A revised law of comparative judgment

In contrast to Thurstone's Law of Comparative Judgment, a model in which a comparison pair and its complement are assumed to give rise to two different distributions of differences is considered. The consequences of this revised model on scaling problems is developed.The research in this article was supported jointly by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology.     

Reliability for the law of comparative judgment

A variance-components analysis is presented for paired comparisons in terms of three components:s, the scale value of the stimuli;d, a deviation from the linear model specified by the law of comparative judgment; andb, a binomial error component. Estimates are given for each of the three variances, _s ² , _d ² , and _b ² . Several coefficients, analogous to reliability coefficients, based on these three variances are indicated. The techniques are illustrated in a replicated comparison of handwriting specimens.This research was jointly supported in part by Princeton University, the Office of Naval Research under contract Nonr-1858(15), and the National Science Foundation under grant NSF G-642, and in part by Educational Testing Service. Reproduction in whole or in part is permitted for any purpose of the United States Government.Thanks are due to Ledyard Tucker and Frederic Lord for valuable suggestions on the development presented here.     

A solution for case iii of the law of comparative judgment

A solution for Case III of the Law of Comparative Judgment, modeled after Thurstone's solution for Case IV but eliminating the restrictive assumption of relatively equal discriminal dispersions, is developed.     

A least-squares solution for case IV of the law of comparative judgment

Case IV of Thurstone's Law of Comparative Judgment is displayed as a system of homogeneous linear equations for which a least-squares solution is presented, using various conditional equations which fix the origin and the unit of measurement. The computational load is, however, quite heavy.     

Ambiguous comparative judgment: Fuzzy set model and data analysis

Abstract:  This paper proposed two types of fuzzy set models for ambiguous comparative judgments, which did not always hold transitivity and comparability properties. The first type of model was a fuzzy theoretical extension of the additive difference model for preference that was used to explain ambiguous preference strength. The second was a fuzzy logic model for explaining ambiguous preference in which preference strength was bounded, such as a probability measure. In both models, multi-attribute weighting parameters and all attribute values were assumed to be asymmetric fuzzy L-R numbers. For each model, a method of parameter estimation using fuzzy regression analysis was proposed. Numerical examples were also provided for comparison. Finally, the theoretical and practical implications of the proposed models were discussed.     

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.     

The law of comparative judgment in the successive intervals and graphic rating scale methods

The law of comparative judgment is applied to the successive intervals and graphic rating scale methods. A procedure for estimating the modal discriminal process and discriminal dispersion of the stimuli, as well as the value of the boundaries of the intervals on the continuum, is given. From the estimated values it is possible to determine the theoretical proportions and to compare them with the actual experimental proportions. The agreement between these values is an indication of the adequacy of the assumptions made.     

An eye fixation analysis of choice and judgment with multiattribute stimuli

The present paper examined the selection of processing heuristics for choice and judgment across a range of stimuli. Eye fixations were monitored while six subjects made choices or judgments of pairs of gambles, vacations, or gifts. Each stimulus was represented by three attributes that varied in the amount of dimensional interdependency. The two heuristics of interest were dimensional and holistic evaluation. Each manifested a characteristic pattern of eye fixations. Dimensional evaluation required alternating fixations from a single attribute of one stimulus to the same attribute of the other stimulus. Holistic evaluation was characterized by transitions from attribute to attribute within a single stimulus. The results demonstrated that when the stimulus attributes were either interdependent (gambles) or dissimilar (gifts), the processing heuristic was determined by stimulus characteristics. When the stimulus dimensions were neither interdependent nor dissimilar (vacations), the selection of a processing strategy was determined by the prescribed task. This study suggests that any global theory of choice or judgment must be validated over a wide range of stimuli.     

A perspective on judgment and choice: mapping bounded rationality

Early studies of intuitive judgment and decision making conducted with the late Amos Tversky are reviewed in the context of two related concepts: an analysis of accessibility, the ease with which thoughts come to mind; a distinction between effortless intuition and deliberate reasoning. Intuitive thoughts, like percepts, are highly accessible. Determinants and consequences of accessibility help explain the central results of prospect theory, framing effects, the heuristic process of attribute substitution, and the characteristic biases that result from the substitution of nonextensional for extensional attributes. Variations in the accessibility of rules explain the occasional corrections of intuitive judgments. The study of biases is compatible with a view of intuitive thinking and decision making as generally skilled and successful.     

A general model for preferential and triadic choice in terms of centralF distribution functions

A model for preferential and triadic choice is derived in terms of weighted sums of centralF distribution functions. This model is a probabilistic generalization of Coombs' (1964) unfolding model and special cases, such as the model of Zinnes and Griggs (1974), can be derived easily from it. This new form extends previous work by Mullen and Ennis (1991) and provides more insight into the same problem that they discussed.     

Discounting: A practical guide to multilevel analysis of choice data

Multilevel modeling provides the ability to simultaneously evaluate the discounting of individuals and groups by examining choices between smaller sooner and larger later rewards. A multilevel logistic regression approach is advocated in which sensitivity to relative reward magnitude and relative delay are considered as separate contributors to choice. Examples of how to fit choice data using multilevel logistic models are provided to help researchers in the adoption of these methods.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

A constructive process view of decision making: Multiple strategies in judgment and choice

A viewpoint that has recently emerged in decision research is that preferences for objects of any complexity are often constructed — not merely revealed — in generating a response to a judgement or choice task. This paper reviews a program of research that traces the constructiveness of preferences to the use of multiple strategies in decision making, contingent on task demands. It is argued that individuals often build strategies opportunistically, changing their processing on the spot depending upon the information they encounter during the course of solving the decision problem.     

A note on the analysis of subjective judgment matrices

Saaty (1977–1983) presents an eigenvector (EV) procedure for analyzing matrices of subjective estimates of the utility of one entity relative to another. The procedure is an especially effective tool for analyzing hierarchical problems where the dependence of the entities at one level on the entities in adjacent levels is estimated subjectively. Despite the absence of a formal proof that the procedure has desirable qualities as an estimator of the underlying relative utilities, the process has gained an active following. This paper derives a comparable estimate, the geometric mean (GM) vector (also known as the logarithmic least squares method or LLSM), that can be applied to hierarchical problems in exactly the same way but is developed from statistical considerations. It is shown to be optimal when the judge's errors are multiplicative with a lognormal distribution. The GM shares the desirable qualities of the EV and is preferable to it in several important respects.