Analysis of asymptotic time complexity of an assumption-free alternative to the log-rank test

Comparison of two time-event survival curves representing two groups of individuals’ evolution in time is relatively usual in applied biostatistics. Although the log-rank test is the suggested tool how to face the above-mentioned problem, there is a rich statistical toolbox used to overcome some of the properties of the log-rank test. However, all of these methods are limited by relatively rigorous statistical assumptions.In this study, we introduce a new robust method for comparing two time-event survival curves. We briefly discuss selected issues of the robustness of the log-rank test and analyse a bit more some of the properties and mostly asymptotic time complexity of the proposed method. The new method models individual time-event survival curves in a discrete combinatorial way as orthogonal monotonic paths, which enables direct estimation of the p-value as it was originally defined. We also gently investigate how the surface of an area, bounded by two survival curves plotted onto a plane chart, is related to the test’s p-value. Finally, using simulated time-event data, we check the robustness of the introduced method in comparison with the log-rank test.Based on the theoretical analysis and simulations, the introduced method seems to be a promising and valid alternative to the log-rank test, particularly in case on how to compare two time-event curves regardless of any statistical assumptions.