Doctor of Philosophy / Department of Statistics / John E. Boyer Jr., Dallas E. Johnson / A crossover design is an experimental design in which each experimental unit receives a series of experimental treatments over time. The order that an experimental unit receives its treatments is called a sequence (example, the sequence AB means that treatment A is given first, and then followed by treatment B). A period is the time interval during which a treatment is administered to the experimental unit. A period could range from a few minutes to several months depending on the study. Sequences usually involve subjects receiving a different treatment in each successive period. However, treatments may occur more than once in any sequence (example, ABAB). Treatments and periods are compared within subjects, i.e. each subject serves as his/her own control. Therefore, any effect that is related to subject differences is removed from treatment and period comparisons. Carryover effects are residual effects from a previous treatment manifesting themselves in subsequent periods. Crossover designs both with and without carryover are traditionally analyzed assuming that the response due to different treatments have equal variances. The effects of unequal variances on traditional tests for treatment and carryover difference were recently considered in crossover designs assuming that the response due to treatments have unequal variances with a compound symmetry correlation structure. The likelihood function for the two treatment/two sequence crossover design has closed form maximum likelihood solutions for the parameters at both the null hypothesis, H0 : sigma_A^2 =\sigma_B^2, and at alternative hypothesis, HA : not H0. Under HA : not H0, the method of moment estimators and the maximum likelihood estimators of sigma_A^2,sigma_B^2 and rho are identical. The dual balanced design, ABA=BAB, which is balanced for carryover effects is also considered. The dual balanced design has a closed form solution that maximizes the likelihood function under the null hypothesis, H0 :sigma_A^2=sigma_B^2, but not for the alternative hypothesis, HA : not H0. Similarly, the three treatment/three sequence crossover design, ABC=BCA=CAB, has a closed form solution that maximizes the likelihood function at the null hypothesis, H0 : sigma_A^2=sigma_B^2 = sigma_C^2, but not for the alternative hypothesis, HA : not H0. An iterative procedure is introduced to estimate the parameters for the two and three treatment crossover designs. To check the performance of the likelihood ratio tests, Type I error rates and power comparisons are explored using simulations.

No description available.

No description available.

Introduction: The association between daily air pollution exposure and risk of mortality is well established. Few studies have investigated in detail the associations beyond a seven day lag. The aim of this thesis was to investigate the change in risk across longer (30 day) periods post exposure for three specific causes of death: pneumonia, chronic obstructive pulmonary disease (COPD), and ischaemic heart disease (IHD). Methods: Daily Scottish mortality data (1980-2011) was matched to measurements from local fixed site pollution (Black smoke, PM10, PM2.5, SO2, & NO2) and temperature monitors. Exposure on subjects' 'day of death' was compared with control days in a time-stratified case-crossover analysis. Exposure effects on 30 days prior to day of death were modelled using distributed lag non-linear, lag stratified, and cubic distributed lag models. Matching hospital admissions data inferred subject location during exposure, further analyses investigated extreme outliers and missing data using multiple imputation techniques. The analysis accounted for several confounders including accurately modelling temperature relationships unique for each cause of death. Results: Of the 919,301 deaths, 20% were classified as being caused by pneumonia, 9.5% as COPD, and 30% as IHD in the 'any' cause of death field. Non-linear effects for temperature and linear effects for the pollutants were present across all 30 days. Temperature-mortality was observed to be U-shaped at shorter lags. Consistently increased risk occurred for longer in cold temperatures with 1oC increase (30 days lag) = %RR -0.35% Pneumonia, -0.62% COPD, and -0.26% IHD. PM2.5 on all three outcomes, and all pollutants on COPD showed the greatest effect sizes. In general, COPD risk only occurred after a delay, peaking between 12-18 days. COPD risk due to PM2.5 was immediate (%RR (95% C.I.) = 1.05% (0.14%,2.01%)) and lasted the full 30 days. Pneumonia risk often reported the shortest lag of 10-15 days, whereas IHD risk occurred 2 days after exposure but lasted the remaining 30 days. There was some evidence especially for pneumonia of a smaller association between air pollution on mortality when subjects included were present in hospital. A simulation study indicated slight improvement in accuracy when 'multiple imputation' was performed compared to 'complete cases' analysis; though both techniques reported similarly underestimated effect estimates. Extreme outliers in the main analysis of pollution exposure did not appear to have a strong influence on the risk. However, large variability between monitor measurements of pollution exposure was present and appeared to be influencing the results. Conclusion: This study provides additional evidence on the link between air pollution, and temperature, and acute mortality. Particular focus was on three causes of death (pneumonia, COPD, and IHD) that are shown to be influenced by air pollution in subtly different ways. Results also indicated that the 'true' effect of air pollution on mortality might be greater than shown by mortality studies which do not use hospital admission location during exposure into account.

Bioequivalence determines if two drugs are alike. The three kinds of bioequivalence are Average, Population, and Individual Bioequivalence. These Bioequivalence criteria can be evaluated using aggregate and disaggregate methods. Considerable work assessing bioequivalence in a frequentist method exists, but the advantages of Bayesian methods for Bioequivalence have been recently explored. Variance parameters are essential to any of theses existing Bayesian Bioequivalence metrics. Usually, the prior distributions for model parameters use either informative priors or vague priors. The Bioequivalence inference may be sensitive to the prior distribution on the variances. Recently, there have been questions about the routine use of inverse gamma priors for variance parameters. In this paper we examine the effect that changing the prior distribution of the variance parameters has on Bayesian models for assessing Bioequivalence and the carry-over effect. We explore our method with some real data sets from the FDA.

The purpose of this study is to focus on constructing a modelling-based learning environment to improve learner performance in grade 6 mathematics classrooms. The purpose emanates from the continued poor performance of learners in mathematics at different school levels, especially grade 6. The teaching and learning of mathematics is explained from an ontological point of departure, focussing on constructivist paradigms. Different types of constructivism are discussed with special attention to the school mathematics domain. The learning, problem based learning, problem solving and learning environment are key components in the discussion. A theoretical perspective on the design of modelling as a powerful learning environment in primary schools mathematics classrooms is provided. Focus is placed on the applicability of the modelling-based learning environment on the South African mathematics curriculum and on study orientation as a key component to help develop an understanding of why learners perform or do not perform in mathematics. A mixed method research design, in which quantitative and qualitative are combined to achieve the outcomes of the research problem, is chosen for this research study project to provide a purposeful research framework. The findings of the research include not only learners’ improvement in dealing with non-routine, mathematical word problems but also in general-routine, mathematical word problems. A second finding shows that the overall SOM pre/post/retention showed good reliability, acceptable construct validity, good practical significance, and large effect but had low to medium effect in individual fields. The univariate analysis for the Crossover design used indicated that the problem solving field had statistical significance and practical significance, and the study milieu and mathematical confidence field might have statistical significance and practical significance. The third finding provided evidence concerning teacher administration, teacher and learner interaction, assessment and homework. The findings from the quantitative and qualitative data-analysis and interpretations, and literature review, guided the researcher in proposing a construct for a modelling-based learning environment as a means to improve learners’ mathematics performance in grade 6 mathematics classes in the John Toalo Gaetswe (JTG) District. The contribution that this study makes is to propose a construct for a modelling-based learning environment to improve learner performance in grade 6 mathematics. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

The purpose of this study is to focus on constructing a modelling-based learning environment to improve learner performance in grade 6 mathematics classrooms. The purpose emanates from the continued poor performance of learners in mathematics at different school levels, especially grade 6. The teaching and learning of mathematics is explained from an ontological point of departure, focussing on constructivist paradigms. Different types of constructivism are discussed with special attention to the school mathematics domain. The learning, problem based learning, problem solving and learning environment are key components in the discussion. A theoretical perspective on the design of modelling as a powerful learning environment in primary schools mathematics classrooms is provided. Focus is placed on the applicability of the modelling-based learning environment on the South African mathematics curriculum and on study orientation as a key component to help develop an understanding of why learners perform or do not perform in mathematics. A mixed method research design, in which quantitative and qualitative are combined to achieve the outcomes of the research problem, is chosen for this research study project to provide a purposeful research framework. The findings of the research include not only learners’ improvement in dealing with non-routine, mathematical word problems but also in general-routine, mathematical word problems. A second finding shows that the overall SOM pre/post/retention showed good reliability, acceptable construct validity, good practical significance, and large effect but had low to medium effect in individual fields. The univariate analysis for the Crossover design used indicated that the problem solving field had statistical significance and practical significance, and the study milieu and mathematical confidence field might have statistical significance and practical significance. The third finding provided evidence concerning teacher administration, teacher and learner interaction, assessment and homework. The findings from the quantitative and qualitative data-analysis and interpretations, and literature review, guided the researcher in proposing a construct for a modelling-based learning environment as a means to improve learners’ mathematics performance in grade 6 mathematics classes in the John Toalo Gaetswe (JTG) District. The contribution that this study makes is to propose a construct for a modelling-based learning environment to improve learner performance in grade 6 mathematics. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014