Concordance Probability and Discriminatory Power in Proportional Hazards Regression

The concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models. We derive an analytic expression for the concordance probability in the Cox proportional hazards model. The proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's c-index based on informative pairs. The asymptotic distribution of the concordance probability estimate is derived using U-statistic theory and the methodology is applied to a predictive model in lung cancer.

Tests for Comparing Mark-Specific Hazards and Cumulative Incidence Functions

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson--Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free.Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.

Additive Hazards Models with Latent Treatment Effectiveness Lag Time

In many clinical trials to evaluate treatment efficacy, it is believed that there may exist latent treatment effectiveness lag times after which medical procedure or chemical compound would be in full effect. In this article, semiparametric regression models are proposed and studied to estimate the treatment effect accounting for such latent lag times. The new models take advantage of the invariance property of the additive hazards model in marginalizing over random effects, so parameters in the models are easy to be estimated and interpreted, while the flexibility without specifying baseline hazard function is kept. Monte Carlo simulation studies demonstrate the appropriateness of the proposed semiparametric estimation procedure. Data collected in the actual randomized clinical trial, which evaluates the effectiveness of biodegradable carmustine polymers for treatment of recurrent brain tumors, are analyzed.