### Abstract:

Testing main effects and interaction effects in factorial designs are basic content
in statistics textbooks and widely used in various fields. In balanced designs there
is general agreement on the appropriate main effect and interaction sums of
squares and these are typically displayed in an analysis of variance (ANOVA). A
number of methods for analyzing unbalanced designs have been developed, but in
general they do not lead to unique results. For example, in SAS one can get three
different main effect sums of squares in an unbalanced design. I, If these results
are viewed from the theory of the general linear model, then it is typically the case
that the different sums of squares all lead to F-tests, but they are testing different
linear hypotheses. In particular, if one clearly specifies the linear hypothesis being
tested, then linear model theory leads to one unique deviation sum of squares. One
exception to this statement is an ANOVA, called an unweighted means ANOVA
(UANOVA) introduced by Yates (1934). The deviation sum of squares in a
UNANOVA typically does not lead to an F-test and hence does not reduce to a
usual deviation sum of squares for some linear hypothesis.
The UANOVA tests have been suggested by several writers as an alternative to
the usual tests. Almost all of these results are limited to the one-way model or a
two-way model with interaction, and hence the UANOVA procedure is not
available for a general linear model. This thesis generalizes the UANOVA test
prescribed in the two-way model with interaction to a general linear model. In
addition, the properties of the test are investigated. It is found, similar to the usual
test, that computation of the UANOVA test statistic does not depend on how the
linear hypothesis is formulated. It is also shown that the numerator of the
UANOVA test is like a linear combination of independent chi-squared random
variables as opposed to a single chi-squared random variable in the usual test. In
addition we show how the Imhof (1961) paper can be used to determine critical
values, p-values and power for the UANOVA test. Comparisons with the usual
test are also included. It is found that neither test is more powerful than the other.
Even so, for most circumstances our general recommendation is that the usual test
is probably superior to the UANOVA test.