Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure

Normality and variance homogeneity assumptions are usually the main concern of parametric procedures such as in testing the equality of central tendency measures. Violation of these assumptions can seriously inflate the Type I error rates, which will cause spurious rejection of null hypotheses. Parametric procedures such as ANOVA and t-test rely heavily on the assumptions which are hardly encountered in real data. Alternatively, nonparametric procedures do not rely on the distribution of the data, but the procedures are less powerful. In order to overcome the aforementioned issues, robust procedures are recommended. S₁ statistic is one of the robust procedures which uses median as the location parameter to test the equality of central tendency measures among groups, and it deals with the original data without having to trim or transform the data to attain normality. Previous works on S₁ showed lack of robustness in some of the conditions under balanced design. Hence, the objective of this study is to improve the original S₁ statistic by substituting median with Hodges-Lehmann estimator. The substitution was also done on the scale estimator using the variance of Hodges-Lehmann as well as several robust scale estimators. To examine the strengths and weaknesses of the proposed procedures, some variables like types of distributions, number of groups, balanced and unbalanced group sizes, equal and unequal variances, and the nature of pairings were manipulated. The findings show that all proposed procedures are robust across all conditions for every group case. Besides, three proposed procedures namely S₁(MADn), S₁(Tn) and S₁(Sn) show better performance than the original S₁ procedure under extremely skewed distribution. Overall, the proposed procedures illustrate the ability in controlling the inflation of Type I error. Hence, the objective of this study has been achieved as the three proposed procedures show improvement in robustness under skewed distributions.