A note on Kendall’s Tau coefficient for gap times in presence of right censoring

In several clinical and epidemiology studies, data from events that occur successively in time in the same individual, are frequently reported. Among these, the most common are recurrent events where each subject may experience a number of failures over the course of follow-up. Examples include repeated hospitalization of patients, recurrences of tumor, recurrent infections, among others. In this work, the interest is to study the correlation between successive recurrent events, gap times, in the presence of right censoring. To measure the association between two gap times we use the Kendall’s τ correlation coefficient, by incorporating suitable bivariate estimators of the joint distribution function of the gap times and of the marginal distribution function of the second gap time, into the integrals that define the probability of concordant pairs and the probability of discordant pairs. Two of the estimators of the joint distribution function of the gap times considered in this work are already known, but we consider also estimators with Kaplan-Meier weights defined by using decision trees and random forests methodology. We conclude that all the estimators perform better in a scenario of negative association. When the association is moderately negative, the performance of the estimator with smoothed weights using random forests is superior. In the case of strong positive association, the best estimator is the presmoothed nonparametric but, in the case of moderate positive association, this estimator has identical performance as the estimator with presmoothed weights using random forests.This work was supported by Portuguese funds through the CMAT - Research Centre of Mathematics of University of Minho - within projects UIDB/00013/2020 and UIDP/00013/2020. Authors thank to referees for their careful reading and for their constructive suggestions

parfm : Parametric Frailty Models in R

Frailty models are getting more and more popular to account for overdispersion and/or clustering in survival data. When the form of the baseline hazard is somehow known in advance, the parametric estimation approach can be used advantageously. Nonetheless, there is no unified widely available software that deals with the parametric frailty model. The new parfm package remedies that lack by providing a wide range of parametric frailty models in R. The gamma, inverse Gaussian, and positive stable frailty distributions can be specified, together with five different baseline hazards. Parameter estimation is done by maximising the marginal log-likelihood, with right-censored and possibly left-truncated data. In the multivariate setting, the inverse Gaussian may encounter numerical difficulties with a huge number of events in at least one cluster. The positive stable model shows analogous difficulties but an ad-hoc solution is implemented, whereas the gamma model is very resistant due to the simplicity of its Laplace transform

Estimation of Exposure Distribution Adjusting for Association Between Exposure Level and Detection Limit

In environmental exposure studies, it is common to observe a portion of exposure measurements to fall below experimentally determined detection limits (DLs). The reverse Kaplan–Meier estimator, which mimics the well‐known Kaplan–Meier estimator for right‐censored survival data with the scale reversed, has been recommended for estimating the exposure distribution for the data subject to DLs because it does not require any distributional assumption. However, the reverse Kaplan–Meier estimator requires the independence assumption between the exposure level and DL and can lead to biased results when this assumption is violated. We propose a kernel‐smoothed nonparametric estimator for the exposure distribution without imposing any independence assumption between the exposure level and DL. We show that the proposed estimator is consistent and asymptotically normal. Simulation studies demonstrate that the proposed estimator performs well in practical situations. A colon cancer study is provided for illustration

Statistical models to capture the association between progression-free and overall survival in oncology trials

In oncology trials, different clinical endpoints can be measured. For the survival analysis of patients, the most traditional primary endpoint is overall survival (OS), which is defined as the time from study entry to death from any cause. Besides, progression-free related measurements such as progression-free survival (PFS) might be also considered. For assessing the performance of therapies, OS is the most reliable endpoint. However, utilizing earlier endpoints such as information from disease progression might lead to a gain in efficiency. However, the gain in efficiency might depend on the relationship between those two endpoints. This thesis explores various statistical models for capturing the association between PFS and OS. The research is partitioned into three topics. At first, it considers methods for quantifying the association between PFS and OS in oncology trials, in terms of Kendall’s τ rank correlation rather than Pearson correlation. Copula-based, non-parametric, and illness-death model–based methods are reviewed. In addition, the approach based on an underlying illness-death model is generalized to allow general parametric models. The simulations suggest that the illness-death model–based method provides good estimates of Kendall’s τ across several scenarios. In some situations, copula-based methods Perform well but their performance is sensitive to the choice of copula. The Clayton copula is most appropriate in scenarios which might realistically reflect an oncology trial, but the use of copula models in practice is questionable. In the second and third topic, the estimation of the group difference faces the issue of non-proportionality for treatments effects. Instead of the standard hazard ratio we use the average hazard ratio for estimating the group difference as it is able to cope with non-proportional hazards well as it considers group difference depending on time. Subsequently, it compares methods for jointly modelling time-to-progression and time-to-death within a Bayesian framework. By incorporating treatment effects, we investigate an illness-death model-based approach and also copula-based approaches. According to the simulations results the Gaussian copula-based model performed the best overall, but the illness-death model-based approach showed a good performance as well. However, in contrast to the good performance of the Clayton copula-based approach in the first topic, the Clayton copula model did not perform well regarding the estimation of AHR. The third topic explores various semi-parametric multi-state model-based methods for gaining efficiency in testing for, and estimating the treatment effects in terms on, overall survival in oncology trials compared to standard methods based on directly applying Cox regression or the log-rank test. The semi-parametric multi-state model-based method fits a Cox model to (a subset of) transition intensities in an illness-death model assuming either a Markov or semi-Markov model and uses AHR to measure treatment effect. In most of the situations, the semi-parametric multi-state model-based methods perform better than the Cox-based approach. The performance of the methods in each topic is investigated by simulations and also illustrated using data from a clinical trial of treatments for advanced ovarian cancer in topic 2 and for colon cancer in topics 1 and 3

NONPARAMETRIC ESTIMATION OF BIVARIATE FAILURE TIME ASSOCIATIONS IN THE PRESENCE OF A COMPETING RISK

There has been much research on the study of associations among paired failure times. Most has either assumed time invariance of association or been based on complex measures or estimators. Little has accommodated failures arising amid competing risks. This paper targets the conditional cause specific hazard ratio, a recent modification of the conditional hazard ratio to accommodate competing risks data. Estimation is accomplished by an intuitive, nonparametric method that localizes Kendall’s tau. Time variance is accommodated through a partitioning of space into “bins” between which the strength of association may differ. Inferential procedures are researched, small sample performance evaluated, and methods applied to investigate familial association in dementia onset. The proposed methodology augments existing methodology with an approach that may be more readily applied and interpreted, thus facilitate dissemination of methodology addressing failure time associations into the substantive literatur

Statistical methods for assays with limits of detection: Serum bile acid as a differentiator between patients with normal colons, adenomas, and colorectal cancer

In analytic chemistry a detection limit (DL) is the lowest measurable amount of an analyte that can be distinguished from a blank; many biomedical measurement technologies exhibit this property. From a statistical perspective, these data present inferential challenges because instead of precise measures, one only has information that the value is somewhere between 0 and the DL (below detection limit, BDL). Substitution of BDL values, with 0 or the DL can lead to biased parameter estimates and a loss of statistical power. Statistical methods that make adjustments when dealing with these types of data, often called left-censored data, are available in many commercial statistical packages. Despite this availability, the use of these methods is still not widespread in biomedical literature. We have reviewed the statistical approaches of dealing with BDL values, and used simulations to examine the performance of the commonly used substitution methods and the most widely available statistical methods. We have illustrated these methods using a study undertaken at the Vanderbilt-Ingram Cancer Center, to examine the serum bile acid levels in patients with colorectal cancer and adenoma. We have found that the modern methods for BDL values identify disease-related differences that are often missed, with statistically naive approaches

Mixture Cure Survival Models with Dependent Censoring

A number of authors have studies the mixture survival model to analyze survival data with nonnegligible cure fractions. A key assumption made by these authors is the independence between the survival time and the censoring time. To our knowledge, no one has studies the mixture cure model in the presence of dependent censoring. To account for such dependence, we propose a more general cure model which allows for dependent censoring. In particular, we derive the cure models from the perspective of competing risks and model the dependence between the censoring time and the survival time using a class of Archimedean copula models. Within this framework, we consider the parameter estimation, the cure detection, and the two-sample comparison of latency distribution in the presence of dependent censoring when a proportion of patients is deemed cured. Large sample results using the martingale theory are obtained. We applied the proposed methodologies to the SEER prostate cancer data

Regression survival analysis with dependent censoring and a change point for the hazard rate: With application to the impact of the Gramm-Leach-Bliley Act to insurance companies' survival

This dissertation is aiming to find out the impact of the Gramm-Leach-Bliley Act on insurance companies' survival. The events of interest are bankruptcy and acquisition, which are correlated and censor each other. A statistical survival analysis method is developed first and then applied to the real insurance companies' survival data. In the methodology development, we first assess the effect of assuming independent censoring on the regression parameter estimates in Cox proportional hazard model. Then we apply the copula function to model the dependent censoring. Next, we propose an extended partial likelihood function maximized with an iteration algorithm to estimate the regression parameters and to derive the marginal survival functions under a dependent censoring setting. Simulations are conducted to demonstrate the method's performance, and sensitivity analyses are performed to assess the impact of the dependent censoring on the regression parameter estimates. In the last part of methodology, we propose a method to test the existence and to identify the location of a change-point in a hazard function. The application of our methodology to real insurance companies' survival data discloses important influence of the GLB Act on insurance companies survival