Winsorized Modified One Step M-estimator in Alexander-Govern Test

Tobi Kingsley Ochuko,Sharipah Soaad Syed Yahaya,Zakiyah Zain ,Shahrum Abdullah

doi:10.5539/mas.v9n11p51

Abstract

This research centres on independent group test of comparing two or more means by using the parametric method, namely the Alexander-Govern test. The Alexander-Govern (AG) test uses mean as a measure of its central tendency. It is a better alternative to the Welch test, James test and the ANOVA, because it has a good control of Type I error rates and produces a high power efficient for a normal data under variance heterogeneity, but not for non-normal data. As a result, trimmed mean was applied on the test under non-normal data for two group condition, but as the number of groups increased above two, the test fails to be robust. Due to this, when the MOM estimator was applied on the test, it was not influenced by the number of groups, but failed to give a good control of Type I error rates under skewed heavy tailed distribution. In this research, the Winsorized MOM estimator was applied in AG test as a measure of its central tendency. 5,000 data sets were simulated and analysed using Statistical Analysis Software (SAS). The result shows that with the pairing of unbalanced sample size with unequal variance of (1:36) and the combination of both balanced and unbalanced sample sizes with both equal and unequal variances, under six group condition, for g = 0.5 and h = 0.5, for both positive and negative pairing condition, the test gives a remarkable control of Type I error rates. In overall, the AGWMOM test has the best control of Type I error rates, across the distributions and across the groups, compared to the AG test, the AGMOM test and the ANOVA.

Highlights

This research focuses on comparing the performance of the Type I error rates and power of the AG test, the AGMOM test, the AGWMOM test, the t-test and the analysis of variance (ANOVA), for two, four and six group conditions
The t-test have two of its Type I error rates fall within the stringent criteria of robustness and only one of its Type I error rates fall within the lenient criteria of robustness, with the combination of balanced sample size with equal variance and the pairing of unbalanced sample size with both equal and unequal variance
While the AGWMOM test have two of its Type I error rates fall within the stringent criteria of robustness and the remaining three of its Type I error rates fall within the lenient criteria of robustness

Summary

Introduction

This research focuses on comparing the performance of the Type I error rates and power of the AG test, the AGMOM test, the AGWMOM test, the t-test and the ANOVA, for two, four and six group conditions. The trimmed mean and Winsorized variance are widely used as alternative to the common mean and variance respectively, due to some good properties, such as having a remarkable control over Type I error rates and the power of the test, when there is a violation under the assumptions of homogeneity of the variance and when the distribution is normal (Wilcox, 1995). The Winsorized MOM was applied in Alexander–Govern test under variance heterogeneity for non-normal data, under a skewed heavy tailed distribution, and it gave the test a remarkable control of Type I error rates and produced a high power efficient for test

The Alexander-Govern Test

The Alexander-Govern Test Statistic

The Modified Alexander-Govern Test

The Variables Used in this Research

The Research Design

Discussion and Conclusion