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MSc, Faculty of Engineering, University of the Witwatersrand, 1978

Reference intervals (RIs) are sets of percentiles that outline the range of laboratory test results belonging to healthy individuals. They are essential for the interpretation of laboratory test results. A wide variety of factors affect the validity of RIs. Among them are the statistical methods used to estimate RIs. However, little investigation has gone into the effect that different statistical methods have on the resulting RIs. This is particularly needed as the complexity of paediatric data makes it difficult to estimate RIs. These difficulties, however, can be addressed using appropriate statistical techniques, provided that there is an outline of scenarios under which these techniques are truly “appropriate”. The objective of this thesis is to provide a thorough investigation into the effect of different statistical methods on RIs. A systematic review was first conducted with a focus on paediatric RIs. The results of this review revealed that critical analysis steps are often overlooked due to complicated paediatric data. Even though a guideline addressing the establishment of RIs is available, there is great heterogeneity in the statistical methods chosen to estimate paediatric RIs. An extensive simulation involving the three most commonly used approaches to estimate RIs (the parametric, non-parametric, and robust methods) was also conducted to investigate and compare the performance of the different methods. The simulation results show that, when data follows a Gaussian distribution, or close to it, the parametric method provides the best estimates. The non-parametric method did not provide the best estimates of RIs (compared to the parametric method) unless data was highly skewed and/or large sample sizes were used. In addition, the bias and MSE associated with the parametric method when data follows a Gaussian distribution was mathematically derived, which may lead to the development of a bias corrected and more precise approach in the future. / Thesis / Master of Science (MSc)

Master of Science / Department of Statistics / Paul I. Nelson / Hotelling’s T2 test is the standard tool for inference about the mean of a multivariate normal population. However, this test may perform poorly when used on samples from multivariate distributions with highly skewed marginal distributions. The goal of our study was to investigate the type I error rate and power properties of Hotelling’s one sample test when sampling from a class of multivariate skew-normal (SN) distributions, which includes the multivariate normal distribution and, in addition to location and scale parameters, has a shape parameter to regulate skewness. Simulation results of tests carried out at nominal type I error rate 0.05 obtained from various levels of shape parameters, sample sizes, number of variables and fixed correlation matrix showed that Hotelling’s one sample test provides adequate control of type I error rates over the entire range of conditions studied. The test also produces suitable power levels for detecting departures from hypothesized values of a multivariate mean vector when data result from a random sample from a multivariate SN. The shape parameter of the SN family appears not to have much of an effect on the robustness of Hotelling’s test. However, surprisingly, it does have a positive impact on power.

Multistage tests (MSTs) have received renewed interest in recent years as an effective compromise between fixed-length linear tests and computerized adaptive test. Most MSTs studies scored the assessments based on item response theory (IRT) methods. Many assessments are currently being developed as mixed-format assessments that administer both standalone items and clusters of items associated with a common stimulus called testlets. By the nature of a testlet, a natural dependency occurs between the items within the testlet that violates the local independence of items. Local independence is a fundamental assumption of the IRT models. Using dichotomous IRT methods on a mixed-format testlet-based assessment knowingly violates local independence. By combining the score points within a testlet, researchers have successfully applied polytomous IRT models. However, the use of such models loses information by not using the unique response patterns provided by each item within a testlet. The three-parameter logistic testlet response theory (3PL-TRT) model is a measurement model developed to retain the uniqueness in response patterns of each item, while accounting for the local dependency exhibited by a testlet, or testlet effect. Because few studies have examined mixed-format MSTs administration under the 3PL-TRT model, the dissertation performed a simulation to investigate the administration of a mixed-format testlet based MSTs under the 3PL-TRT model. Simulee responses were generated based on the 3PL-TRT calibrated item parameters from a real large-scale passage based standardized assessment. The manipulated testing conditions considered four panel designs, two test lengths, three routing procedures, and three conditions of local item dependence. The study found functionally no bias across testing conditions. All conditions showed adequate measurement properties, but a few differences did occur between some of the testing conditions. The measurement precision was impacted by panel design, test length and the magnitude of local item dependence. The three-stage MSTs consistently illustrated slightly lower measurement precision than the two-stage MSTs. As expected, the longer test length conditions had better measurement precision than the shorter test length conditions. Conditions with the largest magnitude of local item dependency showed the worst measurement precision. The routing procedure had little impact on the measurement effectiveness. / text

It has been shown that pair-matching on an ordered baseline with normally distributed measures reduces the variance of the estimated treatment effect (Park and Johnson, 2006). The main objective of this study is to examine if pair-matching improves the power when the distribution is a mixture of two normal distributions. Multiple scenarios with a combination of different sample sizes and parameters are simulated. The power curves are provided for three cases, with and without matching, as follows: analysis of post-intervention data only, adding baseline as a covariate, and classic pre-post comparison. The study shows that the additional variance reduction provided by pair-matching in the pre-post design is limited for high correlation. When correlation is low, there is a significant power increase. It is shown that the baseline pair-matching improves the power when the two means of a mixture normal distribution are widely spread. The pattern becomes clear for low correlation.

<div>Atomistic simulations are used in this project to study the deformation mechanism of polycrystalline and bicrystal of pure Al and Al-Mg alloys. Voronoi Tessellation was used to create three-dimensional polycrystalline models. Monte Carlo and Molecular</div><div>Dynamics simulations were used to achieve both mechanical and chemical equilibrium in all models. The first part of the results showed improved strength, which is included the yield strength and ultimate strength in the applied tensile loading through the addition of 5 at% Mg to nanocrystalline aluminum. By viewing atomic structures, it clearly shows the multiple strengthening mechanisms related to doping in Al-Mg alloys. The strength mechanism of dopants exhibits as dopant pinning grain boundary (GB) migration at the early deformation stage. At the late stage where it is close to the failure of nanocrystalline materials, Mg dopants can stop the initiation of intergranular cracks and also do not let propagation of existing cracks along the GBs. Therefore, the flow stress will improve in Al-Mg alloy compared to pure Al. In the second part of our results, in different bicrystal Al model, Σ 3 model has higher strength than other models. This result indicates that GB structure can affect the strength of the material. When the Mg dopants were added to the Al material, the strength of sigma 5 bicrystal models was improved in the applied shear loading. </div><div><br></div><div>However, it did not happen for Σ 3 model, which shows Mg dopants cannot affect the behavior of this GB significantly. Analysis of GB movements shows that Mg dopants stopped GBs from moving in the Σ 5 models. However, in sigma 3 GBs, displacement of grain boundary planes was not affected by Mg dopants. Therefore, the strength and flow stress are improved by Mg dopants in Σ 5 Al GBs, not in the Σ 3 GB.</div>

In many applications of random-effects models to longitudinal data, such as heart rate variability (HRV) data, a normal-mixture distribution seems to be more appropriate than the normal distribution assumption. While the random-effects methodology is well developed for several distributions in the exponential family, the case of the normal-mixture has not been dealt with adequately in the literature. The models and the estimation methods that have been proposed in the past assume the conditional model (fixing the random-effects) to be normal and allow a mixture distribution for the random effects (Xu and Hedeker, 2001, Xu, 1995). The methods proposed in this dissertation assume the conditional model to be a normal-mixture while the random-effects are assumed to be normal. This is primarily to fit the HRV data, which seems to follow a normal-mixture within subjects. Another advantage of this model is that the estimation becomes much simpler through the use of an EM-algorithm. Existing methods and software such as the PROC MIXED in SAS are exploited to facilitate the estimation procedure.A simulation study is performed to examine the properties of the random-effects model with normal-mixture distribution and the estimation of the parameters using the EM-algorithm. The study shows that the estimates have similar properties to the usual normal random-effects models. The between subject variance parameter seems to require larger numbers of subjects to achieve reasonable accuracy, which is typical in all random-effects models.The HRV data is used to illustrate the random-effects normal-mixture method. These data consist of 9 subjects who completed a series of five speech tasks (Cacioppo et al., 2002). For each of the tasks, a series of RR-intervals was collected during baseline, preparation, and delivery periods. Information about their age and gender were also available. The random-effects mixture model presented in this dissertation treats the subjects as random and models age, gender, task, type, and task × type as fixed-effects. The analysis leads to the conclusion that all the fixed effects are statistically significant. The model further indicates a two-component normal-mixture with the same mixture proportion across individuals fit the data adequately.

Master of Science / Department of Statistics / Paul I. Nelson / Suppose that based on data consisting of independent repetitions of an experiment a researcher wants to predict the outcome of the next independent outcome of the experiment. The researcher models the data as being realizations of independent, identically distributed random variables { Xi, i=1,2,..n} having density f() and the next outcome as the value of an independent random variable Y , also having density f() . We assume that the density f() lies in one of three location-scale families: standard normal (symmetric); Cauchy (symmetric, heavy-tailed); extreme value (asymmetric.). The researcher does not know the values of the location and scale parameters. For f() = f0() lying in one of these families, an exact prediction interval for Y can be constructed using equivariant estimators of the location and scale parameters to form a pivotal quantity based on { Xi, i=1,2,..n} and Y. This report investigates via a simulation study the performance of these prediction intervals in terms of coverage rate and length when the assumption that f() = f0() is correct and when it is not. The simulation results indicate that prediction intervals based on the assumption of normality perform quite well with normal and extreme value data and reasonably well with Cauchy data when the sample sizes are large. The heavy tailed Cauchy assumption only leads to prediction intervals that perform well with Cauchy data and is not robust when the data are normal and extreme value. Similarly, the asymmetric extreme value model leads to prediction intervals that only perform well with extreme value data. Overall, this study indicates robustness with respect to a mismatch between the assumed and actual distributions in some cases and a lack of robustness in others.

Master of Science / Department of Statistics / Perla Reyes / Heterodera glycines, or the soybean cyst nematode (SCN), is a parasite that targets and damages the roots of soybean plants. It is the most yield-limiting pathogen of soybean in the U.S. and the reliable detection and accurate estimation of population densities is crucial to research and management of this pathogen. A study was performed to understand the effects of crop rotation on the prevalence of SCN. Standard sampling procedures in the plant pathology community dictate taking soil samples from potentially infected fields, processing them and counting the number of eggs in one 1 mL subsample via microscope. Suspecting the traditional procedure may lead to invalid results, false negatives in particular, the researcher created and implemented a sampling procedure based on his knowledge of sampling methods and constraints of sampling in the field. Using the data collected, we will discuss the strengths and limitations of the procedure in estimating the population density of SCN in the field. In addition, a simulation study informed by the data will be conducted to determine a sampling strategy that will yield accurate results while still considering the conditions in the field. Knowledge on how the different stages of the sampling procedure for SCN affect the accurate detection of the pathogen would extend to experimental designs and sampling methodologies for other soil dwelling organisms.