An estimating equation for parametric shared frailty models with marginal additive hazards

Summary.  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 some contexts, however, it may be more reasonable to use the marginal additive hazards model. We derive asymptotic properties of the Lin and Ying estimators for the marginal additive hazards model for multivariate failure time data. Furthermore we suggest estimating equations for the regression parameters and association parameters in parametric shared frailty models with marginal additive hazards by using the Lin and Ying estimators. We give the large sample properties of the estimators arising from these estimating equations and investigate their small sample properties by Monte Carlo simulation. A real example is provided for illustration.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

Additive transformation models for clustered failure time data

We propose a class of additive transformation risk models for clustered failure time data. Our models are motivated by the usual additive risk model for independent failure times incorporating a frailty with mean one and constant variability which is a natural generalization of the additive risk model from univariate failure time to multivariate failure time. An estimating equation approach based on the marginal hazards function is proposed. Under the assumption that cluster sizes are completely random, we show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also provide goodness-of-fit test statistics for choosing the transformation. Simulation studies and real data analysis are conducted to examine the finite-sample performance of our estimators.     

Analysis of transformation models with right-truncated data

In many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in right-truncated data. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of right-truncated data. We proposed two estimators for regression coefficients. The first estimator is based on martingale estimating equations. The second estimator is based on the conditional likelihood function given the truncation times. The asymptotic properties of both estimators are derived. The finite sample performance is examined through a simulation study.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.     

Mark-specific additive hazards regression with continuous marks

For survival data, mark variables are only observed at uncensored failure times, and it is of interest to investigate whether there is any relationship between the failure time and the mark variable. The additive hazards model, focusing on hazard differences rather than hazard ratios, has been widely used in practice. In this article, we propose a mark-specific additive hazards model in which both the regression coefficient functions and the baseline hazard function depend nonparametrically on a continuous mark. An estimating equation approach is developed to estimate the regression functions, and the asymptotic properties of the resulting estimators are established. In addition, some formal hypothesis tests are constructed for various hypotheses concerning the mark-specific treatment effects. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a data set from the first HIV vaccine efficacy trial is provided.     

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.     

Uniqueness of permuted treatment estimator distributions in the proportional hazards model

We discuss findings regarding the permutation distributions of treatment effect estimators in the proportional hazards model. For fixed sample size n, we will prove that all uncensored and untied event times yield the same permutation distribution of treatment effect estimators in the proportional hazards model. In other words this distribution is irrelevant with respect to the actual event times. We will show several uniqueness properties under different conditions. These properties are useful for small sample permutation tests and also helpful to large sample cases.     

A class of accelerated means regression models for recurrent event data

In this article, we propose a general class of accelerated means regression models for recurrent event data. The class includes the proportional means model, the accelerated failure time model and the accelerated rates model as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the model parameters, estimating equation approaches are developed and both large and final sample properties of the proposed estimators are established. In addition, some graphical and numerical procedures are presented for model checking. An illustration with multiple-infection data from a clinic study on chronic granulomatous disease is also provided.     

Proportional hazards model with varying coefficients for length-biased data

Length-biased data arise in many important applications including epidemiological cohort studies, cancer prevention trials and studies of labor economics. Such data are also often subject to right censoring due to loss of follow-up or the end of study. In this paper, we consider a proportional hazards model with varying coefficients for right-censored and length-biased data, which is used to study the interact effect nonlinearly of covariates with an exposure variable. A local estimating equation method is proposed for the unknown coefficients and the intercept function in the model. The asymptotic properties of the proposed estimators are established by using the martingale theory and kernel smoothing techniques. Our simulation studies demonstrate that the proposed estimators have an excellent finite-sample performance. The Channing House data is analyzed to demonstrate the applications of the proposed method.     

Empirical Likelihood for Censored Linear Regression and Variable Selection

The linear regression model for right censored data, also known as the accelerated failure time model using the logarithm of survival time as the response variable, is a useful alternative to the Cox proportional hazards model. Empirical likelihood as a non‐parametric approach has been demonstrated to have many desirable merits thanks to its robustness against model misspecification. However, the linear regression model with right censored data cannot directly benefit from the empirical likelihood for inferences mainly because of dependent elements in estimating equations of the conventional approach. In this paper, we propose an empirical likelihood approach with a new estimating equation for linear regression with right censored data. A nested coordinate algorithm with majorization is used for solving the optimization problems with non‐differentiable objective function. We show that the Wilks' theorem holds for the new empirical likelihood. We also consider the variable selection problem with empirical likelihood when the number of predictors can be large. Because the new estimating equation is non‐differentiable, a quadratic approximation is applied to study the asymptotic properties of penalized empirical likelihood. We prove the oracle properties and evaluate the properties with simulated data. We apply our method to a Surveillance, Epidemiology, and End Results small intestine cancer dataset.     

A marginal regression model for multivariate failure time data with a surviving fraction

A marginal regression approach for correlated censored survival data has become a widely used statistical method. Examples of this approach in survival analysis include from the early work by Wei et al. (J Am Stat Assoc 84:1065–1073, 1989) to more recent work by Spiekerman and Lin (J Am Stat Assoc 93:1164–1175, 1998). This approach is particularly useful if a covariate’s population average effect is of primary interest and the correlation structure is not of interest or cannot be appropriately specified due to lack of sufficient information. In this paper, we consider a semiparametric marginal proportional hazard mixture cure model for clustered survival data with a surviving or “cure” fraction. Unlike the clustered data in previous work, the latent binary cure statuses of patients in one cluster tend to be correlated in addition to the possible correlated failure times among the patients in the cluster who are not cured. The complexity of specifying appropriate correlation structures for the data becomes even worse if the potential correlation between cure statuses and the failure times in the cluster has to be considered, and thus a marginal regression approach is particularly attractive. We formulate a semiparametric marginal proportional hazards mixture cure model. Estimates are obtained using an EM algorithm and expressions for the variance–covariance are derived using sandwich estimators. Simulation studies are conducted to assess finite sample properties of the proposed model. The marginal model is applied to a multi-institutional study of local recurrences of tonsil cancer patients who received radiation therapy. It reveals new findings that are not available from previous analyses of this study that ignored the potential correlation between patients within the same institution.     

Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model

Multivariate failure time data arise when the sample consists of clusters and each cluster contains several possibly dependent failure times. The Clayton–Oakes model (Clayton, 1978; Oakes, 1982) for multivariate failure times characterizes the intracluster dependence parametrically but allows arbitrary specification of the marginal distributions. In this paper, we discuss estimation in the Clayton–Oakes model when the marginal distributions are modeled to follow the Cox (1972) proportional hazards regression model. Parameter estimation is based on an approximate generalized maximum likelihood estimator. We illustrate the model's application with example datasets.     

Analyzing Length-biased Data with Semiparametric Transformation and Accelerated Failure Time Models

Right-censored time-to-event data are often observed from a cohort of prevalent cases that are subject to length-biased sampling. Informative right censoring of data from the prevalent cohort within the population often makes it difficult to model risk factors on the unbiased failure times for the general population, because the observed failure times are length biased. In this paper, we consider two classes of flexible semiparametric models: the transformation models and the accelerated failure time models, to assess covariate effects on the population failure times by modeling the length-biased times. We develop unbiased estimating equation approaches to obtain the consistent estimators of the regression coefficients. Large sample properties for the estimators are derived. The methods are confirmed through simulations and illustrated by application to data from a study of a prevalent cohort of dementia patients.     

Analysis of error-prone survival data under additive hazards models: measurement error effects and adjustments

Covariate measurement error occurs commonly in survival analysis. Under the proportional hazards model, measurement error effects have been well studied, and various inference methods have been developed to correct for error effects under such a model. In contrast, error-contaminated survival data under the additive hazards model have received relatively less attention. In this paper, we investigate this problem by exploring measurement error effects on parameter estimation and the change of the hazard function. New insights of measurement error effects are revealed, as opposed to well-documented results for the Cox proportional hazards model. We propose a class of bias correction estimators that embraces certain existing estimators as special cases. In addition, we exploit the regression calibration method to reduce measurement error effects. Theoretical results for the developed methods are established, and numerical assessments are conducted to illustrate the finite sample performance of our methods.     

A class of mixed models for recurrent event data

In this article, we propose a class of mixed models for recurrent event data. The new models include the proportional rates model and Box–Cox transformation rates models as special cases, and allow the effects of covariates on the rate functions of counting processes to be proportional or convergent. For inference on the model parameters, estimating equation approaches are developed. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed procedure is evaluated through simulation studies. A real example with data taken from a clinic study on chronic granulomatous disease (CGD) is also illustrated for the use of the proposed methodology. The Canadian Journal of Statistics 39: 578–590; 2011. © 2011 Statistical Society of Canada     

Non-parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data

Abstract.  This paper describes our studies on non-parametric maximum-likelihood estimators in a semiparametric mixture model for competing-risks data, in which proportional hazards models are specified for failure time models conditional on cause and a multinomial model is specified for the marginal distribution of cause conditional on covariates. We provide a verifiable identifiability condition and, based on it, establish an asymptotic profile likelihood theory for this model. We also provide efficient algorithms for the computation of the non-parametric maximum-likelihood estimate and its asymptotic variance. The success of this method is demonstrated in simulation studies and in the analysis of Taiwan severe acute respiratory syndrome data.     

Marginal Analysis For Clustered Failure Time Data

Clustered failure time data are commonly encountered in biomedical research where the study subjects from the same cluster (e.g., family) share the common genetic and/or environmental factors such that the failure times within the same cluster are correlated. Two approaches that are commonly used to account for the intra-cluster association are frailty models and marginal models. In this paper, we study the marginal proportional hazards model, where the structure of dependence between individuals within a cluster is unspecified. An estimation procedure is developed based on a pseudo-likelihood approach, and a risk set sampling method is proposed for the formulation of the pseudo-likelihood. The asymptotic properties of the proposed estimators are studied, and the related issues regarding the statistical efficiencies are discussed. The performances of the proposed estimator are demonstrated by the simulation studies. A data example from a child vitamin A supplementation trial in Nepal (Nepal Nutrition Intervention Project-Sarlahi, or NNIPS) is used to illustrate this methodology.     

On estimation of linear transformation models with nested case–control sampling

Nested case–control (NCC) sampling is widely used in large epidemiological cohort studies for its cost effectiveness, but its data analysis primarily relies on the Cox proportional hazards model. In this paper, we consider a family of linear transformation models for analyzing NCC data and propose an inverse selection probability weighted estimating equation method for inference. Consistency and asymptotic normality of our estimators for regression coefficients are established. We show that the asymptotic variance has a closed analytic form and can be easily estimated. Numerical studies are conducted to support the theory and an application to the Wilms’ Tumor Study is also given to illustrate the methodology.