Goodman contributed to the theory of scaling by including a category of intrinsically unscalable respondents in addition to the usual scale-type respondents. However, his formulation permits only error-free responses by respondents from the scale types. This paper presents new scaling models which have the properties that: (1) respondents in the scale types are subject to response errors; (2) a test of significance can be constructed to assist in deciding on the necessity for including an intrinsically unscalable class in the model; and (3) when an intrinsically unscalable class is not needed to explain the data, the model reduces to a probabilistic, rather than to a deterministic, form. Three data sets are analyzed with the new models and are used to illustrate stages of hypothesis testing.

A probabilistic model for the validation of behavioral hierarchies is presented. Estimation is by means of iterative convergence to maximum likelihood estimates, and two approaches to assessing the fit of the model to sample data are discussed. The relation of this general probabilistic model to other more restricted models which have been presented previously is explored and three cases of the general model are applied to exemplary data.

The method of scoring is used to obtain maximum likelihood estimates of the parameters in the White and Clark learning hierarchy validation model. From the estimate of the proportion of the population possessing only the superordinate skill in a pair of hierarchical skills, and its variance, the hypothesis of inclusion is tested. An illustrative example of the procedure is given.

This paper presents a strategy for pairwise assessment which may be used to evaluate the nature of both “prerequisite” and “transference” relations existing among a set of traits. This strategy is appropriate for use both within a confirmatory context, in which an attempt is made to establish the validity of some specified set of relations among traits, as well as within an exploratory context, in which a search is made for unconjectured prerequisite and transference relations existing between pairs of traits. Both uses of this strategy are based on a variety of latent class models which are representative of various possible relational states existing between pairs of traits. Thus, the nature of trait relations may be investigated through the use of statistical assessments of both absolute and relative fit attained by these models. An application is presented to exemplify how this strategy may be used within the exploratory context.