On Bayesian Testing of Additive Conjoint Measurement Axioms Using Synthetic Likelihood

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.     

A two-stage algorithm for assessing violations of additivity via axiomatic and numerical conjoint analysis

An algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described. The algorithm first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model. The axioms are examined at a more detailed level than has been previously done. Violations of the axioms are broken down into different types. Finally, a nonmetric scaling of the data can be done based on either or both of two different badness-of-fit scaling measures. The advantages of combining all of these features into one algorithm for improving the diagnostic value of axiomatic conjoint measurement in evaluating additivity are discussed.     

Cross-modal additive conjoint structures and psychophysical scale convergence

I investigated a variety of issues related to the measurement of the magnitude of psychological experience, especially the magnitude of sensations. Different groups of subjects made pair comparisons, magnitude estimations, and category judgments of the "total sensory magnitude" of light and sound stimuli presented conjointly. Another group judged the dissimilarity of pairs of conjoint stimuli. Various axioms, especially double cancellation, were tested on the resulting rank orders of conjoint stimuli. Judgements of the total magnitude of conjoint combinations of sound and light stimuli formed an additive conjoint structure. Dissimilarity judgments gave rise to a closely related lattice structure. Moreover, various scales of the individual attributes (loudness and brightness) calculated from the two types of judgments of the conjoint stimuli displayed substantial convergence, each scale for a given modality being linear with all other scales for that modality.     

Minimal conditions for additive conjoint measurement and qualitative probability

Axioms for additive conjoint measurement and qualitative probability are given. Representation theorems and uniqueness theorems are proved for structures that satisfy these axioms. Both Archimedean and nonarchimedean cases are considered. Approximations of infinite structures by sequences of finite structures are also considered.     

Theory and tests of the conjoint commutativity axiom for additive conjoint measurement

The empirical study of the axioms underlying additive conjoint measurement initially focused mostly on the double cancellation axiom. That axiom was shown to exhibit redundant features that made its statistical evaluation a major challenge. The special case of double cancellation where inequalities are replaced by indifferences–the Thomsen condition–turned out in the full axiomatic context to be equivalent to the double cancellation property but without exhibiting the redundancies of double cancellation. However, it too has some undesirable features when it comes to its empirical evaluation, the chief among them being a certain statistical asymmetry in estimates used to evaluate it, namely two interlocked hypotheses and a single conclusion. Nevertheless, thinking we had no choice, we evaluated the Thomsen condition for both loudness and brightness and, in agreement with other lines of research, we found more support for conjoint additivity than not. However, we commented on the difficulties we had encountered in evaluating it. Thus we sought a more symmetric replacement, which as Gigerenzer and Strube (1983) first noted, is found in the conjoint commutativity axiom proposed by Falmagne (1976, who called it the “commutative rule”). It turns out that, in the presence of the usual structural and other necessary assumptions of additive conjoint measurement, we can show that conjoint commutativity is equivalent to the Thomsen condition, a result that seems to have been overlooked in the literature. We subjected this property to empirical evaluation for both loudness and brightness. In contrast to Gigerenzer and Strube (1983), our data show support for the conjoint commutativity in both domains and thus for conjoint additivity.     

The Normal Distribution Derived from Qualitative Conditions

The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed. Copyright 2001 Academic Press.     

Nontransitive Decomposable Conjoint Measurement

Traditional models of conjoint measurement look for an additive representation of transitive preferences. They have been generalized in two directions. Nontransitive additive conjoint measurement models allow for nontransitive preferences while retaining the additivity feature of traditional models. Decomposable conjoint measurement models are transitive but replace additivity by a mere decomposability requirement. This paper presents generalizations of conjoint measurement models combining these two aspects. This allows us to propose a simple axiomatic treatment that shows the pure consequences of several cancellation conditions used in traditional models. These nontran- sitive decomposable conjoint measurement models encompass a large number of aggregation rules that have been introduced in the literature.     

The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement

The multinomial (Dirichlet) model, derived from de Finetti's concept of exchangeability, is proposed as a general Bayesian framework to test axioms on data, in particular, deterministic axioms characterizing theories of choice or measurement. For testing, the proposed framework does not require a deterministic axiom to be cast in a probabilistic form (e.g., casting deterministic transitivity as weak stochastic transitivity). The generality of this framework is demonstrated through empirical tests of 16 different axioms, including transitivity, consequence monotonicity, segregation, additivity of joint receipt, stochastic dominance, coalescing, restricted branch independence, double cancellation, triple cancellation, and the Thomsen condition. The model generalizes many previously proposed methods of axiom testing under measurement error, is analytically tractable, and provides a Bayesian framework for the random relation approach to probabilistic measurement (J. Math. Psychol. 40 (1996) 219). A hierarchical and nonparametric generalization of the model is discussed.     

Axiomatic and numerical conjoint measurement: An evaluation of diagnostic efficacy

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.Douglas Emery is now at the Krannert Graduate School of Management, Purdue University, West Lafayette, IN, on leave from the University of Calgary.     

The rasch model,the law of comparative judgment and additive conjoint measurement

Relationships between the Rasch model and both the law of comparative judgment and additive conjoint measurement are discussed. The distance between the ability of Persona and the difficult of Itemi is, in the Rasch model, the baseline value corresponding to the probability thata will respond correctly toi, where this probability is interpreted as the area under a logistic curve (which is substantially equivalent to the normal curve) and is thus an application of the law of comparative judgment. Under certain assumptions, the Rasch model is also a special case of additive conjoint measurement and, properly reinterpreted, may be usefully applied in contexts other than individual differences.     

A conjoint grassmann structure for testing the additivity of binocular color mixtures

A measurement structure for binocular color mixtures is developed, based on the principles that explain monocular matches of additive mixtures of color stimuli. Binocular color mixtures are created by presenting different stimuli to two eyes, such that they fuse and result in a single impression of color. The measurement structure makes it possible to compare binocular color mixtures with physical mixtures of color stimuli. The two types of mixture are characterized as compatible, if the binocular mixture creates a color, which is identical to the vectorial sum of the colors of the respective monocular components. The axioms of the measurement structure are experimental question, suitable for testing the addivity of binocular color mixtures.     

Relativistic effects in visual perception of real and apparent motion

Timing in the visual field is regarded as an additive conjoint measurement structure, psychophysical extension of stimulus path as an extensive measurement structure. These two sets of postulates lead to the derivation of an essential maximum for velocity perception. Apparent phi motion perception under strictly stationary stimulus conditions is described as relative motion of the first stimulus' psychophysical mapping with respect to a moving perceptual subsystem. The essential maximum of velocity modifies the relative motion postulate: relativistic dilatation of seen length of path is predicted. Testable properties of the model, comparison with experimental data from "real" motion and apparent phi motion perception are discussed.     

A new solution to the problem of finding all numerical solutions to ordered metric structures

A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous algorithms.This research was supported in part by NIGMS grant GM-01231 to the University of Michigan. Authors' names are in alphabetic order.     

Subjective evaluations of employer attributes by administration students

The paper is an application of the conjoint measurement methodology. An exploratory study was conducted of the trade-off relationships among four organizational attributes—dynamism, societal concern, functional/administrative emphasis, and relative size—as they relate to the selection of an employer by students of administration. Preference ranks of 36 employer concepts obtained from 86 graduate students of business, public, and hospital administration were analyzed according to the additive conjoint measurement model yielding implicit rank-ordering of the four organizational attributes as choice criteria. Irrespective of the program of study, dynamism and societal concern emerged as the most important determinants in employer selection, but differences existed among programs, year of study, and explicitly vs implicitly elicited weights.     

Lethality of suicide attempt rating scale

An 11-point (0-10) scale for measuring the degree of lethality of suicide attempts is presented. The scale has nine example "anchors" and uses the relative lethality of an extensive table of drugs. The psychometric studies show the scale to be of at least equal-interval, possibly even ratio measurement level. The equal-interval quality allows suicidal behaviors to be specified in numeric relationship to each other relieving us of the burden of using unquantified labels such as "mild," "moderate," and "serious." Having a standardized, commonly used reference scale should greatly facilitate our research and clinical efforts to communicate our findings. The scale can be used reliably by nonmedical personnel with no prior training.     

‘Additive difference’ models without additivity and subtractivity

This paper studies conjoint measurement models tolerating intransitivities that closely resemble Tversky's additive difference model while replacing additivity and subtractivity by mere decomposability requirements. We offer a complete axiomatic characterisation of these models without having recourse to unnecessary structural assumptions on the set of objects. This shows the pure consequences of several cancellation conditions that have often been used in the analysis of more traditional conjoint measurement models. Our models contain as particular cases many aggregation rules that have been proposed in the literature.     

The integration of motivational information: A conjoint measurement analysis

Within Anderson's (1974a, b, c, 1978b) information integration theory, the integration of motivational information was investigated by means of conjoint measurement techniques. Eighteen university students were asked to judge hypothetical co-students characterized by three features (intelligence, motivation and the extent to which they study) according to their chances to pass. Both rank order data and ratings were obtained. The orderings of most subjects could be represented very well by an additive model. A polynomial regression procedure was applied to determine the shape of the response function for the ratings. As this function was quite linear for all subjects, the ratings could be said to form an interval scale.     

The number of two-way tables satisfying certain additivity axioms

An expression is given for the number of ways of ranking the cells of an r by c factorial design so as to satisfy independence. For selected values of r and c, estimates are given of the number of rankings that satisfy both independence and double cancellation, and also of the number of rankings allowing an additive representation. These results may be used in at least two ways: first, in evaluating the probability of the satisfaction of certain measurement axioms by chance; and second, in placing a lower bound on the amount of information necessary to establish the ordering of the cells of a factorial design when it is known that these axioms are satisfied.