In this paper we test the composite hypothesis that lifetimes follow an exponential distribution based on observed randomly right censored data. Testing this hypothesis is complicated by the presence of this censoring, due to the fact that not all lifetimes are observed. To account for this complication, we propose modifications to tests based on the empirical characteristic function and Laplace transform. In the full sample case these empirical functions can be expressed as integrals with respect to the empirical distribution function of the lifetimes. We propose replacing this estimate of the distribution function by the Kaplan-Meier estimate. The resulting test statistics can be expressed in easily calculable forms in terms of summations of functionals of the observed data. Additionally, a general framework for goodness-of-fit testing, in the presence of random right censoring, is outlined. A Monte Carlo study is performed, the results of which indicate that the newly modified tests generally outperform the existing tests. A practical application, concerning initial remission times of leukemia patients, is discussed along with some concluding remarks and avenues for future research.
For studying or reducing the bias of functionals of the Kaplan-Meier survival estimator, the jackknifing approach of Stute and Wang (1994) is natural. We have studied the behavior of the jackknife estimate of bias under different configurations of the censoring level, sample size, and the censoring and survival time distributions. The empirical research reveals some new findings about robust calculation of the bias, particularly for higher censoring levels. We have extended their jackknifing approach to cover the case where the largest observation is censored, using the imputation methods for …
In single-arm clinical trials with survival outcomes, the Kaplan-Meier estimator and its confidence interval are widely used to assess survival probability and median survival time. Since the asymptotic normality of the Kaplan-Meier estimator is a common result, the sample size calculation methods have not been studied in depth. An existing sample size calculation method is founded on the asymptotic normality of the Kaplan-Meier estimator using the log transformation. However, the small sample properties of the log transformed estimator are quite poor in small sample sizes (which are typical s…
The setting of a right-censored random sample subject to contamination is considered. In various fields, expert information is often available and used to overcome the contamination. This paper integrates expert knowledge into the product-limit estimator in two different ways with distinct interpretations. Strong uniform consistency is proved for both cases under certain assumptions on the kind of contamination and the quality of expert information, which sheds light on the techniques and decisions that practitioners may take. The nuances of the techniques are discussed -- also with a view tow…
The Kaplan--Meier (KM) estimator, which provides a nonparametric estimate of a survival function for time-to-event data, has wide application in clinical studies, engineering, economics and other fields. The theoretical properties of the KM estimator including its consistency and asymptotic distribution have been extensively studied. We reconstruct the KM estimator as an M-estimator by maximizing a quadratic M-function based on concordance, which can be computed using the expectation--maximization (EM) algorithm. It is shown that the convergent point of the EM algorithm coincides with the trad…
In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $θ(\widehat{F}_n)=\sum_{i,j}K(X_{[i:n]},X_{[j:n]})W_iW_j$ and $θ_U(\widehat{F}_n)=\sum_{i\neq j}K(X_{[i:n]},X_{[j:n]})W_iW_j/\sum_{i\neq j}W_iW_j$, where $\widehat{F}_n$ is the Kaplan-Meier estimator, $\{W_1,\ldots,W_n\}$ are the Kaplan-Meier weights and $K:(0,\infty)^2\to\mathbb R$ is a symmetric kernel. As in the canonical setting of uncensored data, we differentiate between two asymptotic behaviours for $θ(\widehat{F}_n)$ and $θ_U(\widehat{F}_n)$. Additionally, we derive an asymptotic canonical V-statistic repr…
Population attributable fractions aim to quantify the proportion of the cases of an outcome (for example, a disease) that would have been avoided had no individuals in the population been exposed to a given exposure. This quantity thus plays a crucial role in epidemiology and public health, notably to guide policies, interventions or to assess the burden of a disease due to a particular exposure. Various statistical methods have been proposed to estimate attributable fractions using observational data. When time-to-event data are used, several of these formulas yield invalid results. Alternati…
The Kaplan-Meier product-limit estimator is a simple and powerful tool in time to event analysis. An extension exists for populations stratified into cohorts where a population survival curve is generated by weighted averaging of cohort-level survival curves. For making population-level comparisons using this statistic, we analyze the statistics of the area between two such weighted survival curves. We derive the large sample behavior of this statistic based on an empirical process of product-limit estimators. This estimator was used by an interdisciplinary NIH-SSA team in the identification o…
In a longitudinal study, measures of key variables might be incomplete or partially recorded due to drop-out, loss to follow-up, or early termination of the study occurring before the advent of the event of interest. In this paper, we focus primarily on the implementation of a regression model with a randomly censored predictor. We examine, particularly, the use of inverse probability weighting methods in a generalized linear model (GLM), when the predictor of interest is right-censored, to adjust for censoring. To improve the performance of the complete-case analysis and prevent selection bia…
In survey analysis, the estimation of the cumulative distribution function (cdf) is of great interest: it allows for instance to derive quantiles estimators or other non linear parameters derived from the cdf. We consider the case where the response variable is a right censored duration variable. In this framework, the classical estimator of the cdf is the Kaplan-Meier estimator. As an alternative, we propose a nonparametric model-based estimator of the cdf in a finite population. The new estimator uses auxiliary information brought by a continuous covariate and is based on nonparametric media…
Comparing Two Survival Functions at a Fixed Timepoint
Biostatistics Research Branch, National Institute of Allergy and Infectious Diseases（过敏和传染病研究所生物统计研究分支）；Frederick National Laboratory for Cancer Research（弗雷德癌症国家实验室）；Axle Informatics（Axle信息学）
In the context of right-censored and interval-censored data we develop asymptotic formulas to compute pseudo-observations for the survival function and the Restricted Mean Survival Time (RMST). Those formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan-Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived invo…
Kaplan-Meier survival analysis represents the most objective measure of treatment efficacy in oncology, though subjected to potential bias, which is worrisome in an era of precision medicine. Independent of the bias inherent to the design and execution of clinical trials, bias may be the result of patient censoring, or incomplete observation. Unlike disease/progression free survival, overall survival is based on a well defined time point and thus avoids interval censoring, but right-censoring, due to incomplete follow-up, may still be a source of bias. We study three mechanisms of right-censor…
Mendelian diseases are determined by a single mutation in a given gene. However, in the case of diseases with late onset, the age at onset is variable; it can even be the case that the onset is not observed in a lifetime. Estimating the survival function of the mutation carriers and the effect of modifying factors such as the sex, mutation, origin, etc, is a task of importance, both for management of mutation carriers and for prevention. In this work, we present a semi-parametric method based on a proportional to estimate the survival function using pedigrees ascertained through affected indiv…
A reduced-bias nonparametric estimator of the cumulative distribution function (CDF) and the survival function is proposed using infinite-order kernels. Fourier transform theory on generalized functions is utilized to obtain the improved bias estimates. The new estimators are analyzed in terms of their relative deficiency to the empirical distribution function and Kaplan-Meier estimator, and even improvements in terms of asymptotic relative efficiency (ARE) are present under specified assumptions on the data. The deficiency analysis introduces a deficiency rate which provides a continuum betwe…
Inferential challenges that arise when data are censored have been extensively studied under the classical frameworks. In this paper, we provide an alternative generalized inferential model approach whose output is a data-dependent plausibility function. This construction is driven by an association between the distribution of the relative likelihood function at the interest parameter and an unobserved auxiliary variable. The plausibility function emerges from the distribution of a suitably calibrated random set designed to predict that unobserved auxiliary variable. The evaluation of this pla…
Let F be an unknown univariate distribution function to be estimated from a sample containing censored observations and tau be in dom(F). The author has derived a novel nonparametric estimator F_hat for F without making any assumptions regarding the nature of the censoring mechanism or the distribution function F. The distribution of F_hat(tau) can be easily and accurately estimated even for small sample sizes. The estimator F_hat has significantly outperformed the Kaplan Meier estimator in a simulation study with an exponential and a lognormal distribution functions F and a censoring mechanis…
The Aalen-Johansen estimator generalizes the Kaplan-Meier estimator for independently left-truncated and right-censored survival data to estimating the transition probability matrix of a time-inhomogeneous Markov model with finite state space. Such multi-state models have a wide range of applications for modelling complex courses of a disease over the course of time, but the Markov assumption may often be in doubt. If censoring is entirely unrelated to the multi-state data, it has been noted that the Aalen-Johansen estimator, standardized by the initial empirical distribution of the multi-stat…
