Biased extensive measurement: The general case

We develop a theory of biased extensive measurement which allows us to prove the existence of a ratio-scale without transitivity of indifference and with a property of homothetic invariance weaker than independence. These representations, which cover the cases of interval orders and of semiorders, reveal a unique biasing function smaller or equal to 1 that distorts extensive measurement and explains departures from its standard axioms. We interpret this biasing function as characterizing the qualitative influence of the underlying measurement process and we show that it induces a proportional indifference threshold.     

Analysis of multinomial models under inequality constraints: Applications to measurement theory

Multinomial random variables are used across many disciplines to model categorical outcomes. Under this framework, investigators often use a likelihood ratio test to determine goodness-of-fit. If the permissible parameter space of such models is defined by inequality constraints, then the maximum likelihood estimator may lie on the boundary of the parameter space. Under this condition, the asymptotic distribution of the likelihood ratio test is no longer a simple χ² distribution. This article summarizes recent developments in the constrained inference literature as they pertain to the testing of multinomial random variables, and extends existing results by considering the case of jointly independent mutinomial random variables of varying categorical size. This article provides an application of this methodology to axiomatic measurement theory as a means of evaluating properly operationalized measurement axioms. This article generalizes Iverson and Falmagne’s [Iverson, G. J. & Falmagne, J. C. (1985). Statistical issues in measurement. Mathematical Social Sciences, 10, 131-153] seminal work on the empirical evaluation of measurement axioms and provides a classical counterpart to Myung, Karabatsos, and Iverson’s [Myung, J. I., Karabatsos, G. & Iverson, G. J. (2005). A Bayesian approach to testing decision making axioms. Journal of Mathematical Psychology, 49, 205-225] Bayesian methodology on the same topic.     

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.     

A method of weighing qualitative preference axioms

A method is developed for determining the absolute and relative strengths of qualitative preference axioms in normative Bayesian decision theory. These strengths are calculated for the three most common qualitative axioms; transitivity, the sure-thing principle, and dominance. The relative strength of the latter two axioms with respect to transitivity is calculated for special cases, and a bound is derived which is applicable to a larger class of decision problems. Possible implications of this theoretical work for decision heuristics are discussed.     

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.     

Biased extensive measurement: The homogeneous case

In the homogeneous case of one type of objects, we prove the existence of an additive scale unique up to a positive scaling transformation without transitivity of indifference and with a property of homothetic invariance weaker than monotonicity. The representation, which is a particular case of a semiorder representation, reveals a unique positive factor α?1 that biases extensive structures and explains departures from these standard axioms of extensive measurement (α=1). We interpret α as characterizing the qualitative influence of the underlying measurement process and we show that it induces a proportional indifference threshold.     

Linear Measurement Models-Axiomatizations and Axiomatizability

This paper examines, in the scope of representational measurement theory, different axiomatizations and axiomatizability of linear and bilinear representations of ordinal data contexts in real vector spaces. The representation theorems proved in this paper are modifications and generalizations of Scott's characterization of finite linear measurement models. The advantage of these representation theorems is that they use only finitely many axioms, the number of which depends on the size of the given ordinal data context. Concerning the axiomatizability, it is proved by model-theoretic methods that finite linear measurement models cannot be axiomatized by a finite set of first order axioms. Copyright 2000 Academic Press.     

Evaluating the Equal-Interval Hypothesis with Test Score Scales

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.     

Contextualized Personality Questionnaires: A Case for Copulas in Structural Equation Models for Categorical Data

For structural equation models (SEMs) with categorical data, correlated measurement residuals are not easily implemented. The problem lies mainly in the absence of a categorical analogue to the multivariate normal distribution and the absence of closed form formulas in SEMs for categorical data. We present a novel technique to handle measurement residuals that keeps the attractive SEM mainframe intact yet adds flexibility in dependence modeling without excessive computational burden. The technique is based upon the concept of copula functions and is introduced with a data set of ordinal responses originating from a contextualized personality study on aggression. Focus is on models arising in a multitrait-multimethod context, where the flexibility in dependence structures allows for method effects that can vary across the latent trait dimension. The empirical application illustrates that ignoring design-implied correlated measurement residuals can potentially influence study results and conclusions in both a quantitative as well as a qualitative way. Model parameter estimates can be biased, but more important, model inferences can be heavily distorted.     

Łukasiewicz logic and the foundations of measurement

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable us to treat formally errors arising in nominal and ordinal measurements.     

Extensive measurement with incomparability

Standard theories of extensive measurement assume that the objects to be measured form a complete order with respect to the relevant property. In this paper, representation and uniqueness theorems are presented for a theory that departs radically from this completeness assumption. It is first shown that any quasi-order on a countable set can be represented by vectors of real numbers. If such an order is supplemented by a concatenation operator, yielding a relational structure that satisfies a set of axioms similar to the standard axioms for an extensive structure, we obtain a scale possessing the crucial properties of a ratio scale. Incomparability is thus compatible with extensive measurement. The paper ends with a brief discussion on some possible applications and developments of this result.     

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.     

On axiomatizations of public announcement logic

In the literature, different axiomatizations of Public Announcement Logic (PAL) have been proposed. Most of these axiomatizations share a “core set” of the so-called “reduction axioms”. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that the proof system based on this core set of axioms does not completely axiomatize PAL  without additional axioms and rules. In fact, many of the intuitive axioms and rules we took for granted could not be derived from the core set. Moreover, we also propose and advocate an alternative yet meaningful axiomatization of PAL  without the reduction axioms. The completeness is proved directly by a detour method using the canonical model where announcements are treated as merely labels for modalities as in normal modal logics. This new axiomatization and its completeness proof may sharpen our understanding of PAL  and can be adapted to other dynamic epistemic logics.     

Coherent choice functions under uncertainty

We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility.     

A Stale Challenge to the Philosophy of Science

In his article “Is psychology based on a methodological error?” and based on a quite convincing empirical basis, Michael Schwarz offers a methodological critique of one of mainstream psychology’s key test theoretical axioms, i.e., that of the in principle normal distribution of personality variables. It is characteristic of this paper—and at first seems to be a strength of it—that the author positions his critique within a frame of philosophy of science, particularly positioning himself in the tradition of Karl Popper’s critical rationalism. When scrutinizing Schwarz’s arguments, however, we find Schwarz’s critique profound only as an immanent critique of test theoretical axioms. We raise doubts, however, as to Schwarz’s alleged ‘challenge’ to the philosophy of science because the author not at all seems to be in touch with the state of the art of contemporary philosophy of science. Above all, we question the universalist undercurrent that Schwarz’s ‘bio-psycho-social model’ of human judgment boils down to. In contrast to such position, we close our commentary with a plea for a context- and culture sensitive philosophy of science.     

Bounded Revision: Two-Dimensional Belief Change Between Conservative and Moderate Revision

This paper presents the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as far as (and just a little further than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision does not depend on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche–Pearl (DP) axioms for iterated belief change. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result. Bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artif Intell 157:5–47, 2004) that satisfies neither the SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief.     

Scale Imposition as Quantitative Alchemy: Studies on the Transitivity of Neuroticism Ratings

It is common practice in psychology to devise “measurement” procedures by imposing rating scales (e.g., Likert items) onto phenomena and treating the values they produce as quantities. The validity of these procedures goes untested. Validity checks are instead performed on sets of these measurement procedures (i.e., multi-item scales). We present results from three studies suggesting that people cannot be assumed to preserve transitivity when comparing themselves and others on NEO Neuroticism-domain trait items. As transitivity is one of the fundamental axioms of quantitative measurement, these studies challenge the validity of Neuroticism scales at the level of individual scale items.     

Stevens’ Direct Scaling Methods and the Uniqueness Problem

Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant's sensations. Although Stevens' methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens' direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided. Requests for reprints should be sent to thomas.