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1	F-tests in partially balanced and unbalanced mixed linear models Utlaut, Theresa L. 11 February 1999 (has links) 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 F-distribution Linear models (Statistics)
2	Introduction to Probability Theory Chen, Yong-Yuan 25 May 2010 (has links) In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process. Chebyshev¡¦s inequality central limit theorem Poisson process Markov¡¦s inequality multivariate normal distribution multinomial distribution F distribution t distribution chi-square distribution lognormal distribution Beta distribution Cauchy distribution Weibull distribution gamma distribution exponential distribution normal distribution discrete uniform distribution uniform distribution hypergeometric distribution geometric distribution negative binomial distribution Poisson distribution binomial distribution Bernoulli distribution Bayes theorem theorem of total probability axioms of probability multinomial theorem inclusion-exclusion principle set

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F-tests in partially balanced and unbalanced mixed linear models

Introduction to Probability Theory