In education, constructed response (CR) items are increasingly being used in standardized tests. Data obtained from CR items usually include written responses and categorical scores that are used for grading those responses. The primary purpose of this dissertation study is to develop new statistical approaches using structural equation modeling (SEM) and topic modeling for analyzing student written responses to CR items and scores obtained from the responses. SEM is a statistical modeling method used to investigate relationships among observed and latent constructs. Topic modeling is primarily intended to help identify latent structures in textual materials. This dissertation study consists of three research studies, which are connected by the examination of methodological issues dealing with data from CR items. The first study explores the internal consistency of categorical data having different numbers of response categories. This type of categorical data can often be observed in tests with CR items. This study proposes a SEM approach to nonlinear reliability for tests with items having different numbers of ordered categories. A simulation study evaluating the performance of the proposed approach is presented, and an empirical example is provided to illustrate different reliability coefficients. The second study introduces a topic model to analyze students written responses to CR items with associated scores for their responses. The proposed model is designed to detect meaningful homogeneous subgroups with regard to the relationships between topic proportions in examinees responses and scores. The application of the model is demonstrated through data from student responses to CR items and associated scores on a middle grades test of science inquiry practices. The third study deals with students written responses collected over multiple time points. This study proposes a new model that combines a topic model and a growth curve model to analyze the texts of answers to CR items collected under a longitudinal study design. The application of the proposed approach is illustrated using real data from middle grades students. A simulation study evaluating the proposed model under several practical testing conditions is also provided.
