A Marginal Additive Rates Model for Recurrent Event Data with a Terminal Event

Recurrent events data with a terminal event often arise in many longitudinal studies. Most of existing models assume multiplicative covariate effects and model the conditional recurrent event rate given survival. In this article, we propose a marginal additive rates model for recurrent events with a terminal event, and develop two procedures for estimating the model parameters. The asymptotic properties of the resulting estimators are established. In addition, some numerical procedures are presented for model checking. The finite-sample behavior of the proposed methods is examined through simulation studies, and an application to a bladder cancer study is also illustrated.     

An additive marginal regression model for clustered recurrent event in the presence of a terminal event

In this article, an additive rate model is proposed for clustered recurrent event with a terminal event. The subjects are clustered by some property. For the clustered subjects, the recurrent event is precluded by the death. An estimating equation is developed for the model parameter and the baseline rate function. The asymptotic properties of the resulting estimators are established. In addition, a goodness-of-fit test is presented to assess the adequacy of the model. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer data is illustrated.     

Semiparametric model for recurrent event data with excess zeros and informative censoring

Recurrent event data are often encountered in longitudinal follow-up studies in many important areas such as biomedical science, econometrics, reliability, criminology and demography. Multiplicative marginal rates models have been used extensively to analyze recurrent event data, but often fail to fit the data adequately. In addition, the analysis is complicated by excess zeros in the data as well as the presence of a terminal event that precludes further recurrence. To address these problems, we propose a semiparametric model with an additive rate function and an unspecified baseline to analyze recurrent event data, which includes a parameter to accommodate excess zeros and a frailty term to account for a terminal event. Local likelihood procedure is applied to estimate the parameters, and the asymptotic properties of the estimators are established. A simulation study is conducted to evaluate the performance of the proposed methods, and an example of their application is presented on a set of tumor recurrent data for bladder cancer.     

Additive–multiplicative rates model for recurrent events

Recurrent events are frequently encountered in biomedical studies. Evaluating the covariates effects on the marginal recurrent event rate is of practical interest. There are mainly two types of rate models for the recurrent event data: the multiplicative rates model and the additive rates model. We consider a more flexible additive–multiplicative rates model for analysis of recurrent event data, wherein some covariate effects are additive while others are multiplicative. We formulate estimating equations for estimating the regression parameters. The estimators for these regression parameters are shown to be consistent and asymptotically normally distributed under appropriate regularity conditions. Moreover, the estimator of the baseline mean function is proposed and its large sample properties are investigated. We also conduct simulation studies to evaluate the finite sample behavior of the proposed estimators. A medical study of patients with cystic fibrosis suffered from recurrent pulmonary exacerbations is provided for illustration of the proposed method.     

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.     

A class of additive transformation models for recurrent gap times

Abstract

The gap time between recurrent events is often of primary interest in many fields such as medical studies, and in this article, we discuss regression analysis of the gap times arising from a general class of additive transformation models. For the problem, we propose two estimation procedures, the modified within-cluster resampling (MWCR) method and the weighted risk-set (WRS) method, and the proposed estimators are shown to be consistent and asymptotically follow the normal distribution. In particular, the estimators have closed forms and can be easily determined, and the methods have the advantage of leaving the correlation among gap times arbitrary. A simulation study is conducted for assessing the finite sample performance of the presented methods and suggests that they work well in practical situations. Also the methods are applied to a set of real data from a chronic granulomatous disease (CGD) clinical trial.     

Additive mixed effect model for recurrent gap time data

Gap times between recurrent events are often of primary interest in medical and observational studies. The additive hazards model, focusing on risk differences rather than risk ratios, has been widely used in practice. However, the marginal additive hazards model does not take the dependence among gap times into account. In this paper, we propose an additive mixed effect model to analyze gap time data, and the proposed model includes a subject-specific random effect to account for the dependence among the gap times. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite sample behavior of the proposed methods is evaluated through simulation studies, and an application to a data set from a clinic study on chronic granulomatous disease is provided.     

Semiparametric proportional means model for marker data contingent on recurrent event

In many biomedical studies with recurrent events, some markers can only be measured when events happen. For example, medical cost attributed to hospitalization can only incur when patients are hospitalized. Such marker data are contingent on recurrent events. In this paper, we present a proportional means model for modelling the markers using the observed covariates contingent on the recurrent event. We also model the recurrent event via a marginal rate model. Estimating equations are constructed to derive the point estimators for the parameters in the proposed models. The estimators are shown to be asymptotically normal. Simulation studies are conducted to examine the finite-sample properties of the proposed estimators and the proposed method is applied to a data set from the Vitamin A Community Trial.     

Semiparametric additive marginal regression models for multiple type recurrent events

Recurrent event data are often encountered in biomedical research, for example, recurrent infections or recurrent hospitalizations for patients after renal transplant. In many studies, there are more than one type of events of interest. Cai and Schaube (Lifetime Data Anal 10:121-138, 2004) advocated a proportional marginal rate model for multiple type recurrent event data. In this paper, we propose a general additive marginal rate regression model. Estimating equations approach is used to obtain the estimators of regression coefficients and baseline rate function. We prove the consistency and asymptotic normality of the proposed estimators. The finite sample properties of our estimators are demonstrated by simulations. The proposed methods are applied to the India renal transplant study to examine risk factors for bacterial, fungal and viral infections.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

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.     

An Adaptive Two-stage Estimation Method for Additive Models

Abstract.  In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques to refine the first-stage ones. From this procedure, we can reveal a relationship between the regression function spaces and convergence rate, and then provide estimators that are optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n ^−4/5 , the MSE of the order n ^{− 1} can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.     

Asymptotics for a class of dynamic recurrent event models

Asymptotic properties, both consistency and weak convergence, of estimators arising in a general class of dynamic recurrent event models are presented. The class of models take into account the impact of interventions after each event occurrence, the impact of accumulating event occurrences, the induced informative and dependent right-censoring mechanism due to the data-accrual scheme, and the effect of covariate processes on the recurrent event occurrences. The class of models subsumes as special cases many of the recurrent event models that have been considered in biostatistics, reliability, and in the social sciences. The asymptotic properties presented have the potential of being useful in developing goodness-of-fit and model validation procedures, confidence intervals and confidence bands constructions, and hypothesis testing procedures for the finite- and infinite-dimensional parameters of a general class of dynamic recurrent event models, albeit the models without frailties.     

Efficient Estimation of an Additive Quantile Regression Model

Abstract. In this paper, two non‐parametric estimators are proposed for estimating the components of an additive quantile regression model. The first estimator is a computationally convenient approach which can be viewed as a more viable alternative to existing kernel‐based approaches. The second estimator involves sequential fitting by univariate local polynomial quantile regressions for each additive component with the other additive components replaced by the corresponding estimates from the first estimator. The purpose of the extra local averaging is to reduce the variance of the first estimator. We show that the second estimator achieves oracle efficiency in the sense that each estimated additive component has the same variance as in the case when all other additive components were known. Asymptotic properties are derived for both estimators under dependent processes that are strictly stationary and absolutely regular. We also provide a demonstrative empirical application of additive quantile models to ambulance travel times.     

Nonparametric estimation with recurrent competing risks data

Nonparametric estimators of component and system life distributions are developed and presented for situations where recurrent competing risks data from series systems are available. The use of recurrences of components’ failures leads to improved efficiencies in statistical inference, thereby leading to resource-efficient experimental or study designs or improved inferences about the distributions governing the event times. Finite and asymptotic properties of the estimators are obtained through simulation studies and analytically. The detrimental impact of parametric model misspecification is also vividly demonstrated, lending credence to the virtue of adopting nonparametric or semiparametric models, especially in biomedical settings. The estimators are illustrated by applying them to a data set pertaining to car repairs for vehicles that were under warranty.     

Proportional Rate Model with Incomplete Covariate and Complete Auxiliary Information

This paper deals with the analysis of proportional rate model for recurrent event data when covariates are subject to missing. The true covariate is measured only on a randomly chosen validation set, whereas auxiliary information is available for all cohort subjects. To further utilize the auxiliary information to improve study efficiency, we propose an estimated estimating equation for the regression parameters. The resulting estimators are shown to be consistent and asymptotically normal. Both graphical and numerical techniques for checking the adequacy of the model are presented. Simulations are conducted to evaluate the finite sample performance of the proposed estimators. Illustration with a real medical study is provided.     

On Local Polynomial Modelling of the Additive Risk Model

The additive risk model provides an alternative modelling technique for failure time data to the proportional hazards model. In this article, we consider the additive risk model with a nonparametric risk effect. We study estimation of the risk function and its derivatives with a parametric and an unspecified baseline hazard function respectively. The resulting estimators are the local likelihood and the local score estimators. We establish the asymptotic normality of the estimators and show that both methods have the same formula for asymptotic bias but different formula for variance. It is found that, in some special cases, the local score estimator is of the same efficiency as the local likelihood estimator though it does not use the information about the baseline hazard function. Another advantage of the local score estimator is that it has a closed form and is easy to implement. Some simulation studies are conducted to evaluate and compare the performance of the two estimators. A numerical example is used for illustration.     

Adaptive wavelet series estimation in separable nonparametric regression models

It is well-known that multivariate curve estimation suffers from the curse of dimensionality. However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit a typical restriction on the curve such as additivity. We first propose an adaptive and simultaneous estimation procedure for all additive components in additive regression models and discuss rate of convergence results and data-dependent truncation rules for wavelet series estimators. To speed up computation we then introduce a wavelet version of functional ANOVA algorithm for additive regression models and propose a regularization algorithm which guarantees an adaptive solution to the multivariate estimation problem. Some simulations indicate that wavelets methods complement nicely the existing methodology for nonparametric multivariate curve estimation.     

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