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

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.

The Implementation of the Alexander-Govern Test in Factorial Design Analysis

“…Note that this test is able to control the Type I error rate and has high power in the presence of variance heterogeneity for normal data. Nevertheless, it is not resilient to non-normal data (Ochuko et al, 2015c).…”

“…On the other hand, the AG test is used as the alternative to the ANOVA to deal with unequal variance (Abdullah et al, 2011;Jamaluddin, 2015;Ochuko et al, 2015cOchuko et al, , 2015a. The AG test provides excellent control of Type I error rates for variance heterogeneity in normal data.…”

“…Integration of the AG test with some robust estimators, such as trimmed mean, enhances its ability to better control the Type I error rate under non-normal data. It has been tested in several studies using simulation and some real-world data validation (Ochuko et al, 2015c).…”

“…Based on Bradley's criteria, as discussed previously, the acceptable control range of the Type I error rates is between 0.025 and 0.075. Note that values outside the range indicated the test failed to control the Type I error (Ochuko et al, 2015c).…”

“…In the case of unequal variance, also known as heteroscedasticity, the Welch t-test is the common option to the classical t-test. For the case of more than two groups, various tests were introduced, such as the Welch test, James's test, and the Alexander-Govern test (AG test) (Ochuko et al, 2015c). Alexander and Govern (1994) invented the AG test to test the equality of k-independent means in the case of variance heterogeneity (Abdullah et al, 2014).…”