▼ Bowling Green State University, like other universities, administers a mathematics placement test to incoming students to help to identify an appropriate first math course. Students select and take one of three tests depending on their math background, the test is scored and a placement decision is made. The issue here is whether the placement decision can be improved by changing the criteria. The goal of this thesis is to accurately predict a student's grade from a subset of available covariates for students enrolling in Math 126 at BGSU. The hope is that an accurate methodology and model can be found so that it may be considered as a placement system for students into their first math course at BGSU. We focused on the subset of the data in which students took version B of the placement exam and enrolled in Math 126 (Business Calculus). Three regression models were used to fit the data: linear, logistic, and ordinal logistic. Advisors/Committee Members: Zirbel, Craig.

▼ The analysis of variance (ANOVA) is one of the most important and useful techniques for variety of fields such as agriculture, sociology and medicine for comparing different groups or treatments with respect to their means. A set of assumptions such as normal error distribution, homogeneity of variances and independence of observations, has to be made to employ an F test for equality of the treatment means. It is now well established that the violation of the assumption of homogeneity of variances can have severe effects on the inference of the population means, especially in the case of unequal sample sizes. In fact, the conventional ANOVA F provides generally poor control over both Type I and Type II error rates under a wide variety of variance heterogeneity conditions. Therefore, the problem of homogeneity of variances has to be settled before conducting an ANOVA. While a good number of tests are available for testing homogeneity of variances, Bartlett test and four versions of Levene tests are still popular for testing homogeneity of variances in the case of one-way ANOVA setting. It is evident that the Bartlett test is not as robust as Levene tests against the violation of the normality assumption. On the other hand, Levene tests are less powerful than the Bartlett test. In this dissertation, we proposed a transformed version of Bartlett test where the transformation is intended to achieve the normality of the data and independence of the observations to some extent. It is evident from the simulation that the transformed Bartlett test is more robust than the untransformed Bartlett test against the violation of the normality assumption. It also follows that the transformed Bartlett test is a balance between the Bartlett test and the four versions of the Levene tests in terms of Type I error rate and power concern. While, the estimation of location parameter is of concern, a modified version of trimmed mean has been proposed as an alternative to trimmed mean when the distribution is skewed or contains outliers. It is evident from the simulation study that modified trimmed mean outperform trimmed mean in terms of coverage probability of interval estimate. It is also evident that the test based on the modified version of trimmed mean is suitable for estimating Type I error rate and is more powerful than the tests based on the mean and the trimmed mean in the presence of the outliers. Finally, an alternative method of estimating transformation parameter has been proposed by using the chi-squared goodness of fit criteria which could be useful for testing equality of two population means. Advisors/Committee Members: Chen, Hanfeng.

▼ In the area of ROC analysis, confidence intervals and the corresponding confidence bands are usually constructed by way of first oder Normal approximations based on the popular delta method. However, it has been always known that all the information in the data is contained in the corresponding likelihood function. On another note there has recently been a surge in research in empirical data analysis. Of note is the work of Art Owen on empirical likelihood methods. This research endeavors to explore the use of likelihood-based methods in statistical analysis of the ROC curve, specifically in the construction of confidence bands, and compare them with ROC confidence bands based on Normal approximation resultant from the delta method. In the first part we explore the theory behind the construction of likelihood-based and Normal approximation based confidence bands for the ROC curve, under the assumption that the data are sampled from parametric models. In the second part we explore the theory behind the construction of confidence bands for the ROC curve using the Normal approximation for empirical estimators and empirical likelihood-based confidence bands for the ROC curve. The basis for both of the likelihood-based methods is the chi-squared approximation, popularly known as Wilks’ method. It has been observed that the chi-squared approximation to the likelihood function may not be a very good approximation. As a result a Bartlett correction to the chi-squared approximation can be used to adjust the approximation. The Bartlett correction has been known to be plausible for parametric models. However, it has only been recently known that the empirical likelihood function is also Bartlett correctable, under some conditions. This research goes further to explore Bartlett correction of empirical likelihood confidence bands for the ROC curve. Simulation studies were carried out to demonstrate the improvement of the likelihood-based approach over the Normal approximation approach as well as the further improvement of the Bartlett correction over empirical likelihood method. Advisors/Committee Members: Chen, Hanfeng.

▼ Predictor variables are often contaminated with measurement errors in statistical practice. This may be the case due to bad measurement apparatus or just because the true value of the variable cannot be measured precisely. In the framework of general regression models, measurement errors or misclassifications have very serious consequences in many cases as they lead to bias in the estimated parameters that does not disappear as the sample size goes to infinity. In most cases the estimated effect of the contaminated covariate is attenuated. There are some techniques, regression calibration, simulation extrapolation (SIMEX), and the score function method for correcting effect estimates in the presence of measurement error. These widely used approaches have some restricted applications in many situations, for example, SIMEX is a useful tool for correcting effect estimates in the presences of additive measurement error. The method is especially helpful for complex models with a simple measurement error structure. Score function method is employed only for linear measurement error models. In this dissertation, an inference method has been proposed that accounts for the presence of measurement error in the explanatory variables in both linear and nonlinear models. This approach relies on the consideration of the mean and variance function of the observed data and application of the empirical likelihood approach to those functions, which is referred to as quasi likelihood and variance function (QVF). This proposed approach provides the confidence intervals with high inclusion probability of the unknown regression parameters. Moreover, this method is computationally easy to employ to any measurement error model for correcting bias. In addition, general descriptions and comparisons of the existing methods and the suggested estimation framework with some applications in real life data are discussed. A simulation study is conducted to show the performance of the proposed estimation framework. Advisors/Committee Members: Chen, Hanfeng.

▼ Item Response Theory (IRT) models are commonly used in educational and psychological testing. These models are mainly used to assess the latent abilities of examinees and the effectiveness of the test items in measuring this underlying trait. However, model checking in Item Response Theory is still an underdeveloped area. In this dissertation, various model checking strategies from a Bayesian perspective for different Item Response models are presented. In particular, three methods are employed to assess the goodness-of-fit of different IRT models. First, Bayesian residuals and different residual plots are introduced to serve as graphical procedures to check for model fit and to detect outlying items and examinees. Second, the idea of predictive distributions is used to construct reference distributions for different test quantities and discrepancy measures, including the standard deviation of point bi-serial correlations, Bock's Pearson-type chi-square index, Yen's Q1 index, Hosmer-Lemeshow Statistic, Mckinley and Mill's G2 index, Orlando and Thissen's S-G2 and S-X2 indices, Wright and Stone's W-statistic, and the Log-likelihood statistic. The prior, posterior, and partial posterior predictive distributions are discussed and employed. Finally, Bayes factor are used to compare different IRT models in model selection and detection of outlying discrimination parameters. In this topic, different numerical procedures to estimate the Bayes factors for these models are discussed. All of these proposed methods are illustrated using simulated data and Mathematics placement exam data from BGSU. Advisors/Committee Members: Albert, James H.