One-step targeted maximum likelihood estimation for time-to-event outcomes

Researchers in observational survival analysis are interested in not only estimating survival curve nonparametrically but also having statistical inference for the parameter. We consider right-censored failure time data where we observe n independent and identically distributed observations of a vector random variable consisting of baseline covariates, a binary treatment at baseline, a survival time subject to right censoring, and the censoring indicator. We assume the baseline covariates are allowed to affect the treatment and censoring so that an estimator that ignores covariate information would be inconsistent. The goal is to use these data to estimate the counterfactual average survival curve of the population if all subjects are assigned the same treatment at baseline. Existing observational survival analysis methods do not result in monotone survival curve estimators, which is undesirable and may lose efficiency by not constraining the shape of the estimator using the prior knowledge of the estimand. In this paper, we present a one-step Targeted Maximum Likelihood Estimator (TMLE) for estimating the counterfactual average survival curve. We show that this new TMLE can be executed via recursion in small local updates. We demonstrate the finite sample performance of this one-step TMLE in simulations and an application to a monoclonal gammopathy data.     

SMIM: A unified framework of survival sensitivity analysis using multiple imputation and martingale

Censored survival data are common in clinical trial studies. We propose a unified framework for sensitivity analysis to censoring at random in survival data using multiple imputation and martingale, called SMIM. The proposed framework adopts the δ-adjusted and control-based models, indexed by the sensitivity parameter, entailing censoring at random and a wide collection of censoring not at random assumptions. Also, it targets a broad class of treatment effect estimands defined as functionals of treatment-specific survival functions, taking into account missing data due to censoring. Multiple imputation facilitates the use of simple full-sample estimation; however, the standard Rubin's combining rule may overestimate the variance for inference in the sensitivity analysis framework. We decompose the multiple imputation estimator into a martingale series based on the sequential construction of the estimator and propose the wild bootstrap inference by resampling the martingale series. The new bootstrap inference has a theoretical guarantee for consistency and is computationally efficient compared to the nonparametric bootstrap counterpart. We evaluate the finite-sample performance of the proposed SMIM through simulation and an application on an HIV clinical trial.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

Robust covariate-adjusted log-rank statistics and corresponding sample size formula for recurrent events data

Summary .   Recurrent events data are frequently encountered in clinical trials. This article develops robust covariate-adjusted log-rank statistics applied to recurrent events data with arbitrary numbers of events under independent censoring and the corresponding sample size formula. The proposed log-rank tests are robust with respect to different data-generating processes and are adjusted for predictive covariates. It reduces to the Kong and Slud (1997, Biometrika 84, 847–862) setting in the case of a single event. The sample size formula is derived based on the asymptotic normality of the covariate-adjusted log-rank statistics under certain local alternatives and a working model for baseline covariates in the recurrent event data context. When the effect size is small and the baseline covariates do not contain significant information about event times, it reduces to the same form as that of Schoenfeld (1983, Biometrics 39, 499–503) for cases of a single event or independent event times within a subject. We carry out simulations to study the control of type I error and the comparison of powers between several methods in finite samples. The proposed sample size formula is illustrated using data from an rhDNase study.     

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.     

Joint analysis of panel count and interval-censored data using distribution-free frailty analysis

We propose a joint analysis of recurrent and nonrecurrent event data subject to general types of interval censoring. The proposed analysis allows for general semiparametric models, including the Box–Cox transformation and inverse Box–Cox transformation models for the recurrent and nonrecurrent events, respectively. A frailty variable is used to account for the potential dependence between the recurrent and nonrecurrent event processes, while leaving the distribution of the frailty unspecified. We apply the pseudolikelihood for interval-censored recurrent event data, usually termed as panel count data, and the sufficient likelihood for interval-censored nonrecurrent event data by conditioning on the sufficient statistic for the frailty and using the working assumption of independence over examination times. Large sample theory and a computation procedure for the proposed analysis are established. We illustrate the proposed methodology by a joint analysis of the numbers of occurrences of basal cell carcinoma over time and time to the first recurrence of squamous cell carcinoma based on a skin cancer dataset, as well as a joint analysis of the numbers of adverse events and time to premature withdrawal from study medication based on a scleroderma lung disease dataset.     

Analysis of times to repeated events in two-arm randomized trials with noncompliance and dependent censoring

This article develops randomization-based methods for times to repeated events in two-arm randomized trials with noncompliance and dependent censoring. Structural accelerated failure time models are assumed to capture causal effects on repeated event times and dependent censoring time, but the dependence structure among repeated event times and dependent censoring time is unspecified. Artificial censoring techniques to accommodate nonrandom noncompliance and dependent censoring are proposed. Estimation of the acceleration parameters are based on rank-based estimating functions. A simulation study is conducted to evaluate the performance of the developed methods. An illustration of the methods using data from an acute myeloid leukemia trial is provided.     

Cox regression with survival-time-dependent missing covariate values

Analysis with time-to-event data in clinical and epidemiological studies often encounters missing covariate values, and the missing at random assumption is commonly adopted, which assumes that missingness depends on the observed data, including the observed outcome which is the minimum of survival and censoring time. However, it is conceivable that in certain settings, missingness of covariate values is related to the survival time but not to the censoring time. This is especially so when covariate missingness is related to an unmeasured variable affected by the patient's illness and prognosis factors at baseline. If this is the case, then the covariate missingness is not at random as the survival time is censored, and it creates a challenge in data analysis. In this article, we propose an approach to deal with such survival-time-dependent covariate missingness based on the well known Cox proportional hazard model. Our method is based on inverse propensity weighting with the propensity estimated by nonparametric kernel regression. Our estimators are consistent and asymptotically normal, and their finite-sample performance is examined through simulation. An application to a real-data example is included for illustration.     

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.     

Semiparametric regression estimation in the presence of dependent censoring

We propose a semiparametric estimation procedure for estimatingthe regression of an outcome Y, measured at the end of a fixedfollow-up period, on baseline explanatory variables X, measuredprior to start of follow-up, in the presence of dependent censoringgiven X. The proposed estimators are consistent when the dataare ‘missing at random’ but not ‘missing completelyat random’ (Rubin, 1976), and do not require full specificationof the complete data likelihood. Specifically, we assume thatthe probability of censoring at time t is independent of theoutcome Y conditional on the recorded history up to t of a vectorof time-dependent covariates that are correlated with Y. Ourestimators can be used to adjust for dependent censoring andnonrandom noncompliance in randomised trials studying the effectof a treatment on the mean of a response variable of interest.Even with independent censoring, our methods allow the investigatorto increase efficiency by exploiting the correlation of theoutcome with a vector of time-dependent covariates.     

Association tests for traits with variable age at onset

This paper is concerned with testing association between marker genotypes and traits with variable age at onset. Two methods are proposed, one which makes use of both age-at-ascertainment and age-at-onset information, and one which may be applied when only age-at-ascertainment information is available. (Here, by age-at-ascertainment, is meant the subject's age when presence of onset and age at onset are determined; for subjects who have died or are otherwise censored before ascertainment, the censoring time should be used instead). Adjustment for confounding due to population stratification is carried out by conditioning on observed traits and parental genotypes, or, if complete parental genotypes are not available, by conditioning on observed traits and the minimal sufficient statistics under the null hypothesis for the parental genotypes. Proportional hazards regression models and logistic regression models are used to motivate the methods, but correct type I error rates result even if the models are not correct. An illustrative example is described.     

"Smooth" semiparametric regression analysis for arbitrarily censored time-to-event data

Summary .   A general framework for regression analysis of time-to-event data subject to arbitrary patterns of censoring is proposed. The approach is relevant when the analyst is willing to assume that distributions governing model components that are ordinarily left unspecified in popular semiparametric regression models, such as the baseline hazard function in the proportional hazards model, have densities satisfying mild "smoothness" conditions. Densities are approximated by a truncated series expansion that, for fixed degree of truncation, results in a "parametric" representation, which makes likelihood-based inference coupled with adaptive choice of the degree of truncation, and hence flexibility of the model, computationally and conceptually straightforward with data subject to any pattern of censoring. The formulation allows popular models, such as the proportional hazards, proportional odds, and accelerated failure time models, to be placed in a common framework; provides a principled basis for choosing among them; and renders useful extensions of the models straightforward. The utility and performance of the methods are demonstrated via simulations and by application to data from time-to-event studies.     

Applications of a parametric model for informative censoring

In this article, we explore the use of a parametric model (for analyzing survival data) which is defined to allow sensitivity analysis for the presence of informative censoring. The dependence between the failure and the censoring processes is expressed through a parameter delta and a general bias function B(t, theta). We calculate the expectation of the potential bias due to informative censoring, which is an overall measure of how misleading our results might be if censoring is actually nonignorable. Bounds are also calculated for quantities of interest, e.g., parameter of the distribution of the failure process, which do not depend on the choice of the bias function for fixed delta. An application that relates to systematic lupus erythematosus data illustrates how additional information can result in reducing the uncertainty on estimates of the location parameter. Sensitivity analysis on a relative risk parameter is also explored.     

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.     

Use of logrank tests and group sequential methods at fixed calendar times

This paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time. Because the group sequential boundaries of Pocock (1977, Biometrika 64, 191-199) and O'Brien and Fleming (1979, Biometrics 35, 549-556) were constructed on the assumption that each interim analysis provides an equal increment of statistical information, these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times. Nonetheless, our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time.     

Conditioned genome reconstruction: how to avoid choosing the conditioning genome

Genome phylogenies can be inferred from data on the presence and absence of genes across taxa. Logdet distances may be a good method, because they allow expected genome size to vary across the tree. Recently, Lake and Rivera proposed conditioned genome reconstruction (calculation of logdet distances using only those genes present in a conditioning genome) to deal with unobservable genes that are absent from every taxon of interest. We prove that their method can consistently estimate the topology for almost any choice of conditioning genome. Nevertheless, the choice of conditioning genome is important for small samples. For real bacterial genome data, different choices of conditioning genome can result in strong bootstrap support for different tree topologies. To overcome this problem, we developed supertree methods that combine information from all choices of conditioning genome. One of these methods, based on the BIONJ algorithm, performs well on simulated data and may have applications to other supertree problems. However, an analysis of 40 bacterial genomes using this method supports an incorrect clade of parasites. This is a common feature of model-based gene content methods and is due to parallel gene loss.     

Nonparametric comparison of two survival-time distributions in the presence of dependent censoring

When testing the null hypothesis that treatment arm-specific survival-time distributions are equal, the log-rank test is asymptotically valid when the distribution of time to censoring is conditionally independent of randomized treatment group given survival time. We introduce a test of the null hypothesis for use when the distribution of time to censoring depends on treatment group and survival time. This test does not make any assumptions regarding independence of censoring time and survival time. Asymptotic validity of this test only requires a consistent estimate of the conditional probability that the survival event is observed given both treatment group and that the survival event occurred before the time of analysis. However, by not making unverifiable assumptions about the data-generating mechanism, there exists a set of possible values of corresponding sample-mean estimates of these probabilities that are consistent with the observed data. Over this subset of the unit square, the proposed test can be calculated and a rejection region identified. A decision on the null that considers uncertainty because of censoring that may depend on treatment group and survival time can then be directly made. We also present a generalized log-rank test that enables us to provide conditions under which the ordinary log-rank test is asymptotically valid. This generalized test can also be used for testing the null hypothesis when the distribution of censoring depends on treatment group and survival time. However, use of this test requires semiparametric modeling assumptions. A simulation study and an example using a recent AIDS clinical trial are provided.     

Multivariate survival models induced by genetic frailties,with application to linkage analysis

We derive a multivariate survival model for age of onset data of a sibship from an additive genetic gamma frailty model constructed basing on the inheritance vectors, and investigate the properties of this model. Based on this model, we propose a retrospective likelihood approach for genetic linkage analysis using sibship data. This test is an allele-sharing-based test, and does not require specification of genetic models or the penetrance functions. This new approach can incorporate both affected and unaffected sibs, environmental covariates and age of onset or age at censoring information and, therefore, provides a practical solution for mapping genes for complex diseases with variable age of onset. Small simulation study indicates that the proposed method performs better than the commonly used allele-sharing-based methods for linkage analysis, especially when the population disease rate is high. We applied this method to a type 1 diabetes sib pair data set and a small breast cancer data set. Both simulated and real data sets also indicate that the method is relatively robust to the misspecification to the baseline hazard function.     

Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Fundamentals of survival data

Survival data stand out as a special statistical field. This paper tries to describe what survival data is and what makes it so special. Survival data concern times to some events. A key point is the successive observation of time, which on the one hand leads to some times not being observed so that all that is known is that they exceed some given times (censoring), and on the other hand implies that predictions regarding the future course should be conditional on the present status (truncation). In the simplest case, this condition is that the individual is alive. The successive conditioning makes the hazard function, which describes the probability of an event happening during a short interval given that the individual is alive today (or more generally able to experience the event), the most relevant concept. Standard distributions available (normal, log-normal, gamma, inverse Gaussian, and so forth) can account for censoring and truncation, but this is cumbersome. Besides, they fit badly because they are either symmetric or right skewed, but survival time distributions can easily be left-skewed positive variables. A few distributions satisfying these requirements are available, but often nonparametric methods are preferable as they account better conceptually for truncation and censoring and give a better fit. Finally, we compare the proportional hazards regression models with accelerated failure time models.