Case Diagnostics for Factor Analysis of Ordered Categorical Data With Applications to Person-Fit Measurement

Mahalanobis distance (M-distance) case diagnostics are a useful tool for assessing response pattern inconsistency in factor analysis; however, the derivations of these statistics assume continuous variables, which limits their utility in ordinal self-or rater-report data. This research generalizes M-distance diagnostics to categorical factor analysis. We prove that the residual-based M-distance d(r) is equivalent to the person-fit index lco, which motivates the use of the new categorical M-distance d(r)* as a person-fit index. d(r)* is compared and contrasted with zh, a commonly used item response theory person-fit index. A simulation study is used to show that a simple transformation of d(r)* satisfies established criteria for person-fit measures. A sample of responses to the Rosenberg Self-Esteem Scale is used to determine parameters for a simulation study, and real data are analyzed to contrast the use of d(r) and d(r)* as indexes of person-fit in continuous and categorical factor analysis.