## Doctoral Dissertations

#### Abstract

"Data analysts have long been concerned with the problem of giving a complete analysis of the two-way crossed classification design with one observation per cell when interaction is present in the data. Conventional tests of hypotheses and statistical inferences are not applicable in this case since an estimate of the error variance is not available. Transformation of the data to additive data has been one solution to this problem; however, interpretation of the physical units of the transformed data is difficult. Recent contributions to the analysis of this design include a test for interaction, a test for testing differences in treatments and/or blocks when interaction is present, and an estimate of the error variance when interaction is present. Each of these has been accomplished without any type of transformation on the data.

The tests and the estimate of the error variance mentioned above are functions of the characteristic roots of random matrices having certain Wishart distributions. In the case of the test for interaction, the Wishart distribution involved is, under the hypothesis of no interaction, a central distribution and critical points of the test statistic have previously been obtained. For the alternative hypothesis, the random matrix has a noncentral Wishart distribution. Likewise, the test statistic for testing for treatment differences and the error variance estimator are functions of the characteristic roots of a random matrix having a noncentral Wishart distribution.

Up to the present time, the fact that noncentral Wishart distributions are involved in the above mentioned analyses has presented difficult tasks in studying such problems as determining the power of the likelihood ratio test for interaction or determining some properties of the estimator of the error variance. A partial solution to these problems is presented in this thesis. A Monte Carlo technique for generating random matrices having any desired noncentral Wishart distribution is developed and from these matrices the characteristic roots can be obtained and the distributions of the functions of the characteristic roots can then be approximated. With these approximated distribution functions it has been possible to study the power of the likelihood ratio test for interaction, to determine some properties of the error variance estimator, including a discussion of one sided (upper) confidence intervals for the error variance, and to study some related problems.

An additional problem considered in this thesis is concerned with the test for interaction in the two-way crossed classification design with one observation per cell which was presented by Tukey (1949). The power of this test for a particular alternative hypothesis was discussed by Ghosh and Sharma (1963). In this thesis, the distribution of Tukey's test statistic will be given independent of the nature of the alternative hypothesis. From this, a comparison of the power of Tukey's test and the likelihood ratio test will be given for two different types of alternative hypotheses "--Introduction, page 1.

#### Advisor(s)

Johnson, Dallas E., 1938-

#### Committee Member(s)

Rakestraw, Roy M.

Bain, Lee J., 1939-

Gillett, Billy E.

Hume, Merril W.

#### Department(s)

Mathematics and Statistics

#### Degree Name

Ph. D. in Mathematics

#### Publisher

University of Missouri--Rolla

#### Publication Date

1974

#### Pagination

x, 137 pages

#### Note about bibliography

Includes bibliographical references (pages 134-136).

#### Rights

© 1974 Victor John Hegemann, All rights reserved.

#### Document Type

Dissertation - Open Access

#### File Type

text

#### Language

English

#### Subject Headings

Analysis of variance

Factor analysis

Statistical methods

#### Thesis Number

T 3017

#### Print OCLC #

6012266

#### Electronic OCLC #

912515542

#### Link to Catalog Record

#### Recommended Citation

Hegemann, Victor John, "On interaction in the two-way AOV model without replication including power comparisons of two tests for non-additivity" (1974). *Doctoral Dissertations*. 301.

https://scholarsmine.mst.edu/doctoral_dissertations/301