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## 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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