## F-tests in partially balanced and unbalanced mixed linear models

This dissertation considers two approaches for testing hypotheses in
unbalanced mixed linear models. The first approach is to construct a design with
some type of structure or "partial" balance, so that some of the optimal properties of
a completely balanced design hold. It is shown that for a particular type of partially
balanced design certain hypothesis tests are optimal. The second approach is to
study how the unbalancedness of a design affects a hypothesis test in terms of level
and power. Measures of imbalance are introduced and simulation results are
presented that demonstrate the relationship of the level and power of a test and the
measures.
The first part of this thesis focuses on error orthogonal designs which are a
type of partially balanced design. It is shown that with an error orthogonal design
and under certain additional conditions, ANOVA F-tests about certain linear
combinations of the variance components and certain linear combinations of the
fixed effects are uniformly most powerful (UMP) similar and UMP unbiased. The
ANOVA F-tests for the variance components are also invariant, so that the tests are
also UMP invariant similar and UMP invariant unbiased. For certain simultaneous
hypotheses about linear combinations of the fixed effects, the ANOVA F-tests are
UMP invariant unbiased.
The second part of this thesis considers a mixed model with a random
nested effect, and studies the effects of an unbalanced design on the level and
power of a hypothesis test of the nested variance component being equal to zero.
Measures of imbalance are introduced for each of the four conditions necessary to
obtain an exact test. Simulations are done for two different models to determine if
there is a relationship between any of the measures and the level and power for both
a naive test and a test using Satterthwaite's approximation. It is found that a
measure based on the coefficients of the expected mean squares is indicative of
how a test is performing. This measure is also simple to compute, so that it can
easily be employed to determine the validity of the expected level and power. / Graduation date: 1999

 Identifer oai:union.ndltd.org:ORGSU/oai:ir.library.oregonstate.edu:1957/33274 Date 11 February 1999 Creators Utlaut, Theresa L. Contributors Birkes, David S. Source Sets Oregon State University Language en_US Detected Language English Type Thesis/Dissertation

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