In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions that 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 or an estimated p-value.
","abstract_html":"<p>In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions that 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 or an estimated p-value.</p>","abstract_has_math":false,"creators":["Yaacoub, Tony"],"institution":null,"degree_name":"Master of Science in Mathematics (M.S.)","degree_level":"Thesis (open access)","degree_discipline":"Department of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Hani Samawi","Broderick O. Oluyede"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-01-01T08:00:00Z","date_published":"2014-01-01T08:00:00Z","updated_at":"2026-07-24T02:27:52Z","subjects":["ETD","ANOVA","fixed effects one-way design","likelihood ratio test","orthogonal designs","statistically designed experiments","Design of Experiments and Sample Surveys","Statistical Theory"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd/1079","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hani Samawi","Broderick O. Oluyede"]},{"key":"dc:creator","label":"Author","values":["Yaacoub, Tony"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2014-04-12T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (open access)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics (M.S.)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","ANOVA","fixed effects one-way design","likelihood ratio test","orthogonal designs","statistically designed experiments","Design of Experiments and Sample Surveys","Statistical Theory"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd/1079"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions that 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 or an estimated p-value.
"]},{"key":"dc:title","label":"Title","values":["Comparing k Population Means with No Assumption about the Variances"]}]}],"canonical_facts":{"dc:contributor":["Hani Samawi","Broderick O. Oluyede"],"dc:creator":["Yaacoub, Tony"],"dc:date.available":["2014-04-12T07:00:00Z"],"dc:description.abstract":["In the analysis of most statistically designed experiments, it is common to assume equal variances along with the assumptions that 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 or an estimated p-value.
"],"dc:identifier":["https://digitalcommons.georgiasouthern.edu/etd/1079"],"dc:subject":["ETD","ANOVA","fixed effects one-way design","likelihood ratio test","orthogonal designs","statistically designed experiments","Design of Experiments and Sample Surveys","Statistical Theory"],"dc:title":["Comparing k Population Means with No Assumption about the Variances"],"thesis:degree_discipline":["Department of Mathematical Sciences"],"thesis:degree_level":["Thesis (open access)"],"thesis:degree_name":["Master of Science in Mathematics (M.S.)"]},"updated_at":"2026-07-24T02:27:52Z"}