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Abstract
Traditional estimation methods in item response theory are based on parametric methods. The typical process for evaluating person fit involves calculating a variety of statistics to examine model-data fit. Alternatively, graphical displays of nonparametric person response functions (PRFs) and their derivatives can provide insight into where person misfit is occurring. In a standard statistical framework, data are viewed as individual and independent pieces of collected information. Functional data are viewed as a set of measurements along a continuum that, taken together, are to be regarded as a single entity, curve or image (Levitin, Nuzzo, Vines, and Ramsay, 2007, p. 135). The purpose of this study is to introduce Functional Data Analysis (FDA) as a conceptual framework for modeling PRFs, and to detect potential patterns in person responses. Person fit research examines the pattern of individual item responses in comparison with what has been expected on the basis of the measurement model (Emons, Sijtsma, & Meijer, 2005). The present study expands on previously used graphical approaches by studying the use of B-splines to model nonparametric PRFs. A functional clustering approach is employed to identify particular types or categories of misfit. This study uses the HOME data set in Engelhard (2013), and it focuses on 11 items designed to measure the level of learning stimulation in the home (Engelhard, 2013). Simulated data sets were also examined. The data were first analyzed with the Rasch model using the Facets computer software (Linacre, 2013). PRFs were then estimated in R by using splines to nonparametrically model the functional relationship between the probability of endorsing an item and item location using the dichotomous responses. The smoothed curves, residuals, and derivatives were plotted, and compared to theoretical PRFs based on the Rasch model. Differences between nonparametric and theoretical plots indicate person misfit. The same approach was taken with the simulated data to illustrate how to visually detect certain kinds of misfitting person responses. Various levels of guessing were simulated in scenarios corresponding to differing instrument lengths, sample sizes, and respondent proficiencies. The nonparametric PRFs were then clustered to identify persons exhibiting similar response patterns.