation for interval-censored data with measurement error

There has been substantial effort devoted to the analysis of censored failure time with covariates that are subject to measurement error. Previous studies have focused on right-censored survival data, but interval-censored survival data with covariate measurement error are yet to be investigated. Our study is partly motivated by analysis of the HIV clinical trial AIDS Clinical Trial Group (ACTG) 175 data, where the occurrence time of AIDS is interval censored and the covariate CD4 count is subject to measurement error. We assume that the data are realized from a proportional hazards model. A multiple augmentation approach is proposed to convert interval-censored data to right-censored data, and the conditional score approach is then employed to account for measurement error. The proposed approach is easy to implement and can be readily extended to other semiparametric models. Extensive simulations show that the proposed approach has satisfactory finite-sample performance. The ACTG 175 data are then analyzed.     

Residual analysis in linear regression models with an interval-censored covariate

Residual analysis is a useful class of techniques for the evaluation of the goodness of a fitted model. Checking the underlying assumptions is important since most linear regression estimators require a correctly specified regression function and independent and identically distributed errors to be consistent. For uncensored data, the examination of the residuals of the fitted model is a standard tool for checking whether or not the underlying model assumptions hold. Such analysis has not been widely developed for censored data. Hillis (Statistics in Medicine 1995; 14:2023-2036) developed a residual plot for model checking when the response variable of a linear model is right-censored, and Gomez et al. (Statistics in Medicine 2003; 22:409-425) proposed residuals in models with interval-censored covariates. In this paper, we propose a new definition of residuals for linear models that incorporate interval-censored covariates. This definition can be also applied when the response variable is interval-censored. These new residuals are shown to perform better in model checking than other types of residuals in this context. We illustrate them with a data set from an AIDS clinical trial study.     

Prediction trees with soft nodes for binary outcomes

Consider the problem of predicting the occurrence of an event, the onset of diabetes mellitus, say, from a vector of continuous and discrete predictors. We propose a new algorithm for the construction of a tree-structured predictor for the event of interest, which uses a new approach for dealing with continuous predictors. The novelty is that the tree uses splits for continuous variables. This means that at each node an individual goes to the right branch with a certain probability, function of a predictor. The predictor as well as the particular shape of the function is chosen from the data by the proposed algorithm. We evaluate its performance on several real data sets, in particular comparing it with a standard tree-growing algorithm. We also present an analysis of a well-known data set, the Pima Indian diabetes data set, to illustrate the application of the method in biostatistics.     

含有区间删失数据时的生存函数估计及其SAS实现

目的 介绍生存分析中区间删失数据的特点及含区间删失数据生存函数估计的非参数方法,并介绍其在SAS9.2中的程序实现.方法 采用非参数极大似然估计法对Beadle等研究2种疗法对早期乳腺癌患者形体美容效果研究的区间删失数据进行生存函数估计,并利用SAS中的宏程序%EMICM实现其计算.结果 非参数极大似然估计法有效的计算出了2组疗法患者每个个体的生存概率、死亡概率、累积生存概率以及生存率的标准误,绘制出了2组疗法患者的生存曲线.结论 区间删失数据是临床实践中较常见的一类数据,区间删失数据生存函数估计的非参数方法 是常用的一种方法,但其计算相对右删失数据过程复杂,缺乏相应的软件实现,本文提供的SAS宏程序可以有效的实现计算,为实际工作提供帮助.     

Survival curve estimation for informatively coarsened discrete event-time data

Interval-censored, or more generally, coarsened event-time data arise when study participants are observed at irregular time periods and experience the event of interest in between study observations. Such data are often analysed assuming non-informative censoring, which can produce biased results if the assumption is wrong. This paper extends the standard approach for estimating survivor functions to allow informatively interval-censored data by incorporating various assumptions about the censoring mechanism into the model. We include a Bayesian extension in which final estimates are produced by mixing over a distribution of assumed censoring mechanisms. We illustrate these methods with a natural history study of HIV-infected individuals using assumptions elicited from an AIDS expert.     

Generalized log-rank test for mixed interval-censored failure time data

This paper considers the problem of non-parametric treatment comparisons when mixed interval-censored failure time data are available, which often occurs in clinical trials and epidemiological studies. By mixed interval-censored data, we mean that the survival time of interest is observed to belong to an interval or to be right-censored. For the problem, we generalize the most commonly used log-rank test for right-censored survival data. Numerical studies are conducted and reported to evaluate and compare the proposed test with the existing method, which indicate that the presented method works well. We apply the method to a data set arising from an AIDS cohort study, that motivated the study.