Censored Failure Time Research Articles

Deep Neural Network-Based Accelerated Failure Time Models Using Rank Loss.

An accelerated failure time (AFT) model assumes a log-linear relationship between failure times and a set of covariates. In contrast to other popular survival models that work on hazard functions, the effects of covariates are directly on failure times, the interpretation of which is intuitive. The semiparametric AFT model that does not specify the error distribution is sufficiently flexible and robust to depart from the distributional assumption. Owing to its desirable features, this class of model has been considered a promising alternative to the popular Cox model in the analysis of censored failure time data. However, in these AFT models, a linear predictor for the mean is typically assumed. Little research has addressed the non-linearity of predictors when modeling the mean. Deep neural networks (DNNs) have received much attention over the past few decades and have achieved remarkable success in a variety of fields. DNNs have a number of notable advantages and have been shown to be particularly useful in addressing non-linearity. Here, we propose applying a DNN to fit AFT models using Gehan-type loss combined with a sub-sampling technique. Finite sample properties of the proposed DNN and rank-based AFT model (DeepR-AFT) were investigated via an extensive simulation study. The DeepR-AFT model showed superior performance over its parametric and semiparametric counterparts when the predictor was nonlinear. For linear predictors, DeepR-AFT performed better when the dimensions of the covariates were large. The superior performance of the proposed DeepR-AFT was demonstrated using three real datasets.

Regression analysis of doubly censored failure time data with ancillary information.

Doubly censored failure time data occur in many areas and for the situation, the failure time of interest usually represents the elapsed time between two related events such as an infection and the resulting disease onset. Although many methods have been proposed for regression analysis of such data, most of them are conditional on the occurrence time of the initial event and ignore the relationship between the two events or the ancillary information contained in the initial event. Corresponding to this, a new sieve maximum likelihood approach is proposed that makes use of the ancillary information, and in the method, the logistic model and Cox proportional hazards model are employed to model the initial event and the failure time of interest, respectively. A simulation study is conducted and suggests that the proposed method works well in practice and is more efficient than the existing methods as expected. The approach is applied to an AIDS study that motivated this investigation.

Moments of inverse Weibull-geometric distribution based on progressive type-II right censored order statistics

In this article, we derive explicit expressions and some recurrence relations for the single and product moments of progressively Type-II right censored order statistics from the inverse Weibull-geometric distribution, and then we use these results to calculate the means and variances of progressively Type-II right censored order statistics. Also, we discuss the best linear unbiased estimators for the location and scale parameters as well as the best linear unbiased predictors of the censored failure times. Finally, a real data application is presented.

EXPLORING THE MODIFIED GAMMA FRAILTY DISTRIBUTION: AN OPTIMAL DESIGN APPROACH USING PYTHON

A suction/injection controlled mixed convection flow of an incompressible and viscous fluid in a vertical SurvivalAnalysis is pivotal in understanding the effects of covariates on potentially censored failure times and in the joint modelling of clustered data. It is used in the context of incomplete repeated measures and failure times in longitudinal studies. Survival data are often subject to right censoring and to a subsequent loss of information about the effect of explanatory variables. Frailty models are one common approach to handle such data.Three frailty models are used to analyze bivariate time-to-event data. All approaches accommodate right censored lifetime data and account for heterogeneity in the study population. A Modified Gamma Frailty Model is compared with two existing Frailty Models. The survival-analysis was performed using the Python.The newly derived MGF was analyzed using Python which is more robust when sample size is more than forty.The MGF model performs better than the existing models in the presence of clustering. However the CGF is preferable in the absence of clusters in a given data set.

Analysis of the Cox Model with Longitudinal Covariates with Measurement Errors and Partly Interval Censored Failure Times, with Application to an AIDS Clinical Trial.

The online version contains supplementary material available at 10.1007/s12561-023-09372-y.

Estimating subject-specific hazard functions

Abstract The central idea of this paper is to compare mean responses of several subjects in the presence of censoring and subject-specific variation. We develop a semiparametric mixed model for fitting subject-specific hazard curves to a set of censored failure times. A spline-based model and a mixed effects framework for smoothing are used. Efficient estimators of fixed parameters and predictors of the random components are derived and their asymptotic properties studied. This is a generalization of the method proposed by [Cai, T., Hyndman, R. J., & Wand, M. P. (2002). Mixed model-based hazard estimation. Journal of Computational and Graphical Statistics, 11(4), 784–798. https://doi.org/10.1198/106186002862] to incorporate additional subject-specific variation of the hazard function. The results are illustrated using two motivating examples.

The weighted log-rank tests based on stratified clustered survival data: saddle-point p-values and confidence intervals

ABSTRACT Clinical studies sometimes provide clustered data with censored failure times. A crucial factor of the randomized design that lessens selection bias is the random allocation rule. Given this, the weighted rank tests’ p-values for stratified survival clustered sampling based on the random allocation rule are approximated using the double saddle-point approximation technique. For tests of significance and confidence intervals for the treatment effect, this approximation can be utilized. Through simulation experiments, the accuracy of the saddle-point approximation is examined by comparing saddle-point and normal approximations to the exact underlying permutation distribution.

Bayesian Estimation of Transmuted Lomax Mixture Model with an Application to Type-I Censored Windshield Data

Transmuted distributions have been centered of focus for researchers recently due to their flexibility and applicability in statistics. However, the only few contributions have considered estimation for mixture of transmuted lifetime models especially under Bayesian methods has been explored more recently. We have considered the Bayesian estimation of transmuted Lomax mixture model (TLMM) for type-I censored samples. The Bayes estimates (BEs) for informative and non-informative priors. The BEs and posterior risks (PRs) are evaluated using four different loss functions (LFs), two symmetric and two asymmetric, namely the squared error loss function (SELF), precautionary loss function (PLF), weighted balance loss function (WBLF), and general entropy loss function (GELF). Simulations are run using Lindley Approximation method to compare the BEs under various sample sizes and censoring rates. The estimates under informative prior and GELF were found superior to their counterparts. The applicability of the proposed estimates has been illustrated using the analysis of a real data regarding type-I censored failure times of windshields airplanes.

Instrumental variable estimation of the causal hazard ratio.

Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first-stage regression of the treatment on the IV and unmeasured confounders. We propose, to the best of our knowledge, the first consistent estimator of the (population) causal hazard ratio within an instrumental variable framework. A version of our estimator admits a closed-form representation. We derive the asymptotic distribution of our estimator and provide a consistent estimator for its asymptotic variance. Our approach is illustrated via simulation studies and a dataapplication.

Classical and Bayesian estimations of improved Weibull–Weibull distribution for complete and censored failure times data

AbstractIn this article, we demonstrate how to enhance the Weibull–Weibull (WW) distribution introduced in the earlier literature into a better form for fitting monotone and non‐monotone failure rate data, especially the bathtub‐shaped failure rate data with or without a flat region. The model is referred to as an improved WW distribution. The model's flexibility enables it to describe lifetime data with various failure rate functions, including increasing, decreasing, U or V‐shaped, and bathtub‐shaped with a comparatively low and long‐flat segment. We provide a thorough Bayesian analysis of the modified model for complete and right‐censored data. Additionally, we developed maximum likelihood estimators for the model's parameters for both complete and right‐censored data. The Bayesian credible and asymptotic confidence intervals of the estimators are defined, and simulation results validate the estimators' consistency. To illustrate the applications of the improved distribution with the WW and other generalized distributions, we apply one censored and one uncensored failure times data, each with bathtub‐shaped failure rates. The numerical results demonstrate that the improved WW model outperforms the WW distribution and other existing models, as indicated by goodness‐of‐fit statistics and supported by the fitted models' survival and failure rate curves and P‐P plots.

Instrumental variable estimation of the marginal structural Cox model for time-varying treatments.

Robins (1998) introduced marginal structural models, a general class of counterfactual models for the joint effects of time-varying treatments in complex longitudinal studies subject to time-varying confounding. Robins (1998) established the identification of marginal structural model parameters under a sequential randomization assumption, which rules out unmeasured confounding of treatment assignment over time. The marginal structural Cox model is one of the most popular marginal structural models for evaluating the causal effect of time-varying treatments on a censored failure time outcome. In this paper, we establish sufficient conditions for identification of marginal structural Cox model parameters with the aid of a time-varying instrumental variable, in the case where sequential randomization fails to hold due to unmeasured confounding. Our instrumental variable identification condition rules out any interaction between an unmeasured confounder and the instrumental variable in its additive effects on the treatment process, the longitudinal generalization of the identifying condition of Wang & Tchetgen Tchetgen (2018). We describe a large class of weighted estimating equations that give rise to consistent and asymptotically normal estimators of the marginal structural Cox model, thereby extending the standard inverse probability of treatment weighted estimation of marginal structural models to the instrumental variable setting. Our approach is illustrated via extensive simulation studies and an application to estimating the effect of community antiretroviral therapy coverage on HIV incidence.

Saddle-point p-values and confidence intervals based on log-rank tests when dependent subunits of clustered survival data are randomized by random allocation design

In clinical trials, Clustered data with censored failure times often arise from a litter-matched tumorigenesis experiment. Based on such data, the mutual and widely used class of two-sample tests is the weighted log-rank tests. A double saddle-point approximation is used to calculate the p-values of the null permutation distribution of these tests. This approximation is superior to the asymptotic normal approximation. This precision allows us to determine exact confidence intervals for the treatment impact. The findings reveal the efficiency of saddle-point approximation using two real clustered data sets. Extensive simulation studies are carried out to assess the performance of the saddle-point approximation.

Accelerated failure time modeling via nonparametric mixtures.

An accelerated failure time (AFT) model assuming a log-linear relationship between failure time and a set of covariates can be either parametric or semiparametric, depending on the distributional assumption for the error term. Both classes of AFT models have been popular in the analysis of censored failure time data. The semiparametric AFT model is more flexible and robust to departures from the distributional assumption than its parametric counterpart. However, the semiparametric AFT model is subject to producing biased results for estimating any quantities involving an intercept. Estimating an intercept requires a separate procedure. Moreover, a consistent estimation of the intercept requires stringent conditions. Thus, essential quantities such as mean failure times might not be reliably estimated using semiparametric AFT models, which can be naturally done in the framework of parametric AFT models. Meanwhile, parametric AFT models can be severely impaired by misspecifications. To overcome this, we propose a new type of the AFT model using a nonparametric Gaussian-scale mixture distribution. We also provide feasible algorithms to estimate the parameters and mixing distribution. The finite sample properties of the proposed estimators are investigated via an extensive stimulation study. The proposed estimators are illustrated using a realdataset.

Matching distributions for survival data

In studies with survival endpoints, it is often of interest to predict the disease risk or survival probabilities in the presence of censored failure times. One commonly used approach is to model the association between the survival outcome and covariates via a semiparametric regression model and use the fitted model for prediction. In this article, we propose two methods to evaluate or predict the survival rates. The first method estimates survival probabilities by matching survival functions, and the second one is based on matching censored quantiles. Unlike traditional regression approaches, the proposed methods directly match the distribution of linear combinations of the covariates to the entire target distribution or parts of it. To accommodate censoring, we adopt a redistribution‐of‐mass technique for the proposed matching censored quantiles. The asymptotic consistency of the resulting estimators is well established. Simulation studies and an example with real data are also provided to further illustrate the practical utilities of our proposals. The proposed methods have been implemented in an R package.

Evaluation of statistical methods for assessing reliability from degradation data : A simulation study

This paper deals with the evaluation of statistical methods for assessing product reliability from degradation data. The evaluation is based on a simulation study. Benefits of assessing reliability from degradation data over censored failure time data are demonstrated using data from the first simulation run. Bias, precision and coverage probability calculated from the 105 simulated samples are the performance measures used in the evaluation. Analysis of a real gallium arsenide (GaAs) laser data set for telecommunication systems emphasizes the practical advantages of assessing reliability from degradation data.

A Bayesian Survival Model Approach for Business Distress Prediction

The early warning signals of corporate distress and failure have been a major area of concern for shareholders, policy makers and academicians alike. Numerous approaches have been applied to examine firm insolvency ranging from the famous Altman’s Z-score, traditional econometrics, financial ratio analysis to the more contemporary tools of Artificial Intelligence and Machine Learning. The Cox proportional survival hazard model is a commonly applied technique not only in the field of medical sciences for estimating occurrences of a specific event but also in failure prediction of private firms. The study investigates distress prediction of firms in context of emerging nation like India where otherwise the application of Bayesian survival models is limited. A rich panel of firms spanning over ten years and representing varied sectors like manufacturing, services, mining and construction is compiled for the purpose. The study contributes by developing hazard (survival) modelling using Bayesian perspective. The advantage of Bayesian method lies in dealing effectively with censored and small samples over usual frequentist methods. Both standard Cox survival model for censored failure time and Bayesian estimation have been performed to assess and compare their performance. It is found that prediction accuracy of Bayesian Cox model is significantly higher than of the classical Cox model. The study contributes by providing useful insights in detecting early signs of distress in Indian corporate sector that is otherwise scant in literature.

Asymptotic normality of the relative error regression function estimator for censored and time series data

Abstract Consider a survival time study, where a sequence of possibly censored failure times is observed with d-dimensional covariate The main goal of this article is to establish the asymptotic normality of the kernel estimator of the relative error regression function when the data exhibit some kind of dependency. The asymptotic variance is explicitly given. Some simulations are drawn to lend further support to our theoretical result and illustrate the good accuracy of the studied method. Furthermore, a real data example is treated to show the good quality of the prediction and that the true data are well inside in the confidence intervals.

Regression analysis of censored data with nonignorable missing covariates and application to Alzheimer Disease

In this paper, we discuss regression analysis of censored failure time data when there exist missing covariates and more specifically, we will consider interval-censored data, a general form of censored data, and the nonignorable missing. Although many methods have been proposed in the literature for censored data with missing covariates, they only apply to limited situations and it does not seem to exist an established procedure for the situation discussed here. For the analysis, we employ the semiparametric linear transformation model and develop a two-step estimation procedure. In addition, the asymptotic properties of the resulting estimators are established and a Poisson variable-based EM algorithm is provided for the implementation of the proposed estimation procedure. Finally the proposed approach is applied to an Alzheimer Disease study that motivated this investigation.

Log-rank tests for censored clustered data under generalized randomized block design: Saddlepoint approximation

ABSTRACT The weighted log-rank class is the common and widely used class of two-sample tests for clustered data. Clustered data with censored failure times often arise in tumorigenicity investigations and clinical trials. The randomized block design is a significant design that reduces both unintentional bias and selection bias. Accordingly, the p-values of the null permutation distribution of weighted log-rank class for clustered data are approximated using the double saddlepoint approximation technique. Comprehensive simulation studies are carried out to appraise the accuracy of the saddlepoint approximation. This approximation exhibits a significant improvement in precision over the asymptotic approximation. This precision motivates us to determine the approximated confidence intervals for the treatment impact.

Model diagnostics for censored regression via randomized survival probabilities

Residuals in normal regression are used to assess a model's goodness-of-fit (GOF) and discover directions for improving the model. However, there is a lack of residuals with a characterized reference distribution for censored regression. In this article, we propose to diagnose censored regression with normalized randomized survival probabilities (RSP). The key idea of RSP is to replace the survival probability (SP) of a censored failure time with a uniform random number between 0 and the SP of the censored time. We prove that RSPs always have the uniform distribution on (0, 1) under the true model with the true generating parameters. Therefore, we can transform RSPs into normally distributed residuals with the normal quantile function. We call such residuals by normalized RSP (NRSP residuals). We conduct simulation studies to investigate the sizes and powers of statistical tests based on NRSP residuals in detecting the incorrect choice of distribution family and nonlinear effect in covariates. Our simulation studies show that, although the GOF tests with NRSP residuals are not as powerful as a traditional GOF test method, a nonlinear test based on NRSP residuals has significantly higher power in detecting nonlinearity. We also compared these model diagnostics methods with a breast-cancer recurrent-free time dataset. The results show that the NRSP residual diagnostics successfully captures a subtle nonlinear relationship in the dataset, which is not detected by the graphical diagnostics with CS residuals and existing GOF tests.