A Class of Goodness of Fit Tests for a Copula Based on Bivariate Right-Censored Data

An information ratio-based goodness-of-fit test for copula models on censored data

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right-censored data. No formal goodness-of-fit (GOF) test exists for interval-censored or recurrent events data. We develop a general GOF test for copula-based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D-vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type-I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of model fitness.

Cumulative risk regression in case-cohort studies using pseudo-observations

Case-cohort studies are useful when information on certain risk factors is difficult or costly to ascertain. Particularly, a case-cohort study may be well suited in situations where several case series are of interest, e.g. in studies with competing risks, because the same sub-cohort may serve as a comparison group for all case series. Previous analyses of this kind of sampled cohort data most often involved estimation of rate ratios based on a Cox regression model. However, with competing risks this method will not provide parameters that directly describe the association between covariates and cumulative risks. In this paper, we study regression analysis of cause-specific cumulative risks in case-cohort studies using pseudo-observations. We focus mainly on the situation with competing risks. However, as a by-product, we also develop a method by which absolute mortality risks may be analyzed directly from case-cohort survival data. We adjust for the case-cohort sampling by inverse sampling probabilities applied to a generalized estimation equation. The large-sample properties of the proposed estimator are developed and small-sample properties are evaluated in a simulation study. We apply the methodology to study the effect of a specific diet component and a specific gene on the absolute risk of atrial fibrillation.

A novel mathematical model to predict prognosis of burnt patients based on logistic regression and support vector machine

OBJECTIVE

To develop a mathematical model of predicting mortality based on the admission characteristics of 6220 burn cases.

METHODS

Data on all the burn patients presenting to Institute of Burn Research, Southwest Hospital, Third Military Medical University from January of 1999 to December of 2008 were extracted from the departmental registry. The distributions of burn cases were scattered by principal component analysis. Univariate associations with mortality were identified and independent associations were derived from multivariate logistic regression analysis. Using variables independently and significantly associated with mortality, a mathematical model to predict mortality was developed using the support vector machine (SVM) model. The predicting ability of this model was evaluated and verified.

RESULTS

The overall mortality in this study was 1.8%. Univariate associations with mortality were identified and independent associations were derived from multivariate logistic regression analysis. Variables at admission independently associated with mortality were gender, age, total burn area, full thickness burn area, inhalation injury, shock, period before admission and others. The sensitivity and specificity of logistic model were 99.75% and 85.84% respectively, with an area under the receiver operating curve of 0.989 (95% CI: 0.979-1.000; p<0.01). The model correctly classified 99.50% of cases. The subsequently developed support vector machine (SVM) model correctly classified nearly 100% of test cases, which could not only predict adult group but also pediatric group, with pretty high robustness (92%-100%).

CONCLUSION

A mathematical model based on logistic regression and SVM could be used to predict the survival prognosis according to the admission characteristics.

Estimation of time-dependent association for bivariate failure times in the presence of a competing risk

This article targets the estimation of a time-dependent association measure for bivariate failure times, the conditional cause-specific hazards ratio (CCSHR), which is a generalization of the conditional hazards ratio (CHR) to accommodate competing risks data. We model the CCSHR as a parametric regression function of time and event causes and leave all other aspects of the joint distribution of the failure times unspecified. We develop a pseudo-likelihood estimation procedure for model fitting and inference and establish the asymptotic properties of the estimators. We assess the finite-sample properties of the proposed estimators against the estimators obtained from a moment-based estimating equation approach. Data from the Cache County study on dementia are used to illustrate the proposed methodology.

Likelihood ratio procedures and tests of fit in parametric and semiparametric copula models with censored data

Copula models for multivariate lifetimes have become widely used in areas such as biomedicine, finance and insurance. This paper fills some gaps in existing methodology for copula parameters and model assessment. We consider procedures based on likelihood and pseudolikelihood ratio statistics and introduce semiparametric maximum likelihood estimation leading to semiparametric versions. For cases where standard asymptotic approximations do not hold, we propose an efficient simulation technique for obtaining p-values. We apply these methods to tests for a copula model, based on embedding it in a larger copula family. It is shown that the likelihood and pseudolikelihood ratio tests are consistent even when the expanded copula model is misspecified. Power comparisons with two other tests of fit indicate that model expansion provides a convenient, powerful and robust approach. The methods are illustrated on an application concerning the time to loss of vision in the two eyes of an individual.

Regression survival analysis with an assumed copula for dependent censoring: a sensitivity analysis approach

SUMMARY

In clinical studies, when censoring is caused by competing risks or patient withdrawal, there is always a concern about the validity of treatment effect estimates that are obtained under the assumption of independent censoring. Because dependent censoring is nonidentifiable without additional information, the best we can do is a sensitivity analysis to assess the changes of parameter estimates under different assumptions about the association between failure and censoring. This analysis is especially useful when knowledge about such association is available through literature review or expert opinions. In a regression analysis setting, the consequences of falsely assuming independent censoring on parameter estimates are not clear. Neither the direction nor the magnitude of the potential bias can be easily predicted. We provide an approach to do sensitivity analysis for the widely used Cox proportional hazards models. The joint distribution of the failure and censoring times is assumed to be a function of their marginal distributions. This function is called a copula. Under this assumption, we propose an iteration algorithm to estimate the regression parameters and marginal survival functions. Simulation studies show that this algorithm works well. We apply the proposed sensitivity analysis approach to the data from an AIDS clinical trial in which 27% of the patients withdrew due to toxicity or at the request of the patient or investigator.