An incomplete Latin square is a v x v array with an empty n x n subarray with every row and every column containing each symbol at most once and no row or column with an empty cell containing one of the last n symbols. A set of t incomplete mutually orthogonal Latin squares of order v and hole size n is a set of t incomplete Latin squares (containing the same empty subarray on the same set of symbols) with a natural extension to the condition of orthogonality. The existence of such sets have been previously explored only for small values of t. We determine an asymptotic result for the existence of t incomplete mutually orthogonal Latin squares for general t requiring large holes, which we develop from our results on incomplete pairwise balanced designs and incomplete group divisible designs. / Graduate

In longitudinal studies, the exact timing of an event often cannot be observed, and is usually detected at a subsequent visit, which is called interval censoring. Spacing of the visits is important when designing study with interval censored data. In a typical longitudinal study, the spacing of visits is usually the same across all subjects (balanced design). In this dissertation, I propose an unbalanced design: subjects at baseline are divided into a high risk group and a low risk group based on a risk factor, and the subjects in the high risk group are followed more frequently than those in the low risk group. Using a simple setting of a single binary exposure of interest (covariate) and exponentially distributed survival times, I derive the explicit formula for the asymptotic sampling variance of the estimate for the covariate effect. It shows that the asymptotic sampling variance can be simply reduced by increasing the number of examinations in the high risk group. The relative reduction tends to be greater when the baseline hazard rate in the high risk group is much higher than that in the low risk group and tends to be larger when the frequency of assessments in the low risk group is relatively sparse. Numeric simulations are also used to verify the asymptotic results in small samples and evaluate the efficiency of the unbalanced design in more complicated settings. Beyond comparing the asymptotic sampling variances, I further evaluate the power and empirical Type I error from unbalanced design and compare against the traditional balanced design. Data from a randomized clinical trial for type 1 diabetes are further used to test the performance of the proposed unbalanced design, and the parametric analyses of these data confirmed the findings from the theoretical and numerical studies.

Student retention has become a serious topic in the past several decades (Wild and Ebbers, 2002). Problematic, however, is how retention is defined and measured, as well as a lack of multi-institutional studies that support a theoretical model for improving student retention, particularly in community colleges (Bailey & Alfonso, 2005). The Community College Survey of Student Engagement (CCSSE) was launched in 2001. Based on extensive research that pertains to student learning and persistence, CCSSE defined five benchmarks of educational practice. Three of the benchmarks comprise the Retention Index. CCSSE has encouraged additional studies to further validate the relatively new survey instrument. Florida's legislature has a keen interest in the performance of educational institutions which are mandated by statute to participate in system-wide data collection from which accountability measures are drawn, including institutional retention rates. Using institutional level data in simple and multiple linear regressions, this study examined the relationship between the Florida Community Colleges' CCSSE Retention Indices and their retention rate(s) measured by the Florida Accountability Measure. Student level data was also analyzed using a Nested ANOVA to examine mean differences in CCSSE Retention Index scores of students from different racial and gender groups when accounting for the possible influence of institution attended.

A linear space is a set of points and lines such that any pair of points lie on exactly one line together. This is equivalent to a pairwise balanced design PBD(v, K), where there are v points, lines are regarded as blocks, and K ⊆ Z≥2 denotes the set of allowed block sizes. The dimension of a linear space is the maximum integer d such that any set of d points is contained in a proper subspace. Specifically for K = {3, 4, 5}, we determine which values of v admit PBD(v,K) of dimension at least three for all but a short list of possible exceptions under 50. We also observe that dimension can be reduced via a substitution argument. / Graduate / 0405 / jniezen@uvic.ca