Stratified proportional subdistribution hazards model with covariate-adjusted censoring weight for case-cohort studies

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival or competing risks outcomes. A case-cohort study dataset consists of two parts: (a) a random sample and (b) all cases or failures from a specific cause of interest. Clinicians often assess covariate effects on competing risks outcomes. The proportional subdistribution hazards model directly evaluates the effect of a covariate on the cumulative incidence function under the non-covariate-dependent censoring assumption for the full cohort study. However, the non-covariate-dependent censoring assumption is often violated in many biomedical studies. In this article, we propose a proportional subdistribution hazards model for case-cohort studies with stratified data with covariate-adjusted censoring weight. We further propose an efficient estimator when extra information from the other causes is available under case-cohort studies. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show (a) the proposed estimator is unbiased when the censoring distribution depends on covariates and (b) the proposed efficient estimator gains estimation efficiency when using extra information from the other causes. We analyze a bone marrow transplant dataset and a coronary heart disease dataset using the proposed method.     

Proportional Hazards Model for Multivariate Failure Time Data

Multivariate failure time data also referred to as correlated or clustered failure time data, often arise in survival studies when each study subject may experience multiple events. Statistical analysis of such data needs to account for intracluster dependence. In this article, we consider a bivariate proportional hazards model using vector hazard rate, in which the covariates under study have different effect on two components of the vector hazard rate function. Estimation of the parameters as well as base line hazard function are discussed. Properties of the estimators are investigated. We illustrated the method using two real life data. A simulation study is reported to assess the performance of the estimator.     

Comparing Sub-Survival Functions in a Competing Risks Model

In the competing risks literature, one usually compares whether two risks are equal or whether one is more serious. In this paper, we propose tests for the equality of two competing risks against an ordered alternative specified by their sub-survival functions. These tests are naturally developed as extensions of those based on hazard rates and cumulative incidence functions. We note that the interpretation of the new test results is more direct compared to the situation when the hypotheses are framed in terms of their cumulative incidence functions. The proposed tests are of the Kolmogrov–Smirnov type, based on maximum differences between sub-survival functions. Our simulation studies indicate that they are excellent competitors of the existing tests, that are based mainly on differences between cumulative incidence functions. A numerical example will demonstrate the advantages of the proposed tests.     

Regression analysis of case-cohort studies in the presence of dependent interval censoring

The case-cohort design is widely used as a means of reducing the cost in large cohort studies, especially when the disease rate is low and covariate measurements may be expensive, and has been discussed by many authors. In this paper, we discuss regression analysis of case-cohort studies that produce interval-censored failure time with dependent censoring, a situation for which there does not seem to exist an established approach. For inference, a sieve inverse probability weighting estimation procedure is developed with the use of Bernstein polynomials to approximate the unknown baseline cumulative hazard functions. The proposed estimators are shown to be consistent and the asymptotic normality of the resulting regression parameter estimators is established. A simulation study is conducted to assess the finite sample properties of the proposed approach and indicates that it works well in practical situations. The proposed method is applied to an HIV/AIDS case-cohort study that motivated this investigation.     

Bayesian Analysis of the Proportional Hazards Model with Time‐Varying Coefficients

We study a Bayesian analysis of the proportional hazards model with time‐varying coefficients. We consider two priors for time‐varying coefficients – one based on B‐spline basis functions and the other based on Gamma processes – and we use a beta process prior for the baseline hazard functions. We show that the two priors provide optimal posterior convergence rates (up to the term) and that the Bayes factor is consistent for testing the assumption of the proportional hazards when the two priors are used for an alternative hypothesis. In addition, adaptive priors are considered for theoretical investigation, in which the smoothness of the true function is assumed to be unknown, and prior distributions are assigned based on B‐splines.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

A 3-parameter Gompertz distribution for survival data with competing risks,with an application to breast cancer data

The cumulative incidence function is of great importance in the analysis of survival data when competing risks are present. Parametric modeling of such functions, which are by nature improper, suggests the use of improper distributions. One frequently used improper distribution is that of Gompertz, which captures only monotone hazard shapes. In some applications, however, subdistribution hazard estimates have been observed with unimodal shapes. An extension to the Gompertz distribution is presented which can capture unimodal as well as monotone hazard shapes. Important properties of the proposed distribution are discussed, and the proposed distribution is used to analyze survival data from a breast cancer clinical trial.     

Estimating Survival Curves Under Proportional Hazards Model with Covariate Measurement Errors

For the Cox proportional hazards model with additive covariate measurement errors, we propose a corrected cumulative baseline hazard estimator that reduces the bias of the na]ve Breslow estimator. We also derive corresponding modified estimators for the hazard functions and the survival functions of individuals with particular covariate values. Using a Monte Carlo technique developed by Lin et al . (1994), we construct confidence bands for such hazard and survival functions.     

Multiple imputation for competing risks regression with interval-censored data

ABSTRACT

We present here an extension of Pan's multiple imputation approach to Cox regression in the setting of interval-censored competing risks data. The idea is to convert interval-censored data into multiple sets of complete or right-censored data and to use partial likelihood methods to analyse them. The process is iterated, and at each step, the coefficient of interest, its variance–covariance matrix, and the baseline cumulative incidence function are updated from multiple posterior estimates derived from the Fine and Gray sub-distribution hazards regression given augmented data. Through simulation of patients at risks of failure from two causes, and following a prescheduled programme allowing for informative interval-censoring mechanisms, we show that the proposed method results in more accurate coefficient estimates as compared to the simple imputation approach. We have implemented the method in the MIICD R package, available on the CRAN website.     

Estimation of Stratified Mark‐Specific Proportional Hazards Models with Missing Marks

Abstract. An objective of randomized placebo‐controlled preventive HIV vaccine efficacy trials is to assess the relationship between the vaccine effect to prevent infection and the genetic distance of the exposing HIV to the HIV strain represented in the vaccine construct. Motivated by this objective, recently a mark‐specific proportional hazards (PH) model with a continuum of competing risks has been studied, where the genetic distance of the transmitting strain is the continuous ‘mark’ defined and observable only in failures. A high percentage of genetic marks of interest may be missing for a variety of reasons, predominantly because rapid evolution of HIV sequences after transmission before a blood sample is drawn from which HIV sequences are measured. This research investigates the stratified mark‐specific PH model with missing marks where the baseline functions may vary with strata. We develop two consistent estimation approaches, the first based on the inverse probability weighted complete‐case (IPW) technique, and the second based on augmenting the IPW estimator by incorporating auxiliary information predictive of the mark. We investigate the asymptotic properties and finite‐sample performance of the two estimators, and show that the augmented IPW estimator, which satisfies a double robustness property, is more efficient.     

Hazards regression for freemium products and services: a competing risks approach

We propose a competing risks approach to analyse customer behaviours in freemium products and services. The event of interest is when a customer starts to pay for additional features or functionalities. The observation of such an event may be preempted by an event where the customer quits using the product before paying and consuming the additional features or functionalities. One such freemium service is the online game category. The Fine-Gray regression model was implemented for an online game player activity data to study how covariates affect the paying hazard. Some covariates are hypothesized to have different discrete effects at multiple change points. We extend the model to allow for possible change points in the analysis.     

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.     

Inference for the Dependent Competing Risks Model with Masked Causes of Failure

The competing risks model is useful in settings in which individuals/units may die/fail for different reasons. The cause specific hazard rates are taken to be piecewise constant functions. A complication arises when some of the failures are masked within a group of possible causes. Traditionally, statistical inference is performed under the assumption that the failure causes act independently on each item. In this paper we propose an EM-based approach which allows for dependent competing risks and produces estimators for the sub-distribution functions. We also discuss identifiability of parameters if none of the masked items have their cause of failure clarified in a second stage analysis (e.g. autopsy). The procedures proposed are illustrated with two datasets.     

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.     

An Exact Corrected Log-Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract.  This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [ Biometrika 77 (1990) 127] methodology of corrected log-likelihood will be applied to the so-called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non-normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [ Scand. J. Statist. 25 (1998) 573] and Kong & Gu [ Statistica Sinica 9 (1999) 953] are re-established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.     

Estimation of conditional cumulative incidence functions under generalized semiparametric regression models with missing covariates,with application to analysis of biomarker correlates in vaccine trials

This article presents generalized semiparametric regression models for conditional cumulative incidence functions with competing risks data when covariates are missing by sampling design or happenstance. A doubly robust augmented inverse probability weighted (AIPW) complete-case approach to estimation and inference is investigated. This approach modifies IPW complete-case estimating equations by exploiting the key features in the relationship between the missing covariates and the phase-one data to improve efficiency. An iterative numerical procedure is derived to solve the nonlinear estimating equations. The asymptotic properties of the proposed estimators are established. A simulation study examining the finite-sample performances of the proposed estimators shows that the AIPW estimators are more efficient than the IPW estimators. The developed method is applied to the RV144 HIV-1 vaccine efficacy trial to investigate vaccine-induced IgG binding antibodies to HIV-1 as correlates of acquisition of HIV-1 infection while taking account of whether the HIV-1 sequences are near or far from the HIV-1 sequences represented in the vaccine construct.     

Inference for cumulative incidence on discrete failure times with competing risks

In the competing risks analysis, most inferences have been developed based on continuous failure time data. However, failure times are sometimes observed as being discrete. We propose nonparametric inferences for the cumulative incidence function for pure discrete data with competing risks. When covariate information is available, we propose semiparametric inferences for direct regression modelling of the cumulative incidence function for grouped discrete failure time data with competing risks. Simulation studies show that the procedures perform well. The proposed methods are illustrated with a study of contraceptive use in Indonesia.     

A Class of Non-parametric Tests in the Competing Risks Model for Comparing Two Samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D₁, obtained when the process is under control. Our main goal is to test if the failure rates in D₁ are equal to or less than the failure rates in another data set D₂, against undesirable neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.