It is well known that when the input variables of the linear regression model are subject to noise contamination, the model parameters can not be estimated uniquely. This, in the statistical literature, is referred to as the identifiability problem of the errors-in-variables models. Further, in linear regression there is an explicit assumption of the existence of a single linear relationship. The statistical properties of the errors-in-variables models under the assumption that the noise variances are either known or that they can be estimated are well documented. In many situations, however, such information is neither available nor obtainable. Although under such circumstances one can not obtain a unique vector of parameters, the space, Ω, of the feasible solutions can be computed. Additionally, assumption of existence of a single linear relationship may be presumptuous as well. A multi-equation model similar to the simultaneous-equations models of econometrics may be more appropriate. The goals of this dissertation are the following: (1) To present analytical techniques or algorithms to reduce the solution space, Ω, when any type of prior information, exact or relative, is available; (2) The data covariance matrix, Σ, can be examined to determine whether or not Ω is bounded. If Ω is not bounded a multi-equation model is more appropriate. The methodology for identifying the subsets of variables within which linear relations can feasibly exist is presented; (3) Ridge regression technique is commonly employed in order to reduce the ills caused by collinearity. This is achieved by perturbing the diagonal elements of Σ. In certain situations, applying ridge regression causes some of the coefficients to change signs. An analytical technique is presented to measure the amount of perturbation required to render such variables ineffective. This information can assist the analyst in variable selection as well as deciding on the appropriate model; (4) For the situations when Ω is bounded, a new weighted regression technique based on the computed upper bounds on the noise variances is presented. This technique will result in identification of a unique estimate of the model parameters.

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No description available.

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

Graduation date: 1994

The following study utilizes data from the High School and Beyond Study in order to predict mathematics achievement using both student characteristics and school level characteristics. Utilizing Hierarchical Linear Modeling, this study extends the body of literature by exploring how race, socio-economic status, and gender, as well as the percentage of minority students in a school, whether or not the school is Catholic, the proportion of students in the academic track, and the mean socioeconomic status of the school all affect mathematics achievement. Through this methodology, it was possible to see the direct effects of both student level and school level variables on achievement, as well as the cross-level interaction of all of these variables. Findings suggest that there are discrepancies in how different types of students achieve, as well as how those students achieve in varying contexts. Many of the variables were statistically significant in their effect on mathematics achievement. Implications for this research are discussed and considerations for future research are presented.

Data in the form of counts or proportions often exhibit more variability than that predicted by a Poisson or binomial distribution. Many different models have been proposed to account for extra-Poisson or extra-binomial variation. A simple model includes a single heterogeneity factor (dispersion parameter) in the variance. Other models that allow the dispersion parameter to vary between groups or according to a continuous covariate also exist but require a more complicated analysis. This thesis is concerned with (1) understanding the consequences of using an oversimplified model for overdispersion, (2) presenting diagnostic tools for detecting the dependence of overdispersion on covariates in regression settings for counts and proportions and (3) presenting diagnostic tools for distinguishing between some commonly used models for overdispersed data. The double exponential family of distributions is used as a foundation for this work. A double binomial or double Poisson density is constructed from a binomial or Poisson density and an additional dispersion parameter. This provides a completely parametric framework for modeling overdispersed counts and proportions. The first issue above is addressed by exploring the properties of maximum likelihood estimates obtained from incorrectly specified likelihoods. The diagnostic tools are based on a score test in the double exponential family. An attractive feature of this test is that it can be computed from the components of the deviance in the standard generalized linear model fit. A graphical display is suggested by the score test. For the normal linear model, which is a special case of the double exponential family, the diagnostics reduce to those for heteroscedasticity presented by Cook and Weisberg (1983). / Graduation date: 1990

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Statistics and Biostatistics." Includes bibliographical references (p. 59-61).

Project (M.S.)--Brigham Young University. Dept. of Statistics, 2004. / Includes bibliographical references (p. 109-111).

Diss.--Univ. zu Köln, 2009. / Includes bibliographical references.