# Comparing k Population Means with No Assumption about the Equality of the Population Variances

## Location

Room 2904 B

## Session Format

Paper Presentation

## Research Area Topic:

Natural & Physical Sciences - Mathematics

## Abstract

In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions the sample measurements are independent and normally distributed. Under these three assumptions, a likelihood ratio test is used to test for the difference in population means. Typically, the assumption of independence can be justified based on the sampling method used by the researcher. The likelihood ratio test is robust to the assumption of normality. However, the equality of variances is often difficult to justify. It has been found that the assumption of equal variances cannot be made even after transforming the data. Our interest is to develop a method for comparing k population means assuming the data are independent and normally distributed but without assuming equal variances. This is the Behrens-Fisher problem for k=2. We propose a method that uses the exact distribution of the likelihood ratio (test) statistic. The data is used to estimate this exact distribution to obtain an estimated critical value and/or an estimated p-value.

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Presentation Type and Release Option

Presentation (Open Access)

## Start Date

4-16-2016 9:30 AM

## End Date

4-16-2016 10:30 AM

## Recommended Citation

Champ, Charles W., "Comparing k Population Means with No Assumption about the Equality of the Population Variances" (2016). *GS4 Georgia Southern Student Scholars Symposium*. 150.

https://digitalcommons.georgiasouthern.edu/research_symposium/2016/2016/150

Comparing k Population Means with No Assumption about the Equality of the Population Variances

Room 2904 B

In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions the sample measurements are independent and normally distributed. Under these three assumptions, a likelihood ratio test is used to test for the difference in population means. Typically, the assumption of independence can be justified based on the sampling method used by the researcher. The likelihood ratio test is robust to the assumption of normality. However, the equality of variances is often difficult to justify. It has been found that the assumption of equal variances cannot be made even after transforming the data. Our interest is to develop a method for comparing k population means assuming the data are independent and normally distributed but without assuming equal variances. This is the Behrens-Fisher problem for k=2. We propose a method that uses the exact distribution of the likelihood ratio (test) statistic. The data is used to estimate this exact distribution to obtain an estimated critical value and/or an estimated p-value.