Estimation of the association for bivariate doubly censored data

We study the problem of estimating the association between two related survival variables when they follow a copula model and the bivariate doubly censored data is available. A two-stage estimation procedure is proposed and the asymptotic properties of the proposed estimator are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimate.     

Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

A new measure of association for bivariate survival data

Time dependent association measures between variables are of interest in bivariate survival data. Several such measures have been proposed in literature for the modelling and analysis of survival data. In this paper, we introduce a new measure of association for bivariate survival data using product moment residual life function and mean residual life function. Various properties of the proposed measure and its relationship with existing measures are discussed. We also develop a non-parametric estimator of the measure and study its asymptotic properties. The application of the result is illustrated using a real life data. Finally, a stimulation study is carried out to assess the performance of the estimator.     

A class of dynamic piecewise exponential models with random time grid

A novel fully Bayesian approach for modeling survival data with explanatory variables using the Piecewise Exponential Model (PEM) with random time grid is proposed. We consider a class of correlated Gamma prior distributions for the failure rates. Such prior specification is obtained via the dynamic generalized modeling approach jointly with a random time grid for the PEM. A product distribution is considered for modeling the prior uncertainty about the random time grid, turning possible the use of the structure of the Product Partition Model (PPM) to handle the problem. A unifying notation for the construction of the likelihood function of the PEM, suitable for both static and dynamic modeling approaches, is considered. Procedures to evaluate the performance of the proposed model are provided. Two case studies are presented in order to exemplify the methodology. For comparison purposes, the data sets are also fitted using the dynamic model with fixed time grid established in the literature. The results show the superiority of the proposed model.     

Regression Analysis of Longitudinal Data with Time‐Dependent Covariates and Informative Observation Times

Abstract. Longitudinal data frequently occur in many studies, and longitudinal responses may be correlated with observation times. In this paper, we propose a new joint modelling for the analysis of longitudinal data with time‐dependent covariates and possibly informative observation times via two latent variables. For inference about regression parameters, estimating equation approaches are developed and asymptotic properties of the proposed estimators are established. In addition, a lack‐of‐fit test is presented for assessing the adequacy of the model. The proposed method performs well in finite‐sample simulation studies, and an application to a bladder tumour study is provided.     

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.     

Generalized Log-Rank Tests for Interval-Censored Failure Time Data

Abstract.  Several non-parametric test procedures have been proposed for incomplete survival data: interval-censored failure time data. However, most of them have unknown asymptotic properties with heuristically derived and/or complicated variance estimation. This article presents a class of generalized log-rank tests for this type of survival data and establishes their asymptotics. The methods are evaluated using simulation studies and illustrated by a set of real data from a cancer study.     

Statistical Analysis of General Degradation Path Model and Failure Time Data with Multiple Failure Modes

ABSTRACT

This article considers degradation and failure time models with multiple failure modes which used to study the problem of longevity and aging in survival analysis and reliability. Degradation process is modeled using general nonparametric, nonlinear path models. Semi-parametric models for the intensities of the traumatic failures are used supposing that these intensities depend on degradation level. Semi-parametric estimators of various reliability characteristics are proposed and asymptotic properties of the estimators are obtained. The theoretical results are illustrated using simulated data.     

Tests of independence for censored bivariate failure time data

Bivariate failure time data is widely used in survival analysis, for example, in twins study. This article presents a class of chi2-type tests for independence between pairs of failure times after adjusting for covariates. A bivariate accelerated failure time model is proposed for the joint distribution of bivariate failure times while leaving the dependence structures for related failure times completely unspecified. Theoretical properties of the proposed tests are derived and variance estimates of the test statistics are obtained using a resampling technique. Simulation studies show that the proposed tests are appropriate for practical use. Two examples including the study of infection in catheters for patients on dialysis and the diabetic retinopathy study are also given to illustrate the methodology.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Joint modeling and estimation for longitudinal data with informative observation and terminal event times

ABSTRACT

In many longitudinal studies, there may exist informative observation times and a dependent terminal event that stops the follow-up. In this paper, we propose a joint model for analysis of longitudinal data with informative observation times and a dependent terminal event via two latent variables. Estimation procedures are developed for parameter estimation, and asymptotic properties of the proposed estimators are derived. Simulation studies demonstrate that the proposed method performs well for practical settings. An application to a bladder cancer study is illustrated.     

On a deconvolution problem under competing risks

Bagai and Prakasa Rao [Analysis of survival data with two dependent competing risks. Biometr J. 1992;7:801–814] considered a competing risks model with two dependent risks. The two risks are initially independent but dependence arises because of the additive effect of an independent risk on the two initially independent risks. They showed that the ratio of failure rates are identifiable in the nonparametric set-up. In this paper, we consider it as a measurement error/deconvolution problem and suggest a nonparametric kernel-type estimator for the ratio of two failure rates. The local error properties of the proposed estimator are studied. Simulation studies show the efficacy of the proposed estimator.     

Double filter instrumental variable estimation of panel data models with weakly exogenous variables

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.     

Testing the proportional odds model for interval-censored data

This paper discusses the goodness-of-fit test for the proportional odds model for K-sample interval-censored failure time data, which frequently occur in, for example, periodic follow-up survival studies. The proportional odds model has a feature that allows the ratio of two hazard functions to be monotonic and converge to one and provides an important tool for the modeling of survival data. To test the model, a procedure is proposed, which is a generalization of the method given in Dauxois and Kirmani [Dauxois JY, Kirmani SNUA (2003) Biometrika 90:913–922]. The asymptotic distribution of the procedure is established and its properties are evaluated by simulation studies     

Inference for Bivariate Survival Data by Copula Models Adjusted for the Boundary Effect

Copula models describe the dependence structure of two random variables separately from their marginal distributions and hence are particularly useful in studying the association for bivariate survival data. Semiparametric inference for bivariate survival data based on copula models has been studied for various types of data, including complete data, right-censored data, and current status data. This article discusses the boundary effect on these inference procedures, a problem that has been neglected in the previous literature. Specifically, asymptotic distribution of the association estimator on the boundary of parameter space is derived for one-dimensional copula models. The boundary properties are applied to test independence and to study the estimation efficiency. Simulation study is conducted for the bivariate right-censored data and current status data.     

Nonparametric Test for Doubly Interval-Censored Failure Time Data

This paper considers comparison of discrete failure time distributions when the survival time of interest measures elapsed time between two related events and observations on the occurrences of both events could be interval-censored. This kind of data is often referred to as doubly interval-censored failure time data. If the occurrence of the first event defining the survival time can be exactly observed, the data are usually referred to as interval-censored data. For the comparison problem based on interval-censored failure time data, Sun (1996) proposed a nonparametric test procedure. In this paper we generalize the procedure given in Sun (1996) to doubly interval-censored data case and the generalized test is evaluated by simulations.     

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.     

Regression analysis of clustered interval-censored failure time data with linear transformation models in the presence of informative cluster size

This paper discusses regression analysis of clustered interval-censored failure time data, which often occur in medical follow-up studies among other areas. For such data, sometimes the failure time may be related to the cluster size, the number of subjects within each cluster or we have informative cluster sizes. For the problem, we present a within-cluster resampling method for the situation where the failure time of interest can be described by a class of linear transformation models. In addition to the establishment of the asymptotic properties of the proposed estimators of regression parameters, an extensive simulation study is conducted for the assessment of the finite sample properties of the proposed method and suggests that it works well in practical situations. An application to the example that motivated this study is also provided.     

Joint modeling of longitudinal data with a dependent terminal event

ABSTRACT

Longitudinal data often arise in longitudinal follow-up studies, and there may exist a dependent terminal event such as death that stops the follow-up. In this article, we propose a new joint modeling for the analysis of longitudinal data with informative observation times via a dependent terminal event and two latent variables. Estimating equations are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. In addition, a generalization of the joint model with time-varying coefficients for the longitudinal response variable is considered, and goodness-of-fit methods for assessing the adequacy of the model are also provided. The proposed method works well in our simulation studies, and is applied to a data set from a bladder cancer study.     

OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.