Parametric Person–Fit Statistics, Zeta (z), and Generalized Zetas (z1, …, zm)
Several authors have shown an interest in detecting aberrant response patterns that do not fit in IRT models, and have tried to extract the information not contained in the total scores from those aberrant response patterns. Levine and Rubin (1979) and Wright (1977), for example, referred to the identification of “guessing, sleeping, fumbling and plodding” (p. 110) from the plots of residual item scores based on the differences between item responses and the expected responses for an individual based on the Rasch model. Levine and Rubin discussed response patterns that are so typical that a student’s aptitude test score fails to be a completely appropriate measure (p. 269). Trabin and Weiss (1979) also investigated measures to identify unusual response patterns. Sato (1975) proposed a “caution” index, which is intended to identify students whose total scores on a test must be treated with caution. Tatsuoka and Tatsuoka (1982) and Harnisch and Linn (1981) have discussed the relationship of response patterns to instructional experiences and the possible use of item response pattern information to help diagnose the types of errors a student is making. These researchers developed various indices to detect aberrant response patterns. Later, Molenaar and his associates (Molenaar & Hoitjink, 1990; Molenaar & Sijtsma, 1984) developed an index (i.e., person-fit statistics) that is statistically capable of judging an aberrant response pattern and to what extent it deviates from average response patterns. The aim of person-fit statistics is to identify persons whose item score patterns on a test are aberrant when compared with a majority of item score patterns that are expected on the basis of some statistical or psychometric models, mainly Item Response Models. An observed pattern of item scores is called “aberrant” when it is not likely to happen, as compared with a majority of item response patterns. If a IRT parametric model can explain a set of item response patters well, then we say the model fits the data (Meijer, 1994, p.12).