The Partition of Variance (POV) method is a simplistic way to identify large sources of variation in manufacturing systems. This method identifies the variance by estimating the variance of the means (between variance) and the means of the variance (within variance). The project shows that the method correctly identifies the variance source when compared to the ANOVA method. Although the variance estimators deteriorate when varying degrees of non-normality is introduced through simulation; however, the POV method is shown to be a more stable measure of variance in the aggregate. The POV method also provides non-negative, stable estimates for interaction when compared to the ANOVA method. The POV method is shown to be more stable, particularly in low sample size situations. Based on these findings, it is suggested that the POV is not a replacement for more complex analysis methods, but rather, a supplement to them. POV is ideal for preliminary analysis due to the ease of implementation, the simplicity of interpretation, and the lack of dependency on statistical analysis packages or statistical knowledge.

The purpose of this study was to compare perceptions of success and failure, attributions of success and failure, predictions of future success, and reports of out-of-class engagement in composition among middle school band students composing in open task conditions (n = 32) and closed task conditions (n = 31). Two intact band classes at the same middle school were randomly assigned to treatment groups. Both treatment groups composed music once a week for eight weeks during their regular band time. In Treatment A (n = 32), the open task group, students were told to compose music however they wished. In Treatment B (n = 31), the closed task group, students were given specific, structured composition assignments to complete each week. At the end of each session, students were asked to complete a Composing Diary in which they reported what they did each week. Their responses were coded for evidence of perceptions of success and failure as well as out-of-class engagement in composing. At the end of eight weeks, students were given three additional measures: the Music Attributions Survey to measure attributions of success and failure on 11 different subscales; the Future Success survey to measure students' predictions of future success; and the Out-of-Class Engagement Letter to measure students' engagement with composition outside of the classroom. Results indicated that students in the open task group and students in the closed task group behaved similarly. There were no significant differences between treatment groups in terms of perceptions of success or failure as composers, predictions of future success composing music, and reports of out-of-class engagement in composition. Students who felt they failed at composing made similar attributions for their failure in both treatment groups. Students who felt they succeeded also made similar attributions for their success in both treatment groups, with one exception. Successful students in the closed task group rated Peer Influence significantly higher than the successful students in the open task group. The findings of this study suggest that understanding individual student's attributions and offering a variety of composing tasks as part of music curricula may help educators meet students' needs.

This work presents two complementary studies that propose heuristic methods to capture characteristics of data using the ensemble learning method of random forest. The first study is motivated by the problem in education of determining teacher effectiveness in student achievement. Value-added models (VAMs), constructed as linear mixed models, use students’ test scores as outcome variables and teachers’ contributions as random effects to ascribe changes in student performance to the teachers who have taught them. The VAMs teacher score is the empirical best linear unbiased predictor (EBLUP). This approach is limited by the adequacy of the assumed model specification with respect to the unknown underlying model. In that regard, this study proposes alternative ways to rank teacher effects that are not dependent on a given model by introducing two variable importance measures (VIMs), the node-proportion and the covariate-proportion. These VIMs are novel because they take into account the final configuration of the terminal nodes in the constitutive trees in a random forest. In a simulation study, under a variety of conditions, true rankings of teacher effects are compared with estimated rankings obtained using three sources: the newly proposed VIMs, existing VIMs, and EBLUPs from the assumed linear model specification. The newly proposed VIMs outperform all others in various scenarios where the model was misspecified. The second study develops two novel interaction measures. These measures could be used within but are not restricted to the VAM framework. The distribution-based measure is constructed to identify interactions in a general setting where a model specification is not assumed in advance. In turn, the mean-based measure is built to estimate interactions when the model specification is assumed to be linear. Both measures are unique in their construction; they take into account not only the outcome values, but also the internal structure of the trees in a random forest. In a separate simulation study, under a variety of conditions, the proposed measures are found to identify and estimate second-order interactions.