Confidence Intervals for the Relative Risk Ratio Parameter from Survival Data Under a Random Censorship Model in Biomedical and Epidemiologic Studies (By Simulation)

The present paper reports the results of a Monte Carlo simulation study to examine the performance of several approximate confidence intervals for the Relative Risk Ratio (RRR) parameter in an epidemiologic study, involving two groups of individuals. The first group consists of n₁ individuals, called the experimental group, who are exposed to some carcinogen, say radiation, whose effect on the incidence of some form of cancer, say skin cancer, is being investigated. The second group consists of n₂ individuals (called the control group) who are not exposed to the carcinogen. Two cases are considered in which the life times (or time to cancer) in the two groups follow (i) the exponential and (ii) the Weibull distributions. The case when the life times follow a Rayleigh distribution follows as a particular case. A general random censorship model is considered in which the life times of the individuals are censored on the right by random censoring times following (i) the exponential and (ii) the Weibull distributions. The Relative Risk Ratio parameter in the study is defined as the ratio of the hazard rates in the two distributions of the times to cancer. Approximate confidence intervals are constructed for the RRR parameter using its maximum likelihood estimator (m.l.e) and several other methods, including a method due to FIELLER. SPROTT'S (1973) and Cox's (1953) suggestions, as well as the Box-Cox (1964) transformation, are also utilized to construct approximate confidence intervals. The performance of these confidence intervals in small samples is investigated by means of some Monte Carlo simulations based on 500 random samples. Our simulation study indicates that many of these confidence intervals perform quite well in samples of size 10 and 15, in terms of the coverage probability and expected length of the interval.     

Addressing an Idiosyncrasy in Estimating Survival Curves Using Double Sampling in the Presence of Self-Selected Right Censoring

We investigate the use of follow-up samples of individuals to estimate survival curves from studies that are subject to right censoring from two sources: (i) early termination of the study, namely, administrative censoring, or (ii) censoring due to lost data prior to administrative censoring, so-called dropout. We assume that, for the full cohort of individuals, administrative censoring times are independent of the subjects' inherent characteristics, including survival time. To address the loss to censoring due to dropout, which we allow to be possibly selective, we consider an intensive second phase of the study where a representative sample of the originally lost subjects is subsequently followed and their data recorded. As with double-sampling designs in survey methodology, the objective is to provide data on a representative subset of the dropouts. Despite assumed full response from the follow-up sample, we show that, in general in our setting, administrative censoring times are not independent of survival times within the two subgroups, nondropouts and sampled dropouts. As a result, the stratified Kaplan-Meier estimator is not appropriate for the cohort survival curve. Moreover, using the concept of potential outcomes, as opposed to observed outcomes, and thereby explicitly formulating the problem as a missing data problem, reveals and addresses these complications. We present an estimation method based on the likelihood of an easily observed subset of the data and study its properties analytically for large samples. We evaluate our method in a realistic situation by simulating data that match published margins on survival and dropout from an actual hip-replacement study. Limitations and extensions of our design and analytic method are discussed.     

General right censoring and its impact on the analysis of survival data.

This paper concerns general right censoring and some of the difficulties it creates in the analysis of survival data. A general formulation of censored-survival processes leads to the partition of all models into those based on noninformative and informative censoring. Nearly all statistical methods for censored data assume that censoring is noninformative. Topics considered within this class include: the relationships between three models for noninformative censoring, the use of likelihood methods for inferences about the distribution of survival time, the effects of censoring on the K-sample problem, and the effects of censoring on model testing. Also considered are several topics which relate to informative censoring models. These include: problems of nonidentifiability that can be encountered when attempting to assess a set of data for the type of censoring in effect, the consequences of falsely assuming that censoring is noninformative, and classes of informative censoring models.     

A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

Estimation of morbid risk and age at onset with missing information.

Investigators of genetic illnesses are currently employing life-table techniques to estimate the lifetime risk of disease and the age-at-onset distribution. This methodology assumes that onset ages are known for affected individuals and that censoring ages are known for unaffected individuals. We extend these methods to incorporate affected individuals with unknown onset ages and unaffected persons with unknown censoring ages and illustrate how conventional life-table methods can produce seriously biased estimates, particularly of lifetime risk. The methodology is not restricted to genetic illnesses and can be applied to more complex illnesses with unknown etiology. We present an example for Huntington disease, which is generally assumed to be a Mendelian autosomal dominant disease, yielding estimates of lifetime risk of .503 +/- .70 and mean onset age of 47.7 +/- 3.1 years for offspring with a single affected parent. When conventional life-table techniques are employed, these estimates are .238 +/- .032 and 43.2 +/- 2.2.     

Nonparametric estimation of stage occupation probabilities in a multistage model with current status data

Multistage models are used to describe individuals (or experimental units) moving through a succession of "stages" corresponding to distinct states (e.g., healthy, diseased, diseased with complications, dead). The resulting data can be considered to be a form of multivariate survival data containing information about the transition times and the stages occupied. Traditional survival analysis is the simplest example of a multistage model, where individuals begin in an initial stage (say, alive) and move irreversibly to a second stage (death). In this article, we consider general multistage models with a directed tree structure (progressive models) in which individuals traverse through stages in a possibly non-Markovian manner. We construct nonparametric estimators of stage occupation probabilities and marginal cumulative transition hazards. Empirical calculations of these quantities are not possible due to the lack of complete data. We consider current status information which represents a more severe form of censoring than the commonly used right censoring. Asymptotic validity of our estimators can be justified using consistency results for nonparametric regression estimators. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on these limited data compare well with those based on complete data. We also apply our method to a real-life data set arising from a cardiovascular diseases study in Taiwan.     

Spatial dependence modeling of latent susceptibility and time to joint damage in psoriatic arthritis

Important scientific insights into chronic diseases affecting several organ systems can be gained from modeling spatial dependence of sites experiencing damage progression. We describe models and methods for studying spatial dependence of joint damage in psoriatic arthritis (PsA). Since a large number of joints may remain unaffected even among individuals with a long disease history, spatial dependence is first modeled in latent joint-specific indicators of susceptibility. Among susceptible joints, a Gaussian copula is adopted for dependence modeling of times to damage. Likelihood and composite likelihoods are developed for settings, where individuals are under intermittent observation and progression times are subject to type K interval censoring. Two-stage estimation procedures help mitigate the computational burden arising when a large number of processes (i.e., joints) are under consideration. Simulation studies confirm that the proposed methods provide valid inference, and an application to the motivating data from the University of Toronto Psoriatic Arthritis Clinic yields important insights which can help physicians distinguish PsA from arthritic conditions with different dependence patterns.     

Inference on Risk Rates Based on Mortality Data Under Censoring and Competing Risks Using Parametric Models

In this paper we consider the competing risks model where the risks may not be independent. We assume both fixed and random censoring. The random censoring mechanism could have either a parametric or a non-parametric form. The life distributions and the parametric censoring distribution considered are exponential or Weibull. The expressions for the asymptotic confidence intervals for various parameters of interest under different models, using the estimated Fisher information matrix and parametric bootstrap techniques have been derived. Monte Carlo simulation studies for some of these cases have been carried out.     

Semiparametric estimation of proportional mean residual life model in presence of censoring

A mean residual life function is the average remaining life of a surviving subject, as it varies with time. The proportional mean residual life model was proposed by Oakes and Dasu (1990, Biometrika77, 409-410) in regression analysis to study its association with related covariates in absence of censoring. In this article, we develop some semiparametric estimation procedures to take censoring into account. The proposed methodology is evaluated via simulation studies, and further applied to a clinical trial of chemotherapy in postoperative radiotherapy of lung cancer patients.     

A modification of the redistribution to the right algorithm using disease markers

Disease markers are time-dependent covariates which describeprogression towards development of disease. Traditional methodsin survival analysis do not make use of available data on thesemarkers to recover additional information from censored individuals.Using a heuristic modification of the redistribution to theright algorithm (Efron, 1967), a new approach for recoveringinformation for censored individuals using disease markers isproposed. Additionally, the statistical properties of the proposedmethod are examined. There are two possible advantages to thismodification: (i) bias reduction when censoring is informative,and (ii) an increase in efficiency in the case of truly noninformativecensoring.     

Selecting an efficient design for assessing exposure-disease relationships in an assembled cohort

We develop approximate methods to compare the efficiencies and to compute the power of alternative potential designs for sampling from a cohort before beginning to collect exposure data. Our methods require only that the cohort be assembled, meaning that the numbers of individuals Nkj at risk at pairs of event times tk and tj greater than or equal to tk are available. To compute Nkj, one needs to know the entry, follow-up, censoring, and event history, but not the exposure, for each individual. Our methods apply to any "unbiased control sampling design," in which cases are compared to a random sample of noncases at risk at the time of an event. We apply our methods to approximate the efficiencies of the nested case-control design, the case-cohort design, and an augmented case-cohort design, compared to the full cohort design, in an assembled cohort of 17,633 members of an insurance cooperative who were followed for mortality from prostatic cancer. The assumptions underlying the approximation are that exposure is unrelated both to the hazard of an event and to the hazard for censoring. The approximations performed well in simulations when both assumptions held and when the exposure was moderately related to censoring.     

Dependence Functions and Frailties

This note considers association between nonnegative random variables in which the two observed survival times depend on an unobservable random variable via the proportional hazard model. When the random variables are subject to censoring, the conditional hazard functions provides a reasonable means of describing the association between the two variables. A numerical example demonstrating association in disease incidence in ordered pairs of individuals is analysed. Also, examples of distributions satisfying the notions of dependence considered are provided.     

Inverse Probability of Censoring Weighted Estimates of Kendall's τ for Gap Time Analyses

Summary In life history studies, interest often lies in the analysis of the interevent, or gap times and the association between event times. Gap time analyses are challenging however, even when the length of follow‐up is determined independently of the event process, because associations between gap times induce dependent censoring for second and subsequent gap times. This article discusses nonparametric estimation of the association between consecutive gap times based on Kendall's τ in the presence of this type of dependent censoring. A nonparametric estimator that uses inverse probability of censoring weights is provided. Estimates of conditional gap time distributions can be obtained following specification of a particular copula function. Simulation studies show the estimator performs well and compares favorably with an alternative estimator. Generalizations to a piecewise constant Clayton copula are given. Several simulation studies and illustrations with real data sets are also provided.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Estimating incremental cost-effectiveness ratios and their confidence intervals with differentially censored data

With medical costs escalating over recent years, cost analysis is being conducted more and more to assess economic impact of new treatment options. An incremental cost-effectiveness ratio (ICER) is a measure that assesses the additional cost for a new treatment for each additional unit of effectiveness, such as saving 1 year of life. In this article, we consider cost-effectiveness analysis for new treatments evaluated in a randomized clinical trial setting with staggered entries. In particular, the censoring times are different for cost and survival data. We propose a method for estimating the ICER and obtaining its confidence interval when differential censoring exists. Simulation experiments are conducted to evaluate our proposed method. We also apply our methods to a clinical trial example comparing the cost-effectiveness of implanted defibrillators with conventional therapy for individuals with reduced left ventricular function after myocardial infarction.     

Randomization inference with general interference and censoring

Interference occurs between individuals when the treatment (or exposure) of one individual affects the outcome of another individual. Previous work on causal inference methods in the presence of interference has focused on the setting where it is a priori assumed that there is “partial interference,” in the sense that individuals can be partitioned into groups wherein there is no interference between individuals in different groups. Bowers et al. (2012, Political Anal, 21, 97–124) and Bowers et al. (2016, Political Anal, 24, 395–403) consider randomization-based inferential methods that allow for more general interference structures in the context of randomized experiments. In this paper, extensions of Bowers et al. that allow for failure time outcomes subject to right censoring are proposed. Permitting right-censored outcomes is challenging because standard randomization-based tests of the null hypothesis of no treatment effect assume that whether an individual is censored does not depend on treatment. The proposed extension of Bowers et al. to allow for censoring entails adapting the method of Wang et al. (2010, Biostatistics, 11, 676–692) for two-sample survival comparisons in the presence of unequal censoring. The methods are examined via simulation studies and utilized to assess the effects of cholera vaccination in an individually randomized trial of 73 000 children and women in Matlab, Bangladesh.     

A Comparison of Robust Estimators in the Presence of Arbitrarily Right Censored Data

Robust estimation of a location parameter is considered when the data from an unknown symmetric population are subject to arbitrary right-censorship. Comparisons are made between various M-estimators, several L-estimators (trimmed means), and the Kaplan-Meier median. Ten sampling distributions, two uniform censoring distributions, and three sample sizes are examined. A Cauchy censoring distribution is also considered when the sample size is equal to twenty for each of the ten sampling distributions. Performance is based on the estimated mean square error.