In this study, I investigated how proficiently seventh-grade students enrolled in two Southwestern schools solve algebra word problems. I analyzed various factors that could affect this proficiency and explored the differences between English Learners (ELs) and native English Primary students (EPs). I collected the data as part of the Animal Watch project, a computer-based initiative designed to improve the mathematical skills of children from grades 5-8 in the Southwest. A sample of 86 students (26 ELs and 60 EPs), clustered in four different classes, was used for this project. A Generalized Linear Mixed Model (GLMM) approach with the GLIMMIX procedure in SAS 9.3 showed that students from the classes that had a higher percentage of EL students performed better than those in the classes where the EL concentration was lower. Classes with more EL males were better at learning mathematics than classes with more EP females. The results also indicated: (a) a positive correlation between the students' ability to solve algebra word problems on their first attempt and their success ratio in solving all problems, and (b) a negative correlation between the percentage of problems solved correctly and those considered too hard from the very beginning. I conclude my dissertation by making specific recommendations for further research.

Master of Science / Department of Statistics / Leigh Murray / This case study was motivated by the need for effective statistical analysis for a series of poultry feeding experiments conducted in 2006 by Kansas State University researchers in the department of Animal Science. Some of these experiments involved an automated auger feed line system commonly used in commercial broiler houses and continuous, proportion response data. Two of the feed line experiments are considered in this case study to determine if a statistical model using a non-normal response offers a better fit for this data than a model utilizing a normal approximation. The two experiments involve fixed as well as multiple random effects. In this case study, the data from these experiments is analyzed using a linear mixed model and Generalized Linear Mixed Models (GLMM’s) with the SAS Glimmix procedure. Comparisons are made between a linear mixed model and GLMM’s using the beta and binomial responses. Since the response data is not count data a quasi-binomial approximation to the binomial is used to convert continuous proportions to the ratio of successes over total number of trials, N, for a variety of possible N values. Results from these analyses are compared on the basis of point estimates, confidence intervals and confidence interval widths, as well as p-values for tests of fixed effects. The investigation concludes that a GLMM may offer a better fit than models using a normal approximation for this data when sample sizes are small or response values are close to zero. This investigation discovers that these same instances can cause GLMM’s utilizing the beta response to behave poorly in the Glimmix procedure because lack of convergence issues prevent the obtainment of valid results. In such a case, a GLMM using a quasi-binomial response distribution with a high value of N can offer a reasonable and well behaved alternative to the beta distribution.

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Background: Adverse birth outcomes present a difficult and chronic challenge in Ontario, in Canada and in developed countries in general. Increasing proportions of preterm births, significant regional disparities and the high cost of treating all adverse birth outcomes have focused attention on explaining them and developing effective treatments. Methods: Birth outcomes and maternal characteristics for approximately 626,000 births, about 90% of births in 2005–2009, were linked to small geographic areas throughout Ontario. For each of four adverse outcomes: late preterm, moderate to very preterm, small for gestation age and still births, proportions of total births were calculated for the full province and for each small geographic area. Geographic hotspots of elevated rates were identified for each of the different adverse birth outcomes using the local Moran’s I statistic. Data for nine known ecologic and individual risk factors were then linked to the areas. Hierarchical regression analysis was used to model each of the outcomes for the full province and for dispersed local areas. The resulting models for the different outcomes were contrasted. Results: Significant geographic hotspots exist for each of the four outcomes. Hotspots for the different outcomes were found to be largely spatially exclusive. For like outcomes, predictive models differed markedly between local areas (i.e. local groups of hotspots) as well as between full-province and local areas. Ecologic level variables played a strong role in all models; the influence of individual level risk factors was consistently modified by ecologic risk factors except for small for gestational births. Conclusions: The finding of significant hotspots for different adverse birth outcomes indicates that certain geographic areas have aetiologies or patterns of predictors sufficient to create significantly elevated levels of particular outcomes. The finding that hotspots for the different adverse outcomes are largely exclusive implies that the aetiologies are specific; i.e., those that are sufficient to create significantly higher levels for one outcome do not also create significantly higher levels of others. The consistently strong role of ecologic level risk factors in modifying individual level risk factors implies that contextual characteristics are an important part of the aetiology of adverse birth outcomes. Differences in local area models suggest the existence of location-specific (rather than universal) aetiologies. The findings support the need for more careful attention to local context when explaining birth outcomes.

Background: Adverse birth outcomes present a difficult and chronic challenge in Ontario, in Canada and in developed countries in general. Increasing proportions of preterm births, significant regional disparities and the high cost of treating all adverse birth outcomes have focused attention on explaining them and developing effective treatments. Methods: Birth outcomes and maternal characteristics for approximately 626,000 births, about 90% of births in 2005–2009, were linked to small geographic areas throughout Ontario. For each of four adverse outcomes: late preterm, moderate to very preterm, small for gestation age and still births, proportions of total births were calculated for the full province and for each small geographic area. Geographic hotspots of elevated rates were identified for each of the different adverse birth outcomes using the local Moran’s I statistic. Data for nine known ecologic and individual risk factors were then linked to the areas. Hierarchical regression analysis was used to model each of the outcomes for the full province and for dispersed local areas. The resulting models for the different outcomes were contrasted. Results: Significant geographic hotspots exist for each of the four outcomes. Hotspots for the different outcomes were found to be largely spatially exclusive. For like outcomes, predictive models differed markedly between local areas (i.e. local groups of hotspots) as well as between full-province and local areas. Ecologic level variables played a strong role in all models; the influence of individual level risk factors was consistently modified by ecologic risk factors except for small for gestational births. Conclusions: The finding of significant hotspots for different adverse birth outcomes indicates that certain geographic areas have aetiologies or patterns of predictors sufficient to create significantly elevated levels of particular outcomes. The finding that hotspots for the different adverse outcomes are largely exclusive implies that the aetiologies are specific; i.e., those that are sufficient to create significantly higher levels for one outcome do not also create significantly higher levels of others. The consistently strong role of ecologic level risk factors in modifying individual level risk factors implies that contextual characteristics are an important part of the aetiology of adverse birth outcomes. Differences in local area models suggest the existence of location-specific (rather than universal) aetiologies. The findings support the need for more careful attention to local context when explaining birth outcomes.