Simultaneous group sequential analysis of rank-based and weighted Kaplan-Meier tests for paired censored survival data

This research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log-rank (PWLR) and standardized paired weighted Kaplan-Meier (PWKM) tests. During a trial, these tests may alternately assume the role of the more extreme statistic. By monitoring PEMAX, the maximum between the absolute values of the standardized PWLR and PWKM, one combines advantages of rank-based (RB) and non-RB paired testing paradigms. Simulations show that monitoring treatment differences using PEMAX maintains type I error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards (PH) as well as non-PH situations. Hence, PEMAX preserves power more robustly than individually monitored PWLR and PWKM, while maintaining a reasonably simple approach to design and analysis of results. An example from the Early Treatment Diabetic Retinopathy Study (ETDRS) is given.     

Symmetric tests and confidence intervals for survival probabilities and quantiles of censored survival data

We describe existing tests and introduce two new tests concerning the value of a survival function. These tests may be used to construct a confidence interval for the survival probability at a given time or for a quantile of the survival distribution. Simulation studies show that error rates can differ substantially from their nominal values, particularly at survival probabilities close to zero or one. We recommend our new constrained bootstrap test for its good overall performance.     

A class of permutation tests for stratified survival data

We propose a class of permutation tests for stratified survival data. The tests are derived using the framework of Fay and Shih (1998, Journal of the American Statistical Association 93, 387-396), which creates tests by permuting scores based on a functional of estimated distribution functions. Here the estimated distribution function for each possibly right-, left-, or interval-censored observation is based on a shrinkage estimator similar to the nonparametric empirical estimator of Ghosh, Lahiri, and Tiwari (1989, Communications in Statistics--Theory and Methods 18, 121-146), and permutation is carried out within strata. The proposed test with a weighted Mann-Whitney functional is similar to the permutation form of the stratified log-rank test when there is a large strata effect or the sample size in each stratum is large and is similar to the permutation form of the ordinary log-rank test when there is little strata effect. Thus, the proposed test unifies the advantages of both the stratified and ordinary log-rank tests. By changing the functional, we may obtain a stratified Prentice-Wilcoxon test or a difference in means test with similar unifying properties. We show through simulations the advantage of the proposed test over existing tests for uncensored and right-censored data.     

Relative risk trees for censored survival data.

A method is developed for obtaining tree-structured relative risk estimates for censored survival data. The first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used Classification and Regression Tree (CART) algorithm of Breiman et al. (1984, Classification and Regression Trees, Belmont, California: Wadsworth). The performance of the technique is investigated through stimulation and compared to the tree-structured survival methods proposed by Davis and Anderson (1989, Statistics in Medicine 8, 947-961) and Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147-160).     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.