Testing and interval estimation for two-sample survival comparisons with small sample sizes and unequal censoring

Abstract

While the commonly used log-rank test for survival times between 2 groups enjoys many desirable properties, sometimes the log-rank test and its related linear rank tests perform poorly when sample sizes are small. Similar concerns apply to interval estimates for treatment differences in this setting, though their properties are less well known. Standard permutation tests are one option, but these are not in general valid when the underlying censoring distributions in the comparison groups are unequal. We develop 2 methods for testing and interval estimation, for use with small samples and possibly unequal censoring, based on first imputing survival and censoring times and then applying permutation methods. One provides a heuristic justification for the approach proposed recently by Heinze and others (2003, Exact log-rank tests for unequal follow-up. Biometrics 59, 1151-1157). Simulation studies show that the proposed methods have good Type I error and power properties. For accelerated failure time models, compared to the asymptotic methods of Jin and others (2003, Rank-based inference for the accelerated failure time model. Biometrika 90, 341-353), the proposed methods yield confidence intervals with better coverage probabilities in small-sample settings and similar efficiency when sample sizes are large. The proposed methods are illustrated with data from a cancer study and an AIDS clinical trial.

Fig. 1.

Time to virologic failure. (a): Kaplan-Meier estimates for time to virologic failure, by treatment group; (b): the average of 10 p-values for testing $H_0:\beta ={\beta }_0$ for various ${\beta }_0\in (0,1.5)$. Solid: IP_T; Dashed: IP_Z; Dotdashed: a horizontal line at 0.05.

Fig. 2.

Time to virologic failure or death. (a): Kaplan-Meier estimates for time to virologic failure or death, by treatment group; (b): the average of 10 p-values for testing $H_0:\beta ={\beta }_0$ for various ${\beta }_0\in (0,1.5)$. Solid: IP_T; Dashed: IP_Z; Dotdashed: a horizontal line at 0.05.