A formal framework for measuring change in sets of dichotomous data is developed and implications of the principle of specific objectivity of results within this framework are investigated. Building upon the concept of specific objectivity as introduced by G. Rasch, three equivalent formal definitions of that postulate are given, and it is shown that they lead to latent additivity of the parametric structure. If, in addition, the observations are assumed to be locally independent realizations of Bernoulli variables, a family of models follows necessarily which are isomorphic to a logistic model with additive parameters, determining an interval scale for latent trait measurement and a ratio scale for quantifying change. Adding the further assumption of generalizability over subsets of items from a given universe yields a logistic model which allows a multidimensional description of individual differences and a quantitative assessment of treatment effects; as a special case, a unidimensional parameterization is introduced also and a unidimensional latent trait model for change is derived. As a side result, the relationship between specific objectivity and additive conjoint measurement is clarified.

The axioms of additive conjoint measurement provide a means of testing the hypothesis that testing data can be placed onto a scale with equal-interval properties. However, the axioms are difficult to verify given that item responses may be subject to measurement error. A Bayesian method exists for imposing order restrictions from additive conjoint measurement while estimating the probability of a correct response. In this study an improved version of that methodology is evaluated via simulation. The approach is then applied to data from a reading assessment intentionally designed to support an equal-interval scaling.

Synthetic data are used to examine how well axiomatic and numerical conjoint measurement methods, individually and comparatively, recover simple polynomial generators in three dimensions. The study illustrates extensions of numerical conjoint measurement (NCM) to identify and model distributive and dual-distributive, in addition to the usual additive, data structures. It was found that while minimum STRESS was the criterion of fit, another statistic, predictive capability, provided a better diagnosis of the known generating model. That NCM methods were able to better identify generating models conflicts with Krantz and Tversky's assertion that, in general, the direct axiom tests provide a more powerful diagnostic test between alternative composition rules than does evaluation of numerical correspondence. For all methods, dual-distributive models are most difficult to recover, while consistent with past studies, the additive model is the most robust of the fitted models.

This article introduces a Bayesian method for testing the axioms of additive conjoint measurement. The method is based on an importance sampling algorithm that performs likelihood-free, approximate Bayesian inference using a synthetic likelihood to overcome the analytical intractability of this testing problem. This new method improves upon previous methods because it provides an omnibus test of the entire hierarchy of cancellation axioms, beyond double cancellation. It does so while accounting for the posterior uncertainty that is inherent in the empirical orderings that are implied by these axioms, together. The new method is illustrated through a test of the cancellation axioms on a classic survey data set, and through the analysis of simulated data.

In pairwise multidimensional scaling, a spatial representation for a set of objects is determined from comparisons of the dissimilarity of any two objects drawn from the set to the dissimilarity of other pairs of objects drawn from that set. In pairwise conjoint scaling, comparisons among the joint effects produced by pairs of objects, where the objects in a pair are drawn from separate sets, are used to determine numerical representations for the objects in each set. Monte Carlo simulations of both pairwise dissimilarities and pairwise conjoint effects show that Johnson's algorithm can provide good metric recovery in the presence of high levels of error even when only a small percentage of the complete set of pairwise comparisons are tested.

This paper illustrates how to apply the RECIPE design to evaluate multiattributejudgment, reporting an experiment in which participants judged intentions toreceive a new vaccine against COVID-19. The attributes varied were Price of thevaccine, Risks of side effects as reported in trials, and Effectiveness of thevaccine in preventing COVID. The RECIPE design is a union of factorial designsin which each of three attributes is presented alone, in pairs with each of theother attributes, and in a complete factorial with all other information.Consistent with previous research with analogous judgment tasks, the additiveand relative weight averaging models with constant weights could be rejected infavor of a configural weight averaging model in which the lowest-valuedattribute receives additional weight. That is, people are unlikely to acceptvaccination if Price is too high, Risk is too high, or Effectiveness is too low.The attribute with the greatest weight was Effectiveness, followed by Risk ofside-effects, and Price carried the least weight.