## Mathematics Ph.D. Dissertations

#### Title

Transformed Tests for Homogeneity of Variances and Means

#### Date of Award

2006

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics/Mathematical Statistics

#### First Advisor

Hanfeng Chen (Advisor)

#### Abstract

The analysis of variance (ANOVA) is one of the most important and useful techniques for variety of fields such as agriculture, sociology and medicine for comparing different groups or treatments with respect to their means. A set of assumptions such as normal error distribution, homogeneity of variances and independence of observations, has to be made to employ an F test for equality of the treatment means. It is now well established that the violation of the assumption of homogeneity of variances can have severe effects on the inference of the population means, especially in the case of unequal sample sizes. In fact, the conventional ANOVA F provides generally poor control over both Type I and Type II error rates under a wide variety of variance heterogeneity conditions. Therefore, the problem of homogeneity of variances has to be settled before conducting an ANOVA. While a good number of tests are available for testing homogeneity of variances, Bartlett test and four versions of Levene tests are still popular for testing homogeneity of variances in the case of one-way ANOVA setting. It is evident that the Bartlett test is not as robust as Levene tests against the violation of the normality assumption. On the other hand, Levene tests are less powerful than the Bartlett test. In this dissertation, we proposed a transformed version of Bartlett test where the transformation is intended to achieve the normality of the data and independence of the observations to some extent. It is evident from the simulation that the transformed Bartlett test is more robust than the untransformed Bartlett test against the violation of the normality assumption. It also follows that the transformed Bartlett test is a balance between the Bartlett test and the four versions of the Levene tests in terms of Type I error rate and power concern. While, the estimation of location parameter is of concern, a modified version of trimmed mean has been proposed as an alternative to trimmed mean when the distribution is skewed or contains outliers. It is evident from the simulation study that modified trimmed mean outperform trimmed mean in terms of coverage probability of interval estimate. It is also evident that the test based on the modified version of trimmed mean is suitable for estimating Type I error rate and is more powerful than the tests based on the mean and the trimmed mean in the presence of the outliers. Finally, an alternative method of estimating transformation parameter has been proposed by using the chi-squared goodness of fit criteria which could be useful for testing equality of two population means.

#### Recommended Citation

Islam, Md. Khairul, "Transformed Tests for Homogeneity of Variances and Means" (2006). *Mathematics Ph.D. Dissertations*. 55.

https://scholarworks.bgsu.edu/math_diss/55