Length-biased Survival Data Research Articles

A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies.

Prevalent cohort studies in medical research often give rise to length-biased survival data that require special treatments. The recently proposed varying-coefficient partially linear transformation (VCPLT) model has the virtue of providing a more dynamic content of the effects of the covariates on survival times than the well-known partially linear transformation (PLT) model by allowing flexible interactions between the covariates. However, no existing analysis of the VCPLT model has considered length-biased sampling. In this paper, we consider the VCPLT model when the data are length-biased and right censored, thereby extending the reach of this flexible and powerful tool. We develop a martingale estimating function-based approach to the estimation of this model, provide theoretical underpinnings, evaluate finite sample performance via simulations, and showcase its practical appeal via an empirical application using data from two HIV vaccine clinical trials conducted by the U.S. National Institute of Allergy and Infectious Diseases.

Dependence measure for length-biased survival data using kernel density estimation with a regression procedure

In statistical literature, several dependence measures have been extensively established and treated, including Pearson's correlation coefficient, Spearman's ρ and Kendall's τ. In the context of survival analysis with length-biased data, a measure of dependence between survival time and covariates appears to have not received much intention in the literature. The purpose of this paper is to extend Kent's [Information gain and a general measure of correlation. Biometrika. 1983;70(1):163–173.] dependence measure, based on the concept of information gain, to length-biased survival data. Specifically, we develop a new approach to measure the degree of dependence between survival time and several continuous covariates, without censoring, when the relationship is linear. In this regard, kernel density estimation with a regression procedure is proposed. The consistency for all proposed estimators is established. In particular, the performance of the dependence measure for length-biased data is investigated by means of simulations studies.

Inference on quantile residual life for length-biased survival data

Length biased data occurs when a prevalent sampling is used to recruit subject into a study that investigates the time from an initial event to a terminal event. Such data are usually left-truncated and right-censored. While there have been accurate and efficient methods to estimate the survival function, not much work have been done regarding estimation of residual life time distribution or the summary parameters such as the median and quantiles of the residual life. In this paper we propose two ways to estimate the quantiles of the residual life time at fixed time points accounting for the length biased and censored na- ture of the data. We provide the asymptotic properties of these estimators and investigate them through simulation studies. Considering that the variance of these estimators require density estimation, we suggest an alternate approach to obtain the confidence intervals for the residual function. We apply these methods to a breast cancer dataset from National Surgical Adjuvant Breast and Bowel Project (NSABP).

Nonparametric smoothed quantile difference estimation for length-biased and right-censored data

We consider the nonparametric analysis of length-biased and right-censored data (LBRC) by quantile difference. With its desirable properties such as superior robustness and easy interpretation, quantile difference has been widely used in practice, in particular, for missing and survival data. Existing approaches for nonparametric estimation of quantile difference in length-biased survival data, however, exhibit some drawbacks such as non-smoothness and instabilities. To overcome these difficulties, we proposed a smoothed quantile difference estimation approach to improve its estimating efficiency with its validity justified by asymptotic theories. Simulations are also conducted to evaluate the performance of the proposed estimator. An application to the Channing house data is further provided for illustration.

A note on the length-biased Weibull-Gamma frailty survival model

Univariate length-biased survival data has been extensively treated in the statistical literature, and its importance has been shown under cross-sectional sampling of prevalent cases. In this article, we discuss frailty length-biased data models, and derive some specific quantities and relationships.

The estimation for the general additive–multiplicative hazard model using the length-biased survival data

We use the general additive–multiplicative hazard model to analyze the length biased data with right censorship and use the estimating equation method that incorporates the information about length-biased sampling scheme to do the inference. In addition, some graphical and numerical methods are developed for assessing the adequacy of the general additive–multiplicative hazard model. The procedures are derived from cumulative sums of martingale-based residuals over follow-up time and covariate values. The simulations are conducted to insure the good performance of this method. An application to the Oscar data is also illustrated.

Dependence measure for length-biased survival data using copulas

Abstract The linear correlation coefficient of Bravais-Pearson is considered a powerful indicator when the dependency relationship is linear and the error variate is normally distributed. Unfortunately in finance and in survival analysis the dependency relationship may not be linear. In such case, the use of rank-based measures of dependence, like Kendall’s tau or Spearman rho are recommended. In this direction, under length-biased sampling, measures of the degree of dependence between the survival time and the covariates appear to have not received much intention in the literature. Our goal in this paper, is to provide an alternative indicator of dependence measure, based on the concept of information gain, using the parametric copulas. In particular, the extension of the Kent’s [18] dependence measure to length-biased survival data is proposed. The performance of the proposed method is demonstrated through simulations studies.

Semiparametric maximum likelihood estimation for the Cox model with length-biased survival data

Right censored survival data collected on a prevalent cohort with constant incidence rate are termed as length-biased survival data. In this paper, we consider semiparametric estimation for the Cox proportional hazards model with length-biased survival data. Utilizing the special structure of length-biased sampling, we give the semiparametric maximum likelihood estimators for the regression parameter and cumulative hazard function. The estimators are shown to be consistent and asymptotically normal, and the limiting variance of the regression parameter estimator achieves the semiparametric efficiency bound. The asymptotic variance of the estimators can be estimated consistently. Simulation studies show the proposed semiparametric maximum likelihood estimator is more efficient than existing estimators in the literature. An analysis of the prevalent data from Canadian Study of Health and Aging (CSHA) illustrates the proposed method.

Inference under biased sampling and right censoring for a change point in the hazard function

Length-biased survival data commonly arise in cross-sectional surveys and prevalent cohort studies on disease duration. Ignoring biased sampling leads to bias in estimating the hazard-of-failure and the survival-time in the population. We address estimating the location of a possible change-point of an otherwise smooth hazard function when the collected data form a biased sample from the target population and the data are subject to informative censoring. We provide two estimation methodologies, for the location and size of the change-point, adapted to two scenarios of the truncation distribution: known and unknown. While the estimators in the first case show gain in efficiency as compared to those in the second case, the latter is more robust to the form of the truncation distribution. In both cases, the change-point estimators can achieve the rate $\mathcal{O}_{p}(1/n)$. We study the asymptotic properties of the estimates and devise interval-estimators for the location and size of the change, paving the way towards making statistical inference about whether or not a change-point exists. Several simulated examples are discussed to assess the finite sample behavior of the estimators. The proposed methods are then applied to analyze a set of survival data collected on elderly Canadian citizen (aged 65$+$) suffering from dementia.

Simple and fast overidentified rank estimation for right-censored length-biased data and backward recurrence time.

Length-biased survival data subject to right-censoring are often collected from a prevalent cohort. However, informative right censoring induced by the sampling design creates challenges in methodological development. While certain conditioning arguments could circumvent the problem of informative censoring, related rank estimation methods are typically inefficient because the marginal likelihood of the backward recurrence time is not ancillary. Under a semiparametric accelerated failure time model, an overidentified set of log-rank estimating equations is constructed based on the left-truncated right-censored data and backward recurrence time. Efficient combination of the estimating equations is simplified by exploiting an asymptotic independence property between two sets of estimating equations. A fast algorithm is studied for solving non-smooth, non-monotone estimating equations. Simulation studies confirm that the overidentified rank estimator can have a substantially improved estimation efficiency compared to just-identified rank estimators. The proposed method is applied to a dementia study for illustration.

Double bias: estimation of causal effects from length-biased samples in the presence of confounding.

Length bias in survival data occurs in observational studies when, for example, subjects with shorter lifetimes are less likely to be present in the recorded data. In this paper, we consider estimating the causal exposure (treatment) effect on survival time from observational data when, in addition to the lack of randomization and consequent potential for confounding, the data constitute a length-biased sample; we hence term this a double-bias problem. We develop estimating equations that can be used to estimate the causal effect indexing the structural Cox proportional hazard and accelerated failure time models for point exposures in double-bias settings. The approaches rely on propensity score-based adjustments, and we demonstrate that estimation of the propensity score must be adjusted to acknowledge the length-biased sampling. Large sample properties of the estimators are established and their small sample behavior is studied using simulations. We apply the proposed methods to a set of, partly synthesized, length-biased survival data collected as part of the Canadian Study of Health and Aging (CSHA) to compare survival of subjects with dementia among institutionalized patients versus those recruited from the community and depict their adjusted survival curves.

Empirical likelihood‐based confidence intervals for length‐biased data

Logistic or other constraints often preclude the possibility of conducting incident cohort studies. A feasible alternative in such cases is to conduct a cross-sectional prevalent cohort study for which we recruit prevalent cases, that is, subjects who have already experienced the initiating event, say the onset of a disease. When the interest lies in estimating the lifespan between the initiating event and a terminating event, say death for instance, such subjects may be followed prospectively until the terminating event or loss to follow-up, whichever happens first. It is well known that prevalent cases have, on average, longer lifespans. As such, they do not constitute a representative random sample from the target population; they comprise a biased sample. If the initiating events are generated from a stationary Poisson process, the so-called stationarity assumption, this bias is called length bias. The current literature on length-biased sampling lacks a simple method for estimating the margin of errors of commonly used summary statistics. We fill this gap by using the empirical likelihood-based confidence intervals by adapting this method to right-censored length-biased survival data. Both large and small sample behaviors of these confidence intervals are studied. We illustrate our method by using a set of data on survival with dementia, collected as part of the Canadian Study of Health and Aging.

Proportional mean residual life model for right-censored length-biased data

To study disease association with risk factors in epidemiologic studies, cross-sectional sampling is often more focused and less costly for recruiting study subjects who have already experienced initiating events. For time-to-event outcome, however, such a sampling strategy may be length biased. Coupled with censoring, analysis of length-biased data can be quite challenging, due to induced informative censoring in which the survival time and censoring time are correlated through a common backward recurrence time. We propose to use the proportional mean residual life model of Oakes & Dasu (Biometrika77, 409-10, 1990) for analysis of censored length-biased survival data. Several nonstandard data structures, including censoring of onset time and cross-sectional data without follow-up, can also be handled by the proposed methodology.

Covariate Bias Induced by Length-Biased Sampling of Failure Times

Although many authors have proposed different approaches to the analysis of length-biased survival data, a number of issues have not been fully addressed. The most important among these issues is perhaps that regarding inclusion of covariates into the analysis of length-biased lifetime data collected through cross-sectional sampling of a population. One aspect of this problem, which appears to have been neglected in the literature, concerns the effect of length bias on the sampling distribution of the covariates. In most regression analyses, it is conventional to condition on the observed covariate values; however, certain covariate values could be preferentially selected into the sample, being linked to the long-term survivors, who themselves are favored by the sampling mechanism. This observation raises two questions: (1) Does the conditional analysis of covariates lead to biased estimators of regression coefficients?; and (2) does inference through the joint l likelihood of covariates and failure times yield more efficient estimators of the regression parameters? We present a joint likelihood approach and study the large-sample behavior of the resulting maximum likelihood estimators (MLEs). We find that these MLEs are more efficient than their conditional counterparts even though the two MLEs are asymptotically equal. Our results are illustrated using data on survival with dementia, collected as part of the Canadian Study of Health and Aging.