We are concerned with the flexible parametric analysis of bivariate survival data. Elsewhere, we have extolled the virtues of the "power generalized Weibull" (PGW) distribution as an attractive vehicle for univariate parametric survival analysis: it is a tractable, parsimonious, model which interpretably allows for a wide variety of hazard shapes and, when adapted (to give an adapted PGW, or APGW, distribution), covers a wide variety of important special/limiting cases. Here, we additionally observe a frailty relationship between a PGW distribution with one value of the parameter which controls distributional choice within the family and a PGW distribution with a smaller value of the same parameter. We exploit this frailty relationship to propose a bivariate shared frailty model with PGW marginal distributions: these marginals turn out to be linked by the so-called BB9 or "power variance function" copula. This particular choice of copula is, therefore, a natural one in the current context. We then adapt the bivariate PGW distribution, in turn, to accommodate APGW marginals. We provide a number of theoretical properties of the bivariate PGW and APGW models and show the potential of the latter for practical work via an illustrative example involving a well-known retinopathy dataset, for which the analysis proves to be straightforward to implement and informative in its outcomes. The novelty in this article is in the appropriate combination of specific ingredients into a coherent and successful whole.
A setting of a trivairate survival function using semi-competing risks concept is proposed. The Stanford Heart Transplant data is reanalyzed using a trivariate Weibull distribution model with the proposed survival function.
In this paper, we consider survival analysis with right-censored data which is a common situation in predictive maintenance and health field. We propose a model based on the estimation of two-parameter Weibull distribution conditionally to the features. To achieve this result, we describe a neural network architecture and the associated loss functions that takes into account the right-censored data. We extend the approach to a finite mixture of two-parameter Weibull distributions. We first validate that our model is able to precisely estimate the right parameters of the conditional Weibull dis…
A bivariate cure copula model with zero-inflated gamma frailty for right-censored data
Graduate Institute for Advanced Studies, SOKENDAI（SOKENDAI 高级研究研究生院）；Institute of Statistical Mathematics（统计数学研究所）；School of Informatics and Data Science, Hiroshima University（广岛大学信息科学与数据科学学院）
The use of flat or weakly informative priors is popular due to the objective a priori belief in the absence of strong prior information. In the case of the Weibull model the improper uniform, equal parameter gamma and joint Jeffrey's priors for the shape parameter are popular choices. The effects and behaviors of these priors have yet to be established from a modeling viewpoint, especially their ability to reduce to the simpler exponential model. In this work we propose a new principled prior for the shape parameter of the Weibull model, originating from a prior on the distance function, and a…
The Weibull distribution, with shape parameter $k>0$ and scale parameter $λ>0$, is one of the most popular parametric distributions in survival analysis with complete or censored data. Although inference of the parameters of the Weibull distribution is commonly done through maximum likelihood, it is well established that the maximum likelihood estimate of the shape parameter is inadequate due to the associated large bias when the sample size is small or the proportion of censored data is large. This manuscript demonstrates how the Bayesian information-theoretic minimum message length principle…
We develop flexible multi-parameter regression survival models for interval censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multi-parameter Weibull regression survival model, which is wholly parametric, and has non-proportional hazards, is the main focus of the paper. We describe the basic model, develop the interval-censored likelihood and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed re-analysis of data from the Signal Tandmobie…
In this paper, we introduce a new class of bivariate distributions called the bivariate exponentiated extended Weibull distributions. The model introduced here is of Marshall-Olkin type. This new class of bivariate distributions contains several bivariate lifetime models. Some mathematical properties of the new class of distributions are studied. We provide the joint and conditional density functions, the joint cumulative distribution function and the joint survival function. Special bivariate distributions are investigated in some detail. The maximum likelihood estimators are obtained using t…
In reliability and life data analysis, the Weibull distribution is widely used to accommodate more data characteristics by changing the values of the parameters. We frequently observe many zeros or close to zero data points in reliability and life testing experiments. We call this phenomenon a nearly instantaneous failure. Many researchers modified the commonly used univariate parametric models such as exponential, gamma, Weibull, and log-normal distributions to appropriately fit such data having instantaneous failure observations. Researchers also find bivariate correlated life testing data h…
In this paper, we develop a family of bivariate beta distributions that encapsulate both positive and negative correlations, and which can be of general interest for Bayesian inference. We then invoke a use of these bivariate distributions in two contexts. The first is diagnostic testing in medicine, threat detection, and signal processing. The second is system survivability assessment, relevant to engineering reliability, and to survival analysis in biomedicine. In diagnostic testing one encounters two parameters that characterize the efficacy of the testing mechanism, {\it test sensitivity},…
Several gene-based association tests for time-to-event traits have been proposed recently, to detect whether a gene region (containing multiple variants), as a set, is associated with the survival outcome. However, for bivariate survival outcomes, to the best of our knowledge, there is no statistical method that can be directly applied for gene-based association analysis. Motivated by a genetic study to discover gene regions associated with the progression of a bilateral eye disease, Age-related Macular Degeneration (AMD), we implement a novel functional regression method under the copula fram…
To examine the effect of exponential, Weibull and Gompertz distributions on sample size determination for superiority trials (STs) or non-inferiority trials (NTs) with time-to-event data, we present two sample size formulas for STs or NTs based on Weibull and Gompertz distributions, respectively. The formulas are compared with the current exponential formula to examine their performance. The simulation results show that the sample size formula based on the Weibull distribution is the most robust among the three formulas in STs or NTs. We suggest that recognizing the appropriate distribution in…
Recently, Lee and Cha (2015, `On two generalized classes of discrete bivariate distributions', {\it American Statistician}, 221 - 230) proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall-Olkin bivariate Weibull distribution. We study various proper…
We introduce a general, flexible, parametric survival modelling framework which encompasses key shapes of hazard function (constant, increasing, decreasing, up-then-down, down-then-up), various common survival distributions (log-logistic, Burr type XII, Weibull, Gompertz), and includes defective distributions (i.e., cure models). This generality is achieved using four basic distributional parameters: two scale-type parameters and two shape parameters. Generalising to covariate dependence, the scale-type regression components correspond to accelerated failure time (AFT) and proportional hazards…
The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterise the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. We call this model the double-Cox model. The R programs for fitting the double-Cox model are available on Github. We formally introduce the double-Cox model with shared frailty and investigat…
The peculiar properties of the Inverse Weibull (IW) distribution are shown. It is proven that the IW distribution is one of the few models having upside- down bathtub (UBT) shaped hazard function. Three real and typical de generative mechanisms, which lead exactly to the IW random variable, are formulated. So a new approach to proper application of this relatively unknown survival model is supported. However, we consider also the case in which any knowledge about generative mechanism is unavailable. In this hypothesis, we study a procedure based on the Anderson-Darling statistic and log-likeli…
This paper describes the use of survival analysis and simulation to model the lifetime of high voltage instrument transformers in the Dutch transmission sys-tem. To represent asset aging, the non-parametric Kaplan-Meier method is used to enable the fitting of Weibull distribution. Such an approach is implemented on three different voltage levels, namely 110kV, 150kV, and 220/380kV. Real failure and inspection data is used to achieve a realistic failure model of the instrument trans-formers. Failure and maintenance data occurring between 1989 and 2021 have been used for this study. In spite of …
This paper presents a hierarchical Bayesian approach to the estimation of components' reliability (survival) using a Weibull model for each of them. The proposed method can be used to estimation with general survival censored data, because the estimation of a component's reliability in a series (parallel) system is equivalent to the estimation of its survival function with right- (left-) censored data. Besides the Weibull parametric model for reliability data, independent gamma distributions are considered at the first hierarchical level for the Weibull parameters and independent uniform distr…
merlin can do a lot of things. From simple stuff, like fitting a linear regression or a Weibull survival model, to a three-level logistic mixed effects model, or a multivariate joint model of multiple longitudinal outcomes (of different types) and a recurrent event and survival with non-linear effects...the list is rather endless. merlin can do things I haven't even thought of yet. I'll take a single dataset, and attempt to show you the full range of capabilities of merlin, and discuss some future directions for the implementation in Stata.
Survival data is encountered in a range of disciplines, most notably health and medical research. Although Bayesian approaches to the analysis of survival data can provide a number of benefits, they are less widely used than classical (e.g. likelihood-based) approaches. This may be in part due to a relative absence of user-friendly implementations of Bayesian survival models. In this article we describe how the rstanarm R package can be used to fit a wide range of Bayesian survival models. The rstanarm package facilitates Bayesian regression modelling by providing a user-friendly interface (us…
For many diseases it is reasonable to assume that the hazard rate is not constant across time, but also that it changes in different time intervals. To capture this, we work here with a piecewise survival model. One of the major problems in such piecewise models is to determine the time points of change of the hazard rate. From the practical point of view this can provide very important information as it may reflect changes in the progress of a disease. We present piecewise Weibull regression models with covariates. The time points where change occurs are assumed unknown and need to be estimat…
We introduce a general class of continuous univariate distributions with positive support obtained by transforming the class of two-piece distributions. We show that this class of distributions is very flexible, easy to implement, and contains members that can capture different tail behaviours and shapes, producing also a variety of hazard functions. The proposed distributions represent a flexible alternative to the classical choices such as the log-normal, Gamma, and Weibull distributions. We investigate empirically the inferential properties of the proposed models through an extensive simula…
A six parameter distribution so-called the McDonald modified Weibull distribution is defined and studied. The new distribution contains, as special submodels, several important distributions discussed in the literature, such as the beta modified Weibull, Kumaraswamy modified Weibull, McDonald Weibull and modified Weibull distribution,among others. The new distribution can be used effectively in the analysis of survival data since it accommodates monotone, unimodal and bathtub-shaped hazard functions. We derive the moments.We propose the method of maximum likelihood for estimating the model par…
We consider nonparametric inference for event time distributions based on current status data. We show that in this scenario conventional mixture priors, including the popular Dirichlet process mixture prior, lead to biologically uninterpretable results as they unnaturally skew the probability mass for the event times toward the extremes of the observed data. Simple assumptions on dependent censoring can fix the problem. We then extend the discussion to bivariate current status data with partial ordering of the two outcomes. In addition to dependent censoring, we also exploit some minimal know…
Massive Open Online Course (MOOC) platforms incorporate large course catalogs from which individual students may register multiple courses. We performed a network-based analysis of student achievement, considering how course-course interactions may positively or negatively affect student success. Our dataset included 378,000 users and 1,000,000 unique registration events in France Universite Numerique (FUN), a national MOOC platform. We adapt reliability theory to model certificate completion rates with a Weibull survival function, following the intuition that students "survive" in a course fo…
LinkedIn Corporation（领英公司）；University of Maryland（马里兰大学）
Mobile notifications have become a major communication channel for social networking services to keep users informed and engaged. As more mobile applications push notifications to users, they constantly face decisions on what to send, when and how. A lack of research and methodology commonly leads to heuristic decision making. Many notifications arrive at an inappropriate moment or introduce too many interruptions, failing to provide value to users and spurring users' complaints. In this paper we explore unique features of interactions between mobile notifications and user engagement. We propo…
Some new survival distributions are introduced based on a generalised exponential function. This class of distributions includes heavy-tailed generalisations of exponential, Weibull and gamma distributions. Properties of the distributions are described, and R code is available for computation of pdf, quantiles, inverse quantiles, random numbers, etc. A use of these distributions for robust inference is suggested, and this is exemplified with a Monte-Carlo study.
An evaluation of FBST, Fully Bayesian Significance Test, restricted to survival models is the main objective of the present paper. A Survival distribution should be chosen among the tree celebrated ones, lognormal, gamma, and Weibull. For this discrimination, a linear mixture of the three distributions, for which the mixture weights are defined by a Dirichlet distribution of order three, is an important tool: the FBST is used to test the hypotheses defined on the mixture weights space. Another feature of the paper is that all three distributions are reparametrized in that all the six parameter…
A Real Data-Driven, Robust Survival Analysis on Patients who Underwent Deep Brain Stimulation for Parkinson's Disease by Utilizing Parametric, Non-Parametric, and Semi-Parametric Approaches
Nokia Bell Labs France（诺基亚贝尔实验室法国）；Inria（法国国家信息与自动化研究所）；Safran Tech（赛峰集团技术公司（Safran Tech））
In this paper, we make an experimental comparison of semi-parametric (Cox proportional hazards model, Aalen's additive regression model), parametric (Weibull AFT model), and machine learning models (Random Survival Forest, Gradient Boosting with Cox Proportional Hazards Loss, DeepSurv) through the concordance index on two different datasets (PBC and GBCSG2). We present two comparisons: one with the default hyper-parameters of these models and one with the best hyper-parameters found by randomized search.
Estimation of reliability and hazard rate is one of the most important problems raised in many applications especially in engineering studies as well as human lifetime. In this regard, different methods of estimation have been used. Each method exploits various tools and suffers from problems such as complexity of computations, low precision, and so forth. This study is employed the E-Bayesian method, for estimating the parameter and survival functions of the Weibull Generalized Exponential distribution. The estimators are obtained under squared error and LINEX loss functions under progressive…
Acyclic phase-type (PH) distributions have been a popular tool in survival analysis, thanks to their natural interpretation in terms of ageing towards its inevitable absorption. In this paper, we consider an extension to the bivariate setting for the modelling of joint lifetimes. In contrast to previous models in the literature that were based on a separate estimation of the marginal behavior and the dependence structure through a copula, we propose a new time-inhomogeneous version of a multivariate PH class (mIPH) that leads to a model for joint lifetimes without that separation. We study pro…
Statistics of punctuation in experimental literature -- the remarkable case of "Finnegans Wake" by James Joyce
Institute of Nuclear Physics, Polish Academy of Sciences（波兰科学院核物理研究所）；Faculty of Computer Science and Telecommunications, Cracow University of Technology（克拉科夫理工大学计算机与电信学院）
We develop two new classes of tests for the Weibull distribution based on Stein's method. The proposed tests are applied in the full sample case as well as in the framework of random right censoring. We investigate the finite sample performance of the new tests using a comprehensive Monte Carlo study. In both the absence and presence of censoring, it is found that the newly proposed classes of tests outperform competing tests against the majority of the distributions considered. In the cases where censoring is present we consider various censoring distributions. Some remarks on the asymptotic …
Technion – Israel Institute of Technology（以色列理工学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Sheba Medical Center, Tel Aviv University（谢巴医疗中心，特拉维夫大学）；London School of Hygiene & Tropical Medicine（伦敦卫生与热带医学院）；New York University（纽约大学）；Tel Aviv University（特拉维夫大学）
Handling missing values plays an important role in the analysis of survival data, especially, the ones marked by cure fraction. In this paper, we discuss the properties and implementation of stochastic approximations to the expectation-maximization (EM) algorithm to obtain maximum likelihood (ML) type estimates in situations where missing data arise naturally due to right censoring and a proportion of individuals are immune to the event of interest. A flexible family of three parameter exponentiated-Weibull (EW) distributions is assumed to characterize lifetimes of the non-immune individuals a…
It is standard practice for covariates to enter a parametric model through a single distributional parameter of interest, for example, the scale parameter in many standard survival models. Indeed, the well-known proportional hazards model is of this kind. In this paper we discuss a more general approach whereby covariates enter the model through more than one distributional parameter simultaneously (e.g., scale and shape parameters). We refer to this practice as "multi-parameter regression" (MPR) modelling and explore its use in a survival analysis context. We find that multi-parameter regress…
