Empirical likelihood methods for comparison of survival functions

The use of empirical likelihood in survival analysis was initiated by Thomas and Grunkemeier (1975) who derived pointwise confidence intervals for the survival function. Since the breakthrough work of Owen (1988, 1990) the method has been applied to a variety of statistical problems. The goal of our research is to develop the approach for the comparison of survival functions for k-sample problems in survival analysis. We derive an empirical likelihood simultaneous confidence band for the ratio of two survival functions based on independent right-censored data. Earlier authors have studied such bands for the difference of two survival functions, but the ratio provides a more appropriate comparison in some applications, e.g., in comparing two treatments in biomedical settings. Our approach also works for the difference of two cumulative hazard functions. A test for equality of corresponding hazard functions is also constructed, and consistency against any fixed alternative is established. We develop a Monte Carlo simulation method to approximate the null distribution of the test statistic. Cumulative hazard ratios appear to be more tractable than ratios of survival functions or differences of cumulative hazard functions in the k-sample setting. However, the band for the ratio of survival functions is more stable and narrower than the band for the ratio of cumulative hazard functions. A goodness-of-fit test is developed for checking proportional hazards in k-sample problems. For the comparison of two distributions in the random censorship model (independent competing risks model without censoring), we construct empirical likelihood confidence bands for the ratio of the two cumulative hazards and the ratio of two survival functions. Goodness-of-fit tests for the Koziol-Green model and the equality of the corresponding hazard functions are also developed.; We extend our approach to adjust for covariate effects. All the corresponding results are established under quite general conditions. The proposed methods are illustrated with a real data from a Mayo Clinic trial involving a treatment for primary biliary cirrhosis (PBC) of the liver.