Estimation of the marginal survival time in the presence of dependent competing risks using inverse probability of censoring weighted (IPCW) methods

In medical studies, there is interest in inferring the marginal distribution of a survival time subject to competing risks. The Kyushu Lipid Intervention Study (KLIS) was a clinical study for hypercholesterolemia, where pravastatin treatment was compared with conventional treatment. The primary endpoint was time to events of coronary heart disease (CHD). In this study, however, some subjects died from causes other than CHD or were censored due to loss to follow-up. Because the treatments were targeted to reduce CHD events, the investigators were interested in the effect of the treatment on CHD events in the absence of causes of death or events other than CHD. In this paper, we present a method for estimating treatment group-specific marginal survival curves of time-to-event data in the presence of dependent competing risks. The proposed method is a straightforward extension of the Inverse Probability of Censoring Weighted (IPCW) method to settings with more than one reason for censoring. The results of our analysis showed that the IPCW marginal incidence for CHD was almost the same as the lower bound for which subjects with competing events were assumed to be censored at the end of all follow-up. This result provided reassurance that the results in KLIS were robust to competing risks.     

Proportional cause-specific reversed hazards model

The proportional reversed hazards model explains the multiplicative effect of covariates on the baseline reversed hazard rate function of lifetimes. In the present study, we introduce a proportional cause-specific reversed hazards model. The proposed regression model facilitates the analysis of failure time data with multiple causes of failure under left censoring. We estimate the regression parameters using a partial likelihood approach. We provide Breslow's type estimators for the cumulative cause-specific reversed hazard rate functions. Asymptotic properties of the estimators are discussed. Simulation studies are conducted to assess their performance. We illustrate the applicability of the proposed model using a real data set.     

A Semi‐parametric Transformation Frailty Model for Semi‐competing Risks Survival Data

In the analysis of semi‐competing risks data interest lies in estimation and inference with respect to a so‐called non‐terminal event, the observation of which is subject to a terminal event. Multi‐state models are commonly used to analyse such data, with covariate effects on the transition/intensity functions typically specified via the Cox model and dependence between the non‐terminal and terminal events specified, in part, by a unit‐specific shared frailty term. To ensure identifiability, the frailties are typically assumed to arise from a parametric distribution, specifically a Gamma distribution with mean 1.0 and variance, say, σ². When the frailty distribution is misspecified, however, the resulting estimator is not guaranteed to be consistent, with the extent of asymptotic bias depending on the discrepancy between the assumed and true frailty distributions. In this paper, we propose a novel class of transformation models for semi‐competing risks analysis that permit the non‐parametric specification of the frailty distribution. To ensure identifiability, the class restricts to parametric specifications of the transformation and the error distribution; the latter are flexible, however, and cover a broad range of possible specifications. We also derive the semi‐parametric efficient score under the complete data setting and propose a non‐parametric score imputation method to handle right censoring; consistency and asymptotic normality of the resulting estimators is derived and small‐sample operating characteristics evaluated via simulation. Although the proposed semi‐parametric transformation model and non‐parametric score imputation method are motivated by the analysis of semi‐competing risks data, they are broadly applicable to any analysis of multivariate time‐to‐event outcomes in which a unit‐specific shared frailty is used to account for correlation. Finally, the proposed model and estimation procedures are applied to a study of hospital readmission among patients diagnosed with pancreatic cancer.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Bayesian Estimation of Survival Functions under Stochastic Precedence

When estimating the distributions of two random variables, X and Y, investigators often have prior information that Y tends to be bigger than X. To formalize this prior belief, one could potentially assume stochastic ordering between X and Y, which implies Pr(X < or = z) > or = Pr(Y < or = z) for all z in the domain of X and Y. Stochastic ordering is quite restrictive, though, and this article focuses instead on Bayesian estimation of the distribution functions of X and Y under the weaker stochastic precedence constraint, Pr(X < or = Y) > or = 0.5. We consider the case where both X and Y are categorical variables with common support and develop a Gibbs sampling algorithm for posterior computation. The method is then generalized to the case where X and Y are survival times. The proposed approach is illustrated using data on survival after tumor removal for patients with malignant melanoma.     

Multivariate Parametric Spatiotemporal Models for County Level Breast Cancer Survival Data

In clustered survival settings where the clusters correspond to geographic regions, biostatisticians are increasingly turning to models with spatially distributed random effects. These models begin with spatially oriented frailty terms, but may also include further region-level terms in the parametrization of the baseline hazards or various covariate effects (as in a spatially-varying coefficients model). In this paper, we propose a multivariate conditionally autoregressive (MCAR) model as a mixing distribution for these random effects, as a way of capturing correlation across both the regions and the elements of the random effect vector for any particular region. We then extend this model to permit analysis of temporal cohort effects, where we use the term temporal cohort to mean a group of subjects all of whom were diagnosed with the disease of interest (and thus, entered the study) during the same time period (say, calendar year). We show how our spatiotemporal model may be efficiently fit in a hierarchical Bayesian framework implemented using Markov chain Monte Carlo (MCMC) computational techniques. We illustrate our approach in the context of county-level breast cancer data from 22 annual cohorts of women living in the state of Iowa, as recorded by the Surveillance, Epidemiology, and End Results (SEER) database. Hierarchical model comparison using the Deviance Information Criterion (DIC), as well as maps of the fitted county-level effects, reveal the benefit of our approach.     

An empirical saddlepoint approximation based method for smoothing survival functions under right censoring

The Kaplan–Meier (KM) estimator is ubiquitously used for estimating survival functions, but it provides only a discrete approximation at the observation times and does not deliver a proper distribution if the largest observation is censored. Using KM as a starting point, we devise an empirical saddlepoint approximation‐based method for producing a smooth survival function that is unencumbered by choice of tuning parameters. The procedure inverts the moment generating function (MGF) defined through a Riemann–Stieltjes integral with respect to an underlying mixed probability measure consisting of the discrete KM mass function weights and an absolutely continuous exponential right‐tail completion. Uniform consistency, and weak and strong convergence results are established for the resulting MGF and its derivatives, thus validating their usage as inputs into the saddlepoint routines. Relevant asymptotic results are also derived for the density and distribution function estimates. The performance of the resulting survival approximations is examined in simulation studies, which demonstrate a favourable comparison with the log spline method (Kooperberg & Stone, 1992) in small sample settings. For smoothing survival functions we argue that the methodology has no immediate competitors in its class, and we illustrate its application on several real data sets. The Canadian Journal of Statistics 47: 238–261; 2019 © 2019 Statistical Society of Canada     

Tests for Equality of Survival Distributions Against Non-Location Alternatives

We propose new two andk-sample tests for evaluating the equality of survival distributions against alternatives that include crossing of survival functions, and proportional and monotone hazard ratios. The tests allow for right censored data. The asymptotic power against local alternatives is investigated. Simulation results demonstrate that the new tests are more powerful than known tests when survival functions cross. We apply the tests to a well known study of chemo- and radio-therapy conducted by the Gastrointestinal Tumor Study Group. TheP-values for both proposed tests are much smaller than for other known tests.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Inverse Probability of Censoring Weighted U‐statistics for Right‐Censored Data with an Application to Testing Hypotheses

Abstract. A right‐censored version of a U ‐statistic with a kernel of degree m 1 is introduced by the principle of a mean preserving reweighting scheme which is also applicable when the dependence between failure times and the censoring variable is explainable through observable covariates. Its asymptotic normality and an expression of its standard error are obtained through a martingale argument. We study the performances of our U ‐statistic by simulation and compare them with theoretical results. A doubly robust version of this reweighted U ‐statistic is also introduced to gain efficiency under correct models while preserving consistency in the face of model mis‐specifications. Using a Kendall's kernel, we obtain a test statistic for testing homogeneity of failure times for multiple failure causes in a multiple decrement model. The performance of the proposed test is studied through simulations. Its usefulness is also illustrated by applying it to a real data set on graft‐versus‐host‐disease.     

Forward Simulation Markov Chain Monte Carlo with Applications to Stochastic Epidemic Models

For many stochastic models, it is difficult to make inference about the model parameters because it is impossible to write down a tractable likelihood given the observed data. A common solution is data augmentation in a Markov chain Monte Carlo (MCMC) framework. However, there are statistical problems where this approach has proved infeasible but where simulation from the model is straightforward leading to the popularity of the approximate Bayesian computation algorithm. We introduce a forward simulation MCMC (fsMCMC) algorithm, which is primarily based upon simulation from the model. The fsMCMC algorithm formulates the simulation of the process explicitly as a data augmentation problem. By exploiting non‐centred parameterizations, an efficient MCMC updating schema for the parameters and augmented data is introduced, whilst maintaining straightforward simulation from the model. The fsMCMC algorithm is successfully applied to two distinct epidemic models including a birth–death–mutation model that has only previously been analysed using approximate Bayesian computation methods.     

Non‐Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise‐Deterministic Markov Processes

This paper presents a non‐parametric method for estimating the conditional density associated to the jump rate of a piecewise‐deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.     

Estimation methods for time‐dependent AUC models with survival data

The performance of clinical tests for disease screening is often evaluated using the area under the receiver‐operating characteristic (ROC) curve (AUC). Recent developments have extended the traditional setting to the AUC with binary time‐varying failure status. Without considering covariates, our first theme is to propose a simple and easily computed nonparametric estimator for the time‐dependent AUC. Moreover, we use generalized linear models with time‐varying coefficients to characterize the time‐dependent AUC as a function of covariate values. The corresponding estimation procedures are proposed to estimate the parameter functions of interest. The derived limiting Gaussian processes and the estimated asymptotic variances enable us to construct the approximated confidence regions for the AUCs. The finite sample properties of our proposed estimators and inference procedures are examined through extensive simulations. An analysis of the AIDS Clinical Trials Group (ACTG) 175 data is further presented to show the applicability of the proposed methods. The Canadian Journal of Statistics 38:8–26; 2010 © 2009 Statistical Society of Canada