Nonparametric analysis of recurrent gap time data

Abstract

Recurrent event data are frequently encountered in longitudinal studies. In many applications, the times between successive recurrent events (gap times) are often of interest and lead to problems that have received much attention recently. In this article, using the approach of inverse probability-of-censoring weights (IPCW), we propose nonparametric estimators for the estimation of the bivariate distribution and survival functions for gap times of recurrent event data. We also consider the estimation of Kendall’s tau for two gap times by expressing it as an integral functional of the bivariate survival function. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate their finite sample performance.     

Promotion Time Cure Rate Model with Bivariate Random Effects

In this article, we consider the inclusion of random effects in both the survival function for at-risk subjects and the cure probability assuming a bivariate normal distribution for those effects in each cluster. For parameter estimation, we implemented the restricted maximum likelihood (REML) approach. We consider Weibull and Piecewise Exponential distributions to model the survival function for non-cured individuals. Simulation studies are performed, and based on a real database we evaluate the performance of our proposed model. Effect of different follow-up times and the effect of considering independent random effects instead of bivariate random effects are also studied.     

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.     

An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring

In this note, we consider estimating the bivariate survival function when both survival times are subject to random left truncation and one of the survival times is subject to random right censoring. Motivated by Satten and Datta [2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. 55, 207–210], we propose an inverse-probability-weighted (IPW) estimator. It involves simultaneous estimation of the bivariate survival function of the truncation variables and that of the censoring variable and the truncation variable of the uncensored components. We prove that (i) when there is no censoring, the IPW estimator reduces to NPMLE of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131] and (ii) when there is random left truncation and right censoring on only one of the components and the other component is always observed, the IPW estimator reduces to the estimator of Gijbels and Gürler [1998. Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statist. Sin. 1219–1232]. Based on Theorem 3.1 of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627], we prove that the IPW estimator is consistent under certain conditions. Finally, we examine the finite sample performance of the IPW estimator in some simulation studies. For the special case that censoring time is independent of truncation time, a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627]. For the special case (i), a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by Huang et al. (2001. Nonnparametric estimation of marginal distributions under bivariate truncation with application to testing for age-of-onset application. Statist. Sin. 11, 1047–1068).     

The bivariate alpha-skew-normal distribution

In this paper, we propose a new bivariate distribution, namely bivariate alpha-skew-normal distribution. The proposed distribution is very flexible and capable of generalizing the univariate alpha-skew-normal distribution as its marginal component distributions; it features a probability density function with up to two modes and has the bivariate normal distribution as a special case. The joint moment generating function as well as the main moments are provided. Inference is based on a usual maximum-likelihood estimation approach. The asymptotic properties of the maximum-likelihood estimates are verified in light of a simulation study. The usefulness of the new model is illustrated in a real benchmark data.     

On a Bivariate Pólya-Aeppli Distribution

In this article, we use the bivariate Poisson distribution obtained by the trivariate reduction method and compound it with a geometric distribution to derive a bivariate Pólya-Aeppli distribution. We then discuss a number of properties of this distribution including the probability generating function, correlation structure, probability mass function, recursive relations, and conditional distributions. The generating function of the tail probabilities is also obtained. Moment estimation of the parameters is then discussed and illustrated with a numerical example.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

The IPW estimator for the joint distribution function of the gap times from recurrent event data

Lee et al. in 2016 proposed a nonparametric estimator of the joint distribution of the gap time between transplant and the first infection and the following gap times between consecutive infections. In this article, we propose an alternative estimator based on the inverse-probability weighted (IPW) approach. Asymptotic properties of the proposed estimator are established . Simulation results indicate that the IPW estimator performs as well as the estimator proposed by Lee et al. We also propose an IPW estimator for estimating the joint distribution function of the gap times between consecutive recurrent events beyond the first episode.     

Some alternative bivariate Kumaraswamy models

In this article we discuss various strategies for constructing bivariate Kumaraswamy distributions. As alternatives to the Nadarajah et al. (2011) bivariate model, four different models are introduced utilizing a conditional specification approach, a conditional survival function approach, and an Arnold–Ng bivariate beta distribution construction approach. Distributional properties for such bivariate distributions are investigated. Parameter estimation strategies for the models are discussed, as are the consequences of fitting two of the models to a particular data set involving the proportion of foggy days at two different airports in Colombia.     

Reliability analysis for gap null gate based on model comparison criterion

The length of the gap is the key factor affecting its reliability. Based on the mechanism of the gap null gate, this paper regards the two endpoint thresholds of the gap length as bivariate random variables and establishes successful response models. Score test statistic is presented to test the correlation coefficient. The DIC criterion is also provided to compare the models. With the experimental data of the gap null gate, we build Probit model and Logit model as the successful response models, and prove that the correlation coefficients in the both models can be regarded as 0. By comparing the DIC value, we find that the Probit model is more suitable to describe the distribution of the endpoint thresholds of the reliability window. Finally, both the point estimation and interval estimation results of the reliability window are given to illustrate the feasibility of the method shown in the paper.     

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.     

A new mixed bivariate geometric-exponential life distribution with applications to series systems

In this paper, a new mixed bivariate geometric-exponential distribution is introduced. We present the properties and numerical characteristics of the distribution. In addition, the applications of this new bivariate distribution are given in a two-unit series system. Through an intensive Monte-Carlo simulation study, we assess the precision of the proposed estimation methods for the parameter inference. A real data is analysed for illustrative purpose.     

Robust inference for bivariate point processes

In the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures.     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

Simulation from the bessel distribution with applications

The Bessel distribution, introduced recently by Yuan and Kalbfleisch (Ann. Inst. Math. Statist., 2000), can be useful in many applications. In particular, this distribution appears in two Bayesian estimation problems, namely, estimation of the noncentrality parameter of a noncentral chi-square distribution and of the parameters of Downton's bivariate exponential distribution. Implementation of Markov chain Monte Carlo algorithms requires generation of observations from the Bessel distribution. In this paper we propose and compare exact simulation schemes generating Bessel variates based on certain properties of the distribution as well as the rejection method.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

Bayesian inference for bisexual galton-watson processes

In this work, an approach to the Bayesian estimation in a bisexual Galton-Watson process is considered. First we study an important parametric case assuming offspring distribution belonging to the bivariate series power family of distributions and then, we continue to investigate the nonparametric case. In both situations, Bayes estimators under weighted squared error loss function, for means, variances and covariance of the off spring distribution are obtained. For the superadditive case, the Bayes estimation of the asymptotic growth rate is also considered. Illustrative examples are given.     

Maximum likelihood estimation of Burr XII distribution parameters under random censoring

In this paper, we consider the maximum likelihood estimation of the parameters of Burr XII distribution using randomly right censored data. We provide necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates. Under such conditions, it is shown that the maximum likelihood estimates are strongly consistent for the true values of the parameters and are asymptotically bivariate normal. An application to leukemia free-survival times for allogeneic and autologous transplant patients is given.     

Non parametric analysis of gap times for bivariate recurrent event data with cure fraction

Recurrent event data often arise in longitudinal studies. In many applications, subjects may experience two different types of events alternatively over time or a pair of subjects may experience recurrent events of the same type. Medical advances have made it possible for some patients to be cured such that the disease of interest does not recur. In this article, we consider non parametric analysis of bivariate recurrent event data with cure fraction. Using the inverse-probability weighted (IPW) approach, we propose non parametric estimators for the proportion of cured patients and for the joint distribution functions of bivariate recurrence times of the uncured ones. The asymptotic properties of the proposed estimators are established. Simulation study indicates that the proposed estimators perform well in finite samples.     

Robustness and monotonicity properties of generalized correlation coefficients

A new class of generalized correlation coefficients that contains the Pearson and Kendall statistics as special cases was defined by Chinchilli et al. (2005) and applied to the estimation of correlations coefficients within the context of 2×2 cross-over designs for clinical trials. In this paper, we determine the infinitesimal robustness and local stability properties of these generalized correlation coefficients by deriving their corresponding influence functions. For cases in which the population distribution is a bivariate normal or a mixture of bivariate normal distributions we obtain explicit formulas, and establish monotonicity and sign-reverse rule properties of the generalized correlation coefficients.