Graphical representation of survival curves in the presence of time-dependent categorical covariates with application to liver transplantation

Graphical representation of survival curves is often used to illustrate associations between exposures and time-to-event outcomes. However, when exposures are time-dependent, calculation of survival probabilities is not straightforward. Our aim was to develop a method to estimate time-dependent survival probabilities and represent them graphically. Cox models with time-dependent indicators to represent state changes were fitted, and survival probabilities were plotted using pre-specified times of state changes. Time-varying hazard ratios for the state change were also explored. The method was applied to data from the Adult-to-Adult Living Donor Liver Transplantation Cohort Study (A2ALL). Survival curves showing a ‘split’ at a pre-specified time t allow for the qualitative comparison of survival probabilities between patients with similar baseline covariates who do and do not experience a state change at time t. Time since state change interactions can be visually represented to reflect changing hazard ratios over time. A2ALL study results showed differences in survival probabilities among those who did not receive a transplant, received a living donor transplant, and received a deceased donor transplant. These graphical representations of survival curves with time-dependent indicators improve upon previous methods and allow for clinically meaningful interpretation.     

An index of local sensitivity to non-ignorability for parametric survival models with potential non-random missing covariate: an application to the SEER cancer registry data

Several survival regression models have been developed to assess the effects of covariates on failure times. In various settings, including surveys, clinical trials and epidemiological studies, missing data may often occur due to incomplete covariate data. Most existing methods for lifetime data are based on the assumption of missing at random (MAR) covariates. However, in many substantive applications, it is important to assess the sensitivity of key model inferences to the MAR assumption. The index of sensitivity to non-ignorability (ISNI) is a local sensitivity tool to measure the potential sensitivity of key model parameters to small departures from the ignorability assumption, needless of estimating a complicated non-ignorable model. We extend this sensitivity index to evaluate the impact of a covariate that is potentially missing, not at random in survival analysis, using parametric survival models. The approach will be applied to investigate the impact of missing tumor grade on post-surgical mortality outcomes in individuals with pancreas-head cancer in the Surveillance, Epidemiology, and End Results data set. For patients suffering from cancer, tumor grade is an important risk factor. Many individuals in these data with pancreas-head cancer have missing tumor grade information. Our ISNI analysis shows that the magnitude of effect for most covariates (with significant effect on the survival time distribution), specifically surgery and tumor grade as some important risk factors in cancer studies, highly depends on the missing mechanism assumption of the tumor grade. Also a simulation study is conducted to evaluate the performance of the proposed index in detecting sensitivity of key model parameters.     

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Abstract

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.     

The inverse power Lindley distribution in the presence of left-censored data

In this study, classical and Bayesian inference methods are introduced to analyze lifetime data sets in the presence of left censoring considering two generalizations of the Lindley distribution: a first generalization proposed by Ghitany et al. [Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), pp. 20–33], denoted as a power Lindley distribution and a second generalization proposed by Sharma et al. [The inverse Lindley distribution: A stress–strength reliability model with application to head and neck cancer data, J. Ind. Prod. Eng. 32 (2015), pp. 162–173], denoted as an inverse Lindley distribution. In our approach, we have used a distribution obtained from these two generalizations denoted as an inverse power Lindley distribution. A numerical illustration is presented considering a dataset of thyroglobulin levels present in a group of individuals with differentiated cancer of thyroid.     

Statistical Inferences for the Lifetime Performance Index of the Products with the Gompertz Distribution under Censored Samples

Manufacturers often apply process capability indices in the quality control. This study constructs statistical analysis methods of assessing the lifetime performance index of Gompertz products under progressively type II right censored samples. The maximum likelihood estimator of the index is inferred by data transformation and then utilized to develop a hypothesis testing procedure and a confidence interval to assess product performance. We also give one example and some Monte Carlo simulations to assess the behavior of the testing procedure and confidence interval. The results show that our proposed method can effectively evaluate whether the lifetime of products meet the requirement.     

Simultaneous confidence bands and regions for log-location-scale distributions with censored data

In many areas of application, especially life testing and reliability, it is often of interest to estimate an unknown cumulative distribution (cdf). A simultaneous confidence band (SCB) of the cdf can be used to assess the statistical uncertainty of the estimated cdf over the entire range of the distribution. Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] presented an approach to construct an SCB for the cdf of a continuous random variable. For the log-location-scale family of distributions, they gave explicit forms for the upper and lower boundaries of the SCB based on expected information. In this article, we extend the work of Cheng and Iles [1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25 (1), 77–86] in several directions. We study the SCBs based on local information, expected information, and estimated expected information for both the “cdf method” and the “quantile method.” We also study the effects of exceptional cases where a simple SCB does not exist. We describe calibration of the bands to provide exact coverage for complete data and type II censoring and better approximate coverage for other kinds of censoring. We also discuss how to extend these procedures to regression analysis.     

Joint modeling of censored longitudinal and event time data

Censoring of a longitudinal outcome often occurs when data are collected in a biomedical study and where the interest is in the survival and or longitudinal experiences of a study population. In the setting considered herein, we encountered upper and lower censored data as the result of restrictions imposed on measurements from a kinetic model producing “biologically implausible” kidney clearances. The goal of this paper is to outline the use of a joint model to determine the association between a censored longitudinal outcome and a time to event endpoint. This paper extends Guo and Carlin's [6] paper to accommodate censored longitudinal data, in a commercially available software platform, by linking a mixed effects Tobit model to a suitable parametric survival distribution. Our simulation results showed that our joint Tobit model outperforms a joint model made up of the more naïve or “fill-in” method for the longitudinal component. In this case, the upper and/or lower limits of censoring are replaced by the limit of detection. We illustrated the use of this approach with example data from the hemodialysis (HEMO) study [3] and examined the association between doubly censored kidney clearance values and survival.     

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.     

Likelihood Ratio Confidence Bands in Non–parametric Regression with Censored Data

Let ( X , Y ) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a non-parametric likelihood ratio approach. This approach was introduced by Thomas & Grunkemeier (1975 ), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0, 1], it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

EM algorithm-based likelihood estimation for a generalized Gompertz regression model in presence of survival data with long-term survivors: an application to uterine cervical cancer data

In this paper we develop a regression model for survival data in the presence of long-term survivors based on the generalized Gompertz distribution introduced by El-Gohary et al. [The generalized Gompertz distribution. Appl Math Model. 2013;37:13–24] in a defective version. This model includes as special case the Gompertz cure rate model proposed by Gieser et al. [Modelling cure rates using the Gompertz model with covariate information. Stat Med. 1998;17:831–839]. Next, an expectation maximization algorithm is then developed for determining the maximum likelihood estimates (MLEs) of the parameters of the model. In addition, we discuss the construction of confidence intervals for the parameters using the asymptotic distributions of the MLEs and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study. Finally, the proposed methodology was applied to a database on uterine cervical cancer.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Likelihood inference for a step-stress partially accelerated life test model with Type-I progressively hybrid censored data from Weibull distribution

Recently, progressively hybrid censoring schemes have become quite popular in life testing and reliability studies. In this article, the point and interval maximum-likelihood estimations of Weibull distribution parameters and the acceleration factor are considered. The estimation process is performed under Type-I progressively hybrid censored data for a step-stress partially accelerated test model. The biases and mean square errors of the maximum-likelihood estimators are computed to assess their performances in the presence of censoring developed in this article through a Monte Carlo simulation study.     

Survival trees with time-dependent covariates: Application to estimating changes in the incubation period of AIDS

Tree-structured methods for exploratory data analysis have previously been extended to right-censored survival data. We further extend these methods to allow for truncation and time-dependent covariates. We apply the new methods to a data set on incubation times of acquired immunodeficiency syndrome (AIDS), using calendar time as a time-dependent covariate. Contrary to expectation, we find that rates of progression to AIDS appear to be faster after August 1989 than before.     

Reliability of a soccer player based on the bivariate Rayleigh distribution with right censored and ignorable missing data

In this paper, we study the performance of a soccer player based on analysing an incomplete data set. To achieve this aim, we fit the bivariate Rayleigh distribution to the soccer dataset by the maximum likelihood method. In this way, the missing data and right censoring problems, that usually happen in such studies, are considered. Our aim is to inference about the performance of a soccer player by considering the stress and strength components. The first goal of the player of interest in a match is assumed as the stress component and the second goal of the match is assumed as the strength component. We propose some methods to overcome incomplete data problem and we use these methods to inference about the performance of a soccer player.     

Analysis of count data with covariate dependence in both mean and variance

Extended Poisson process modelling is generalised to allow for covariate-dependent dispersion as well as a covariate-dependent mean response. This is done by a re-parameterisation that uses approximate expressions for the mean and variance. Such modelling allows under- and over-dispersion, or a combination of both, in the same data set to be accommodated within the same modelling framework. All the necessary calculations can be done numerically, enabling maximum likelihood estimation of all model parameters to be carried out. The modelling is applied to re-analyse two published data sets, where there is evidence of covariate-dependent dispersion, with the modelling leading to more informative analyses of these data and more appropriate measures of the precision of any estimates.     

Multiple imputation of censored survival data in the presence of missing covariates using restricted mean survival time

Missing covariates data with censored outcomes put a challenge in the analysis of clinical data especially in small sample settings. Multiple imputation (MI) techniques are popularly used to impute missing covariates and the data are then analyzed through methods that can handle censoring. However, techniques based on MI are available to impute censored data also but they are not much in practice. In the present study, we applied a method based on multiple imputation by chained equations to impute missing values of covariates and also to impute censored outcomes using restricted survival time in small sample settings. The complete data were then analyzed using linear regression models. Simulation studies and a real example of CHD data show that the present method produced better estimates and lower standard errors when applied on the data having missing covariate values and censored outcomes than the analysis of the data having censored outcome but excluding cases with missing covariates or the analysis when cases with missing covariate values and censored outcomes were excluded from the data (complete case analysis).