An Empirical Investigation of Unscalable Components in Scaling Models

Guttman (1947) developed a scaling method in which the items measuring an attribute can be ordered according to the strength of the attribute. The Guttman scaling model assumes that every member of the population belongs to a scale type and does not allow for response errors. The Proctor (1970) and the intrusion-omission (Dayton and Macready, 1976) models introduced the notion that observed response patterns deviate from Guttman scale types because of response error. The Goodman (1975) model posited that part of the population is intrinsically unscalable. The extended Proctor and intrusion-omission (Dayton and Macready, 1980) models, commonly called extended Goodman models, include both response error and an intrinsically unscalable class (IUC). An alternative approach to the Goodman and extended Goodman models is the two-point mixture index of fit developed by Rudas, Clogg, and Lindsay (1994). The index, pi-star, is a descriptive measure used to assess fit when the data can be summarized in a contingency table for a hypothesized model. It is defined as the smallest proportion of cases that must be deleted from the observed frequency table to result in a perfect fit for the postulated model. In addition to contingency tables, pi-star can be applied to latent class models, including scaling models for dichotomous data. This study investigates the unscalable components in the extended Goodman models and the two-point mixture where the hypothesized model is the Proctor or intrusion-omission model. The question of interest is whether the index of fit associated with the Proctor or intrusion-omission model provides a potential alternative to the IUC proportion for the extended Proctor or intrusion-omission model, or in other words, whether or not pi-star and the IUC proportion are comparable. Simulation results in general did not support the notion that pi-star and the IUC proportion are comparable. Six-variable extended models outperformed their respective two-point mixture models with regard to the IUC proportion across almost every combination of condition levels. This is also true for the four-variable case except the pi-star models showed overall better performance when the true IUC proportion is small. A real data application illustrates the use of the models studied.