Joint Distribution Of Survival Times Research Articles

S26 Comparison of Machine and Deep Learning Methods With the Mayo Clinical Risk Score in Predicting 1-Year Mortality in Primary Biliary Cholangitis Patients

Introduction: To compare machine learning (ML) and deep learning (DL) approaches to predict survival outcome in patients at 1- year with Primary Biliary Cholangitis (PBC) and to compare with the Mayo Clinical risk score Methods: The data is taken from a randomized trial in PBC patients conducted between 1974 and 1984 by the Mayo Clinic. The first 312 cases in the dataset were included out of a total of 424 PBC patients referred. The rest that contained subjects with >30% missing data, were excluded. We evaluated the following models: 1) The 5-covariate Mayo model: log(bilirubin), albumin, log(prothrombin time), edema, and age, 2) 17-covariate Cox proportional hazards model by componentwise likelihood based boosting with stepnumber 10 and penalty number 100, 3) 17-covariate Deephit: DeepHit: trains a neural network to learn the estimated joint distribution of survival time and event, while capturing the right-censored nature inherent in survival data. Analysis was done with frac 0.3, relu activation, 0.1 dropout, 100L epochs and a batch size of 32L, 4) 17-covariate Multitask logistic regression model with ranking based feature selection to predict survival using a logistic regression model and the parameters from each model are estimated simultaneously in the maximization of the joint likelihood function, 5) random survival forest with 1000 trees and 6) Support vector machine. For the ML & DL models all 17-covariates were include in the analysis. Analyses were done in RStudio, and missing values were imputed using missRanger package.1 The data is split into 80 percent training and 20 precent validation. All models were 5-fold cross validated. Prediciton statistics was calculated for each model developed. Finally, the ML & DL models were compared with the 5-covariate model of the Mayo Clinic Risk score. Results: Multitask logistic regression model showed the highest AUC (1.00) with a Harrell’s C-statistic of 0.90. Cox boost and random survival forest had equal AUCs (0.87) and C-indices (0.84). But the 5-covariate model of Mayo clinic Risk score had a higher AUC when compared to cox boost, random survival forest, support vector machine or even Deep hit. Conclusion: Machine learning methods offered limited improvement over the Mayo Clinical risk score except for the Multitask logistic regression model in predicting PBC survival outcome at 1-year (Table). Table 1. - Performance of Risk Scores from the 5-Covariate Model in comparison to the machine learning and deep learning models at 1-year S.No. Model specification AUC 95% CI Harrell’s C-statistic 1. 5-covariate model of Mayo clinic Risk score 0.88 0.85-0.91 0.79 2. Cox boost 0.87 0.84-0.90 0.84 3. Deep hit 0.70 0.66-0.74 0.30 4. Multitask logistic regression 1.00 1-1 0.90 5. Random survival forest 0.87 0.84-0.90 0.84 6. Support Vector Machine 0.56 0.52-0.61 0.43

On Huberized calibration regression for censored medical cost data.

Authors have observed that the distribution of medical expenditures has features that do not lend it to parametric modeling and can present significant challenges for least-squares-type estimators, even on a logarithmic scale. In this note, we discuss caveats and extensions of coefficient estimation in the bivariate accelerated lifetime model of medical cost and survival time on covariates. We consider the setting where medical cost is observed only when the event occurs and potential right-censoring of the event time induces a dependent censoring mechanism on cost. We adopt Huang's (2002) estimation framework using the weighted log-rank estimating equations and investigate his proposal for robust mark-scale coefficient estimation. Due to modeling restrictions on the joint distribution of survival time and cost, we conclude that his robust mark-scale coefficient estimator would benefit from a time-scale adjustment. We use basic principles from robust estimation to define a new weighted marked process that subsequently leads to a new time-corrected robust regression calibration estimator. Our simulation studies illustrate how the proposed estimator has desirable operating characteristics, including reduced sensitivity to extreme values in the cost distribution, smaller finite sample bias and variance than earlier proposals. We illustrate the method in an analysis of lifetime medical cost data from a lung cancer study conducted by the Southwest Oncology Group.

결측이 있는 이산형 공변량에 대한 Cox비례위험모형의 패턴-혼합 모델

공변량에 결측이 발생한 Cox 비례위험 모형을 적합할 때, 결측이 발생하는 개체를 모두 제거한 후 분석을 실시한다면 정보 손실에 의해 비효율적이고 결측의 발생 메커니즘이 완전 임의 결측(missing completely at random; MCAR)이 아니라면 모수의 추정값에 편향이 발생할 수 있다. Cox 비례위험 회귀모형의 공변량에 결측이 있는 경우 적용할 수 있는 여러 가지 방법들이 제안되어져 왔으나 이 분석들은 선택모델(selection model)에 기반하고 있다. 본 연구에서는 Little (1993)이 제안한 패턴-혼합 모델(pattern-mixture model)을 사용하여 Cox 비례위험 회귀모형에서 생존시간과 결측 메커니즘의 결합분포를 모델화 하고, 여러 가지 제약에 근거한 생존 분석의 결과를 비교하였다. 모의실험을 통해서 패턴-혼합 모델의 제약(restrictions)에 따른 모수 추정의 민감도를 확인하였고 결측을 무시한 채 분석한 결과 및 선택모형에 근거한 분석결과와 비교하였다. 패턴-혼합 모델의 제약에 따라 공변량의 결측으로 인한 모수 추정의 민감성 정도를 쥐백혈병 자료 예제를 통해 설명하였다. When fitting a Cox proportional hazards model with missing covariates, it is inefficient to exclude observations with missing values in the analysis. Furthermore, if the missing-data mechanism is not Missing Completely At Random(MCAR), it may lead to biased parameter estimation. Many approaches have been suggested to handle the Cox proportional hazards model when covariates are sometimes missing, but they are based on the selection model. This paper suggest an approach to handle Cox proportional hazards model with missing covariates by using the pattern-mixture model (Little, 1993). The pattern-mixture model is expressed by the joint distribution of survival time and the missing-data mechanism. In the pattern-mixture model, many models can be considered by setting up various restrictions, and different results under various restrictions indicate the sensitivity of the model due to missing covariates. A simulation study was conducted to show the sensitivity of parameter estimation under different restrictions in a pattern-mixture model. The proposed approach was also applied to mouse leukemia data.

Modelling the joint distribution of competing risks survival times using copula functions

The problem of modelling the joint distribution of survival times in a competing risks model, using copula functions, is considered. In order to evaluate this joint distribution and the related overall survival function, a system of non-linear differential equations is solved, which relates the crude and net survival functions of the modelled competing risks, through the copula. A similar approach to modelling dependent multiple decrements was applied by Carriere [Carriere, J., 1994. Dependent decrement theory. Transactions, Society of Actuaries XLVI, 45–65] who used a Gaussian copula applied to an incomplete double-decrement model which makes it difficult to calculate any actuarial functions and draw relevant conclusions. Here, we extend this methodology by studying the effect of complete and partial elimination of up to four competing risks on the overall survival function, the life expectancy and life annuity values. We further investigate how different choices of the copula function affect the resulting joint distribution of survival times and in particular the actuarial functions which are of importance in pricing life insurance and annuity products. For illustrative purposes, we have used a real data set and used extrapolation to prepare a complete multiple-decrement model up to age 120. Extensive numerical results illustrate the sensitivity of the model with respect to the choice of copula and its parameter(s).

On Default Correlation

This article studies the problem of default correlation. It introduces a random variable called “time-until-default” to denote the survival time of each defaultable entity or financial instrument, and defines the default correlation between two credit risks as the correlation coefficient between their survival times. The author explains why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. He shows that the current approach to default correlation through asset correlation is equivalent to using a normal copula function. Numerical examples illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.

On Default Correlation: A Copula Function Approach

This paper studies the problem of default correlation. We first introduce a random variable called time-until-default to denote the survival time of each defaultable entity or financial instrument, and define the default correlation between two credit risks as the correlation coefficient between their survival times. Then we argue why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. The definition and some basic properties of copula functions are given. We show that the current CreditMetrics approach to default correlation through asset correlation is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.

Nonparametric estimation of the joint distribution of survival time and mark variables

SUMMARY In many applications, variables of interest are marks of the endpoint which are not observed when the survival time is censored. This paper focuses on nonparametric estimation of the joint distribution and summaries of survival time and mark variables. We establish a representation of the joint distribution function through the cumulative markspecific hazard function, which is analogous to the product integral representation of univariate survival function. We identify a basic data structure common to various applications, propose nonparametric estimators and show that they maximise the likelihood. We formulate the problem in the marked point process framework and study both finite and large-sample properties of the estimators. We show that the joint distribution function estimator is nearly unbiased, uniformly strongly consistent and asymptotically normal. We also derive asymptotic variances for the estimators and propose sample-based variance estimates. Numerical studies demonstrate that both the estimators and their variance estimates perform well for practical sample sizes. We outline an application strategy.

Multilevel models for longitudinal variables prognostic for survival.

The issue of modelling the joint distribution of survival time and of prognostic variables measured periodically has recently become of interest in the AIDS literature but is of relevance in other applications. The focus of this paper is on clinical trials where follow-up measurements of potentially prognostic variables are often collected but not routinely used. These measurements can be used to study the biological evolution of the disease of interest; in particular the effect of an active treatment can be examined by comparing the time profiles of patients in the active and placebo group. It is proposed to use multilevel regression analysis to model the individual repeated observations as function of time and, possibly, treatment. To address the problem of informative drop-out--which may arise if deaths (or any other censoring events) are related to the unobserved values of the prognostic variables--we analyse sequentially overlapping portions of the follow-up information. An example arising from a randomized clinical trial for the treatment of primary biliary cirrhosis is examined in detail.