Survival Modelling For Data From Combined Cohorts: Opening the Door to Meta Survival Analyses and Survival Analysis using Electronic Health Records.

Non-parametric estimation of the survival function using observed failure time data depends on the underlying data generating mechanism, including the ways in which the data may be censored and/or truncated. For data arising from a single source or collected from a single cohort, a wide range of estimators have been proposed and compared in the literature. Often, however, it may be possible, and indeed advantageous, to combine and then analyze survival data that have been collected under different study designs. We review non-parametric survival analysis for data obtained by combining the most common types of cohort. We have two main goals: (i) To clarify the differences in the model assumptions, and (ii) to provide a single lens through which some of the proposed estimators may be viewed. Our discussion is relevant to the meta analysis of survival data obtained from different types of study, and to the modern era of electronic health records.

Stacked survival models for residual lifetime data.

When modelling the survival distribution of a disease for which the symptomatic progression of the associated condition is insidious, it is not always clear how to measure the failure/censoring times from some true date of disease onset. In a prevalent cohort study with follow-up, one approach for removing any potential influence from the uncertainty in the measurement of the true onset dates is through the utilization of only the residual lifetimes. As the residual lifetimes are measured from a well-defined screening date (prevalence day) to failure/censoring, these observed time durations are essentially error free. Using residual lifetime data, the nonparametric maximum likelihood estimator (NPMLE) may be used to estimate the underlying survival function. However, the resulting estimator can yield exceptionally wide confidence intervals. Alternatively, while parametric maximum likelihood estimation can yield narrower confidence intervals, it may not be robust to model misspecification. Using only right-censored residual lifetime data, we propose a stacking procedure to overcome the non-robustness of model misspecification; our proposed estimator comprises a linear combination of individual nonparametric/parametric survival function estimators, with optimal stacking weights obtained by minimizing a Brier Score loss function.

Nested case-control study designs for left-truncated survival data.

The determination of risk factors for disease incidence has been the subject of much epidemiologic research. With this goal a common study design entails the follow-up of an initially disease-free cohort, keeping track of the dates of disease incidence (onset) and ascertaining covariate (putative risk factor) information on the full cohort. However, the collection of certain covariate information on all study subjects may be prohibitively expensive and, therefore, the nested case-control study has commonly been used. The high cost of full covariate information on all subjects also arises when determining risk factors for "failure," death say, "following" disease onset, in particular, in a prevalent cohort study with follow-up; in such a study a cohort of subjects with existing disease is followed. We here adapt nested case-control designs to the setting of a prevalent cohort study with follow-up, a topic previously not addressed in the literature. We provide the partial likelihood under risk set sampling and state the asymptotic properties of the estimated covariate effects and baseline cumulative hazard. We address the following design questions in the context of prevalent cohort studies with follow-up: How many subjects should be included in the sampled risk sets for efficient estimation? In what way is the proportion of censored subjects associated with the benefit of a nested case-control design? What proportion of overall variance is attributable to risk set sampling? This work is motivated by the anticipated analysis of data on survival with Parkinson's Disease, being collected as part of the ongoing Canadian Longitudinal Study on Aging.

La détermination des facteurs de risque pour l'incidence d'une maladie est le sujet de nombreuses études épidémiologiques. À cet effet, un plan d'expérience commun consiste à suivre une cohorte initialement en santé en prenant note de la date à laquelle la maladie se manifeste (début) et en évaluant les covariables (facteurs de risque présumés) pour la cohorte en entier. Lorsque la collecte de certaines covariables pour tous les individus s'avère trop onéreuse, une étude cas-témoins peut eˆtre considérée. Un problème similaire de couˆts élevés pour la collecte d'information sur tous les sujets peut également se présenter lorsque les facteurs de risque pour un « échec ¼, disons le décès, doivent eˆtre déterminés après le début de la maladie. Une telle situation peut survenir dans une étude sur une cohorte prévalente avec suivi, mais la question n'a pas encore été traitée dans la littérature. Les auteurs développent la vraisemblance partielle pour un échantillonnage dans l'ensemble à risque et décrivent les propriétés asymptotiques des estimés de l'effet des covariables et de la fonction cumulative du risque de base. Ils répondent à certaines questions émergeant d'un plan d'expérience pour une cohorte prévalente avec suivi, notamment le nombre de sujets à inclure dans l'ensemble de risque pour obtenir une estimation efficace, les façons dont la proportion de sujets censurés est liée aux bénéfices d'un plan cas-témoin imbriqué, et la proportion de variance globale attribuable à l'échantillonnage dans l'ensemble à risque. Le développement de ces méthodes est motivé par l'analyse imminente de données de survie de patients atteints de la maladie de Parkinson dont la collecte est en cours dans le cadre de l'é tude longitudinale canadienne sur le vieillissement.

The survival function NPMLE for combined right-censored and length-biased right-censored failure time data: properties and applications.

Many cohort studies in survival analysis have imbedded in them subcohorts consisting of incident cases and prevalent cases. Instead of analysing the data from the incident and prevalent cohorts alone, there are surely advantages to combining the data from these two subcohorts. In this paper, we discuss a survival function nonparametric maximum likelihood estimator (NPMLE) using both length-biased right-censored prevalent cohort data and right-censored incident cohort data. We establish the asymptotic properties of the survival function NPMLE and utilize the NPMLE to estimate the distribution for time spent in a Montreal area hospital.

Benefits of combining prevalent and incident cohorts: An application to myotonic dystrophy.

It is frequently of interest to estimate the time that individuals survive with a disease, that is, to estimate the time between disease onset and occurrence of a clinical endpoint such as death. Epidemiologic survival data are commonly collected from either an incident cohort, whose members' disease onset occurs after the study baseline date, or from a cohort with prevalent disease that is followed forward in time. Incident cohort survival data are limited by study termination, while prevalent cohort data provide biased (left-truncated) survival data. In this article, we investigate the advantages of a study design featuring simultaneous follow-up of prevalent and incident cohorts to the estimation of the survivor function. Our analyses are supported by simulations and illustrated using data on survival after myotonic dystrophy diagnosis from the United Kingdom Clinical Practice Research Datalink (CPRD). We demonstrate that the NPMLE using combined incident and prevalent cohort data estimates the true survivor function very well, even for moderate sample sizes, and ameliorates the disadvantages of using a purely incident or prevalent cohort.

Testing the assumptions for the analysis of survival data arising from a prevalent cohort study with follow-up.

In a prevalent cohort study with follow-up subjects identified as prevalent cases are followed until failure (defined suitably) or censoring. When the dates of the initiating events of these prevalent cases are ascertainable, each observed datum point consists of a backward recurrence time and a possibly censored forward recurrence time. Their sum is well known to be the left truncated lifetime. It is common to term these left truncated lifetimes "length biased" if the initiating event times of all the incident cases (including those not observed through the prevalent sampling scheme) follow a stationary Poisson process. Statistical inference is then said to be carried out under stationarity. Whether or not stationarity holds, a further assumption needed for estimation of the incident survivor function is the independence of the lifetimes and their accompanying truncation times. That is, it must be assumed that survival does not depend on the calendar date of the initiating event. We show how this assumption may be checked under stationarity, even though only the backward recurrence times and their associated (possibly censored) forward recurrence times are observed. We prove that independence of the lifetimes and truncation times is equivalent to equality in distribution of the backward and forward recurrence times, and exploit this equivalence as a means of testing the former hypothesis. A simulation study is conducted to investigate the power and Type 1 error rate of our proposed tests, which include a bootstrap procedure that takes into account the pairwise dependence between the forward and backward recurrence times, as well as the potential censoring of only one of the members of each pair. We illustrate our methods using data from the Canadian Study of Health and Aging. We also point out an equivalence of the problem presented here to a non-standard changepoint problem.

Bayesian change-point analyses in ecology.

Ecological and biological processes can change from one state to another once a threshold has been crossed in space or time. Threshold responses to incremental changes in underlying variables can characterize diverse processes from climate change to the desertification of arid lands from overgrazing. Simultaneously estimating the location of thresholds and associated ecological parameters can be difficult: ecological data are often 'noisy', which can make the identification of the locations of ecological thresholds challenging. We illustrate this problem using two ecological examples and apply a class of statistical models well-suited to addressing this problem. We first consider the case of estimating allometric relationships between tree diameter and height when the trees have distinctly different growth modes across life-history stages. We next estimate the effects of canopy gaps and dense understory vegetation on tree recruitment in transects that transverse both canopy and gap conditions. The Bayesian change-point models that we present estimate both threshold locations and the slope or level of ecological quantities of interest, while incorporating uncertainty in the change-point location into these estimates. This class of models is suitable for problems with multiple thresholds and can account for spatial or temporal autocorrelation.

A formal test for the stationarity of the incidence rate using data from a prevalent cohort study with follow-up.

In a prevalent cohort study with follow-up, the incidence process is not directly observed since only the onset times of prevalent cases can be ascertained. Assessing the "stationarity" of the underlying incidence process can be important for at least three reasons, including an improvement in efficiency when estimating the survivor function. We propose, for the first time, a formal test for stationarity using data from a prevalent cohort study with follow-up. The test makes use of a characterization of stationarity, an extension of this characterization developed in this paper, and of a test for matched pairs of right censored data. We report the results from a power study assuming varying degrees of departure from the null hypothesis of stationarity. The test is also applied to data obtained as part of the Canadian Study of Health and Aging (CSHA) to verify whether the incidence rate of dementia amongst the elderly in Canada has remained constant.

Checking stationarity of the incidence rate using prevalent cohort survival data.

When survival data are collected as part of a prevalent cohort study with follow-up, the recruited cases have already experienced their initiating event, say onset of a disease, and consequently the incidence process is only partially observed. Nevertheless, there are good reasons for interest in certain features of the underlying incidence process, for example whether or not it is stationary. Indeed, the well known relationship between incidence and prevalence, often used by epidemiologists, requires stationarity of the incidence rate for its validity. Also, the statistician can exploit stationarity of the incidence process by improving the efficiency of estimators in a prevalent cohort survival analysis. In addition, whether the incident rate is stationary is often in itself of central importance to medical and other researchers. We present here a necessary and sufficient condition for stationarity of the underlying incidence process, which uses only survival observations, possibly right censored, from a prevalent cohort study with follow-up. This leads to a simple graphical means of checking for the stationarity of the underlying incidence times by comparing the plots of two Kaplan-Meier estimates that are based on partially observed incidence times and follow-up survival data. We use our method to discuss the incidence rate of dementia in Canada between 1971 and 1991.

A Bayesian A-optimal and model robust design criterion.

Suppose that the true model underlying a set of data is one of a finite set of candidate models, and that parameter estimation for this model is of primary interest. With this goal, optimal design must depend on a loss function across all possible models. A common method that accounts for model uncertainty is to average the loss over all models; this is the basis of what is known as Läuter's criterion. We generalize Läuter's criterion and show that it can be placed in a Bayesian decision theoretic framework, by extending the definition of Bayesian A-optimality. We use this generalized A-optimality to find optimal design points in an environmental safety setting. In estimating the smallest detectable trace limit in a water contamination problem, we obtain optimal designs that are quite different from those suggested by standard A-optimality.