S1 statistic with Hodges-Lehmann in two groups case for skewed distribution

Abstract

This study is focused on the modification of S1 statistic, a procedure for testing the equality of groups, by replacing median with Hodges-Lehmann estimator as the location measure in two groups case under skewed distribution. The modification is also extended to the default scale estimator of Hodges-Lehmann, S1(HL) and robust scale estimator, MADn, S1(MADn). The purpose of the modifications is to improve the robustness of the statistic. To test the strengths and weaknesses of S1(HL) and S1(MADn), a simulation study was conducted. Several variables such as the shape of distributions, balanced and unbalanced group sizes, equal and unequal variances and nature of pairings were manipulated to create various conditions for the data. Since the distribution of S1 statistic is unknown, bootstrap method was used for data generation. According to Guo and Luh, a test statistic is considered robust if its empirical error rate does not exceed 0.075 when α = 0.05. Refer to the results obtained, S1(HL) and S1(MADn) can be considered as robust due to all error rates fall below 0.075 for balanced and unbalanced design in each distribution. t-test and Mann-Whitney are the chosen methods for comparison purpose. For balanced design, t-test and Mann-Whitney shown better results compare to S1(HL) and S1(MADn). However, for unbalanced design especially in negative pairing, S1(MADn) is the most appropriate method to be used across three distributions.