abstract: It is common in the analysis of data to provide a goodness-of-fit test to assess the performance of a model. In the analysis of contingency tables, goodness-of-fit statistics are frequently employed when modeling social science, educational or psychological data where the interest is often directed at investigating the association among multi-categorical variables. Pearson's chi-squared statistic is well-known in goodness-of-fit testing, but it is sometimes considered to produce an omnibus test as it gives little guidance to the source of poor fit once the null hypothesis is rejected. However, its components can provide powerful directional tests. In this dissertation, orthogonal components are used to develop goodness-of-fit tests for models fit to the counts obtained from the cross-classification of multi-category dependent variables. Ordinal categories are assumed. Orthogonal components defined on marginals are obtained when analyzing multi-dimensional contingency tables through the use of the QR decomposition. A subset of these orthogonal components can be used to construct limited-information tests that allow one to identify the source of lack-of-fit and provide an increase in power compared to Pearson's test. These tests can address the adverse effects presented when data are sparse. The tests rely on the set of first- and second-order marginals jointly, the set of second-order marginals only, and the random forest method, a popular algorithm for modeling large complex data sets. The performance of these tests is compared to the likelihood ratio test as well as to tests based on orthogonal polynomial components. The derived goodness-of-fit tests are evaluated with studies for detecting two- and three-way associations that are not accounted for by a categorical variable factor model with a single latent variable. In addition the tests are used to investigate the case when the model misspecification involves parameter constraints for large and sparse contingency tables. The methodology proposed here is applied to data from the 38th round of the State Survey conducted by the Institute for Public Policy and Michigan State University Social Research (2005) . The results illustrate the use of the proposed techniques in the context of a sparse data set. / Dissertation/Thesis / Ph.D. Mathematics 2011

The focus of this research was to compare the student membership population proportions, by race and gender, of Brevard County Public School students with the actual participation in select choice programs offered to Brevard County public high school students. This study was based on an analysis of the scores of 1,152 eighth-grade students who received a score of 4 or 5 on the 2008 Florida Comprehensive Assessment Test (FCAT) mathematics and a score of 4 or 5 on the 2008 Florida Comprehensive Assessment Test (FCAT) reading and their participation in high school advanced academic courses. The advanced academic choice programs selected for this study consisted of the four Florida articulated accelerated college credit seeking programs: Advanced Placement (AP), Dual-Enrollment (DE), International Baccalaureate (IB) Diploma Programme, and the Cambridge Advanced International Certificate of Education (AICE). The proportion comparison consisted of student membership data and eighth-grade FCAT scores from 2007-2008 and the student membership data and high school course load data from the 2008-2009, 2009-2010, 2010-2011, and 2011-2012 academic school years. Chi-square goodness-of-fit tests were run to analyze the proportions by race and gender of the sample groups and student membership populations. For each respective year involved in this study, there was a statistically significant difference in the race and gender proportions of the samples and the student membership populations.