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Joint Modeling of Mixed Longitudinal Responses in Educational Research

Creative Commons 'BY' version 4.0 license

The availability of assessment measurements that can be used effectively in pedagogical intervention studies is surprisingly limited. As a result, typical educational studies often report anecdotal evidence about the effectiveness of a new teaching method. It is difficult to perform a randomized experiment as students self-select into courses and breakout discussions based on their schedules. In the few studies that attempt to perform some kind of experiment, the efficacy is assessed by a single exam or possibly a pre- and post-test. However, consider the wealth of information collected during a typical course: homework, exams, clicker questions, as so on. The difficulty in analyzing data of this form is the lack of a natural joint distribution. While there may be some merit in looking into each outcome type separately, using all of the available data can answer new questions about how students learn throughout a course.

This dissertation focuses on two areas of (statistics) education: discovering new pedagogical methods for teaching statistics at the introductory college level and creating novel methods to analyze such data. We motivate the need for our novel statistical methodology by discussing a pseudo-experiment performed in an introductory statistics course on using different simulation methods to teach sampling distributions. Due to the form of the available data, we were limited to three exams as our response, which changed our scientific question of interest. This prompted the development of two overarching methods for analyzing mixed response type longitudinal data. We also considered a second educational study, which provides more data on students through the learning management system (LMS), which is becoming more popular in education research. First, we examine a frequentist approach. We transform non-linear responses via the penalized quasi-likelihood technique and use Newton-Raphson and the expectation-maximization algorithm to estimate the fixed and random effects, respectively. We also considered a second method in which we instead treat all responses as discrete and model the proportion correct. Under the Bayesian school of thought, we consider a similar multilevel structure as done in the frequentist setting and test two different assumptions on the response. In the first, we suppose the responses are univariate, and in the second, we presume the responses are, in fact, multivariate, and we have only observed one outcome type at each response. Following multiple simulation studies and testing these modeling approaches on the applied dataset, we directly compare all of the methods described in this dissertation. Following this, we lay out our recommendations for modeling longitudinal data with mixed-type responses.

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