Right-censored Survival Times Research Articles

Adjusting for Time-Varying Treatment Switches in Randomized Clinical Trials: The Danger of Extrapolation and How to Address It

When choosing estimands and estimators in randomized clinical trials, caution is warranted, as intercurrent events, such as, due to patients who switch treatment after disease progression, are often strongly associated with patients’ time-varying prognostic factors. Consequently, for patients who did experience intercurrent events, there are typically no comparable patients who did not. Statistical analyses may then easily lure one into making large implicit extrapolations, which often go unnoticed. We will illustrate this problem of implicit extrapolations using a real oncology case study, with a right-censored time-to-event endpoint, in which patients can cross over from the control to the experimental treatment after disease progression, for ethical reasons. We address this by developing an estimator for the survival risk ratio contrasting the survival probabilities at each time t if all patients would take experimental treatment with the survival probabilities at those times t if all patients would take control treatment up to time t, using randomization as an instrumental variable to avoid reliance on no unmeasured confounders assumptions. This doubly robust estimator can handle time-varying treatment switches and right-censored survival times. Insight into the rationale behind the estimator is provided and the approach is demonstrated by reanalyzing the oncology trial.

Description of survival data extended to the case of competing risks: a teaching approach based on frequency tables

Survival analysis is a powerful statistical tool to study failure-time data. In introductory courses students learn how to describe right-censored survival time data using the product-limit estimator of the survival function on a given end-point relying on a product of conditional survival probabilities. In the case of a composite end-point, the next step is to account for the presence of competing risks. The complement to one of the survival function is decomposed into the sum of cause-specific incidences, which are obtained as sum of unconditional probabilities due to the single competing risk. However, this algebraic decomposition is not straightforward, given the difference between the structure of the involved estimators. In addition, one is tempted to use the Kaplan-Meier estimator, leading to an erroneous decomposition of the overall incidence. Here we discuss a simple reinterpretation of the Kaplan-Meier formula in terms of sum of non-conditional probabilities of developing the end-point in time, adjusted for the presence of censoring. This approach could be used for describing survival data through simple frequency tables which are directly generalized to the case of competing risks. In addition, it makes clear how the estimation of the single cause-specific incidence through the Kaplan-Meier estimator, simply considering the occurrence of competing events as censored data, leads to an overestimation of the cause-specific incidence. Two examples are provided to support the explanation: the first one, could help to clarify the procedure described by the formulas; the second one, simulates real data in order to present graphically the results.

A Kernel Log-Rank Test of Independence for Right-Censored Data

We introduce a general nonparametric independence test between right-censored survival times and covariates, which may be multivariate. Our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert–Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, finding sufficient conditions to ensure our test correctly rejects the null hypothesis under any alternative. The test statistic can be computed straightforwardly, and the rejection threshold is obtained via an asymptotically consistent Wild Bootstrap procedure. Extensive investigations on both simulated and real data suggest that our testing procedure generally performs better than competing approaches in detecting complex nonlinear dependence.

Inference about time-dependent prognostic accuracy measures in the presence of competing risks

BackgroundEvaluating a candidate marker or developing a model for predicting risk of future conditions is one of the major goals in medicine. However, model development and assessment for a time-to-event outcome may be complicated in the presence of competing risks. In this manuscript, we propose a local and a global estimators of cause-specific AUC for right-censored survival times in the presence of competing risks.MethodsThe local estimator - cause-specific weighted mean rank (cWMR) - is a local average of time-specific observed cause-specific AUCs within a neighborhood of given time t. The global estimator - cause-specific fractional polynomials (cFPL) - is based on modelling the cause-specific AUC as a function of t through fractional polynomials.ResultsWe investigated the performance of the proposed cWMR and cFPL estimators through simulation studies and real-life data analysis. The estimators perform well in small samples, have minimal bias and appropriate coverage.ConclusionsThe local estimator cWMR and the global estimator cFPL will provide computationally efficient options for assessing the prognostic accuracy of markers for time-to-event outcome in the presence of competing risks in many practical settings.

Gaussian process regression for survival time prediction with genome-wide gene expression.

Predicting the survival time of a cancer patient based on his/her genome-wide gene expression remains a challenging problem. For certain types of cancer, the effects of gene expression on survival are both weak and abundant, so identifying non-zero effects with reasonable accuracy is difficult. As an alternative to methods that use variable selection, we propose a Gaussian process accelerated failure time model to predict survival time using genome-wide or pathway-wide gene expression data. Using a Monte Carlo expectation-maximization algorithm, we jointly impute censored log-survival time and estimate model parameters. We demonstrate the performance of our method and its advantage over existing methods in both simulations and real data analysis. The real data that we analyze were collected from 513 patients with kidney renal clear cell carcinoma and include survival time, demographic/clinical variables, and expression of more than 20 000 genes. In addition to the right-censored survival time, our method can also accommodate left-censored or interval-censored outcomes; and it provides a natural way to combine multiple types of high-dimensional -omics data. An R package implementing our method is available in the Supplementary material available at Biostatistics online.

Nonexistence of an unbiased estimating function for the Cox model

It is proved under broad regularity conditions that there is no unbiased estimating function depending only on the structural parameter (the coefficients) in the Cox proportional hazards model with nuisance survival hazard parameters and possibly right-censored survival times.

Modeling clustered long-term survivors using marginal mixture cure model.

There is a great deal of recent interests in modeling right-censored clustered survival time data with a possible fraction of cured subjects who are nonsusceptible to the event of interest using marginal mixture cure models. In this paper, we consider a semiparametric marginal mixture cure model for such data and propose to extend an existing generalized estimating equation approach by a new unbiased estimating equation for the regression parameters in the latency part of the model. The large sample properties of the regression effect estimators in both incidence and the latency parts are established. The finite sample properties of the estimators are studied in simulation studies. The proposed method is illustrated with a bone marrow transplantation data and a tonsil cancer data.

Predicting survival time for metastatic castration resistant prostate cancer: An iterative imputation approach.

In this paper, we present our winning method for survival time prediction in the 2015 Prostate Cancer DREAM Challenge, a recent crowdsourced competition focused on risk and survival time predictions for patients with metastatic castration-resistant prostate cancer (mCRPC). We are interested in using a patient's covariates to predict his or her time until death after initiating standard therapy. We propose an iterative algorithm to multiply impute right-censored survival times and use ensemble learning methods to characterize the dependence of these imputed survival times on possibly many covariates. We show that by iterating over imputation and ensemble learning steps, we guide imputation with patient covariates and, subsequently, optimize the accuracy of survival time prediction. This method is generally applicable to time-to-event prediction problems in the presence of right-censoring. We demonstrate the proposed method's performance with training and validation results from the DREAM Challenge and compare its accuracy with existing methods.

Trend-constrained corrected score for proportional hazards model with covariate measurement error.

In many medical research studies, survival time is typically the primary outcome of interest. The Cox proportional hazards model is the most popular method to investigate the relationship between covariates and possibly right-censored survival time. However, in many clinical trials, the true covariates may not always be accurately measured due to natural biological fluctuation or instrument error. It is well know that for regression analysis in general, naively using mismeasured covariates in conventional inference procedures may incur substantial estimation bias. In the presence of covariate measurement error, several functional modeling methods have been proposed under the situation where the distribution of the measurement error is known. Among them are parametric corrected score and conditional score. Although both methods are consistent, each suffers from severe problem of multiple roots or absence of appropriate root when the measurement error is substantial. The problem persists even when the sample size is practically large. We conduct a detailed investigation on the pathological behaviors of parametric corrected score and propose an approach of incorporating additional estimating functions to remedy these pathological behaviors. The estimation and inference are then accomplished by means of quadratic inference function. Extensive simulation studies are conducted to evaluate the performance of proposed method.

Semiparametric likelihood inference for left-truncated and right-censored data.

This paper proposes a new estimation procedure for the survival time distribution with left-truncated and right-censored data, where the distribution of the truncation time is known up to a finite-dimensional parameter vector. The paper expands on the Vardis multiplicative censoring model (Vardi, 1989. Multiplicative censoring, renewal processes, deconvolution and decreasing density: non-parametric estimation. Biometrika 76: , 751-761), establishes the connection between the likelihood under a generalized multiplicative censoring model and that for left-truncated and right-censored survival time data, and derives an Expectation-Maximization algorithm for model estimation. A formal test for checking the truncation time distribution is constructed based on the semiparametric likelihood ratio test statistic. In particular, testing the stationarity assumption that the underlying truncation time is uniformly distributed is performed by embedding the null uniform truncation time distribution in a smooth alternative (Neyman, 1937. Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift 20: , 150-199). Asymptotic properties of the proposed estimator are established. Simulations are performed to evaluate the finite-sample performance of the proposed methods. The methods and theories are illustrated by analyzing the Canadian Study of Health and Aging and the Channing House data, where the stationarity assumption with respect to disease incidence holds for the former but not the latter.

Doubly robust estimation of attributable fractions in survival analysis.

The attributable fraction is a commonly used measure that quantifies the public health impact of an exposure on an outcome. It was originally defined for binary outcomes, but an extension has recently been proposed for right-censored survival time outcomes; the so-called attributable fraction function. A maximum likelihood estimator of the attributable fraction function has been developed, which requires a model for the outcome. In this paper, we derive a doubly robust estimator of the attributable fraction function. This estimator requires one model for the outcome, and one joint model for the exposure and censoring. The estimator is consistent if either model is correct, not necessarily both.

Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models

Structured additive regression (STAR) provides a general framework for complex Gaussian and non-Gaussian regression models, with predictors comprising arbitrary combinations of nonlinear functions and surfaces, spatial effects, varying coefficients, random effects, and further regression terms. The large flexibility of STAR makes function selection a challenging and important task, aiming at (1) selecting the relevant covariates, (2) choosing an appropriate and parsimonious representation of the impact of covariates on the predictor, and (3) determining the required interactions. We propose a spike-and-slab prior structure for function selection that allows to include or exclude single coefficients as well as blocks of coefficients representing specific model terms. A novel multiplicative parameter expansion is required to obtain good mixing and convergence properties in a Markov chain Monte Carlo simulation approach and is shown to induce desirable shrinkage properties. In simulation studies and with (real) benchmark classification data, we investigate sensitivity to hyperparameter settings and compare performance to competitors. The flexibility and applicability of our approach are demonstrated in an additive piecewise exponential model with time-varying effects for right-censored survival times of intensive care patients with sepsis. Geoadditive and additive mixed logit model applications are discussed in an extensive online supplement.

Independent Screening for Single-Index Hazard rate Models with Ultrahigh Dimensional Features

SummaryIn data sets with many more features than observations, independent screening based on all univariate regression models leads to a computationally convenient variable selection method. Recent efforts have shown that, in the case of generalized linear models, independent screening may suffice to capture all relevant features with high probability, even in ultrahigh dimension. It is unclear whether this formal sure screening property is attainable when the response is a right-censored survival time. We propose a computationally very efficient independent screening method for survival data which can be viewed as the natural survival equivalent of correlation screening. We state conditions under which the method admits the sure screening property within a class of single-index hazard rate models with ultrahigh dimensional features and describe the generally detrimental effect of censoring on performance. An iterative variant of the method is also described which combines screening with penalized regression to handle more complex feature covariance structures. The methodology is evaluated through simulation studies and through application to a real gene expression data set.

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.

Comparing Survival Curves Using an Easy to Interpret Statistic

Here, I describe a statistic for comparing two survival curves that has a clear and obvious meaning and has a long history in biostatistics. Suppose we are comparing survival times associated with two treatments A and B. The statistic operates in such a way that if it takes on the value 0.95, then the interpretation is that a randomly chosen patient treated with A has a 95% chance of surviving longer than a randomly chosen patient treated with B. This statistic was first described in the 1950s, and was generalized in the 1960s to work with right-censored survival times. It is a useful and convenient measure for assessing differences between survival curves. Software for computing the statistic is readily available on the Internet.

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.

A pattern‐mixture model for the analysis of censored quality‐of‐life data

We propose a pattern-mixture model for describing the joint distribution of incomplete repeated measurements of quality of life (QoL) and right-censored survival times. The model assumes that the survival times follow a multinomial distribution and that the quality of life outcome follows a multivariate normal distribution conditional on the survival time. We estimate the model using a Bayesian approach by importance sampling. We then use simulated parameters to create multiple imputations of the censored QoL outcomes, which can then be used to calculate individual values of quality-adjusted life-years (QALYs). We apply the method to data from the Randomized Evaluation of Mechanical Assistance in the Treatment of Congestive Heart Failure (REMATCH) clinical trial.

Variance components analysis for pedigree-based censored survival data using generalized linear mixed models (GLMMs) and Gibbs sampling in BUGS.

Complex human diseases are an increasingly important focus of genetic research. Many of the determinants of these diseases are unknown and there is often a strong residual covariance between relatives even when all known genetic and environmental factors have been taken into account. This must be modeled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariance structure itself. Analysis is straightforward for multivariate normally distributed traits, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including right censored survival times. This includes age-at-onset and age-at-death data and a variety of other censored traits. Markov chain Monte Carlo (MCMC) methods, including Gibbs sampling, provide a convenient framework within which such GLMMs may be fitted. In this paper, we use BUGS ("Bayesian inference using Gibbs sampling": a readily available, generic Gibbs sampler) to fit GLMMs for right-censored survival times in nuclear and extended families. We discuss parameter interpretation and statistical inference, and show how to circumvent a number of important theoretical and practical problems. Using simulated data, we show that model parameters are consistent. We further illustrate our methods using data from an ongoing cohort study. Finally, we propose that the random effects associated with a genetic component of variance (e.g., sigma(2)(A)) in a GLMM may be regarded as an adjusted "phenotype" and used as input to a conventional model-based or model-free linkage analysis. This provides a simple way to conduct a linkage analysis for a trait reflected in a right-censored survival time while comprehensively adjusting for observed confounders at the level of the individual and latent environmental effects shared across families.

Semiparametric Inference in the Proportional Odds Regression Model

For fitting the proportional odds regression model with right-censored survival times, we introduce some weighted empirical odds functions. These functions are solutions of some self-consistency equations and have a nice martingale representation. From these functions, several classes of new regression estimators, such as the pseudo–maximum likelihood estimator, martingale residual-based estimators, and minimum distance estimators, are derived. These estimators have desirable properties such as easy computation, asymptotic normality via a martingale analysis, and reliable asymptotic covariance estimation in closed form. Extensive numerical studies show that the minimum L 2 distance estimators have very good finite-sample behaviors compared to existing methods. Results of some simulation studies and applications to a real dataset are given. The weighted odds function–based approach also provides inference on the baseline odds function and some measures for lack-of-fit analysis.