Functional Data Analysis and Person Response Functions

ABSTRACT The purpose of this study is to illustrate the use of functional data analysis (FDA) as a general methodology for analyzing person response functions (PRFs). Applications of FDA to psychometrics have included the estimation of item response functions and latent distributions, as well as differential item functioning. Although FDA has been suggested for modeling PRFs, there has been relatively little research stressing this application. FDA offers an approach for diagnosing person responses that may be due to guessing and other sources of within-person multidimensionality. PRFs provide graphical displays that can be used to highlight unusual response patterns, and to identify persons that are not responding as expected to a set of test items. In addition to examining individual PRFs, functional clustering techniques can be used to identify subgroups of persons that may be exhibiting categories of misfit such as guessing. A small simulation study is conducted to illustrate how FDA can be used to identify persons exhibiting different levels of guessing behavior (5%, 10%, 15% and 20%). The methodology is also applied to real data from a 3rd grade science assessment used in a southeastern state. FDA offers a promising methodology for evaluating whether or not meaningful scores have been obtained for a person. Typical indices of psychometric quality, such as standard errors of measurement and person fit indices, are not sufficient for representing certain types of aberrance in person response patterns. Nonparametric graphical methods for estimating PRFs that are based FDA provide a rich source of validity evidence regarding the meaning and usefulness of each person’s score.