Nested models of relative judgment: Applications to a similarity averaging model

A theoretical analysis of the Eisler and Ekman (1959) model of similarity judgments for unidimensional continua is presented, based on a general model of relative judgment. This general model assumes that judgments are mediated by perceived relations of pairs of stimuli, that there exists a transformation of the judgmental response that is a function of the sensory ratio of the two stimuli, and that response bias operates in a multiplicative manner. Three structural conditions are presented, each imposing constraints on the structure of observed judgments. The structural conditions define threenested models of relative judgment, with the second a weakened version of the first, and the third a weakened version of the second. The special virtue of the general model is that it is applicable to a variety of judgmental tasks (e.g., ratio estimation, similarity, pair comparison), the key being derivation of theresponse transformation conforming to the structural conditions. The structural conditions thus constitute necessary conditions for several different judgmental models. The theory was first applied with success to ratio estimation judgments (Fagot, 1978), and this paper applies the general model to the Eisler and Ekman similarity “averaging” model. Empirical tests were carried out on published data for pitch, darkness, visual area, and heaviness judgments. Although the strong form of the model presented by Eisler and Ekman was rejected, weakened versions were generally supported by the data. These results were similar to those obtained for ratio estimation (Fagot, 1978), and are interpreted to be very promising for the general model of relative judgment.     

A theory of bidirectional judgments

This paper is concerned with a scaling theory for “bidirectional” judgments, for which the order of judgment is reversible, as in fractional and multiple ratio estimation judgments. With the assumption that judgments are mediated by perceived relations of pairs of stimuli, the theory is developed for judgments of comparison stimuli in relation to standards, taking explicit account of the location of the comparison stimulus relative to the standard. The theory of bidirectional judgments, based on a theory of relative judgment by Fagot (1978, 1979), entails a partial nesting of models characterized by a progressive weakening of the constraints placed on the structure of the data. The weakest model, the relative bias/directional standard (RBDS) model, allows each standard to have two biasing effects, depending on the location of the standard above or below the comparison stimulus. Tests of the theory were carried out on the ratio estimation of brightness and weight data of Engen and Levy (1955) and the part-sum estimation data of Goude (1962). Only the RBDS model was found acceptable for all three data sets     

Measurement and the mental map

In order to test between subtractive and ratio theories of stimulus comparison, judges were asked to estimate “ratios” and “differences” of easterliness and westerliness of U.S. cities. “Difference” judgments fit the subtractive model, and “ratio” judgments fit the ratio model. However, “ratios” and “differences” were monotonically related, contrary to the theory that judges compute both relations on a common scale. Results are consistent with the theory that there is but one operation for both “ratios” and “differences.” To assume that the single operation is a ratio requires the complex interpretation that easterliness and westerliness are nonlinearly related. A simpler interpretation is provided by a subtractive theory, in which all four types of judgments are monotonically related to subjective differences on a single cognitive map.     

Nonmetric tests of ratio vs. subtractive theories of stimulus comparison

Theories of stimulus comparison were tested by examining ordinal properties of data obtained with six scaling tasks. Subjects judged simple “ratios” or “differences” of stimulus pairs constructed from a factorial design. In four additional tasks, the same judges also compared relations between pairs of stimulus pairs, judging “ratios of ratios,” “ratios of differences,” “differences of ratios,” and “differences of differences.” The data were consistent with a subtractive theory, which asserts that two stimuli are compared by subtraction, regardless of the task, but that judges can compare two stimulus differences by either a ratio or a difference. All six tasks could be related by the subtractive theory using a single set of scale values. Other simple theories, including the theory that “ratio” judgments can be represented by a ratio model, could not reproduce the six rank orders (of the six sets of data) using a single set of scale values.     

Subtractive versus Ratio Model of “Fair” Allocation: Can the Group Level Analyses Be Misleading?

Two models of equity judgments are ratio and subtraction. Proponents of the former assume a linear relationship between the subjective feelings of equity and their overt expressions; those of the latter assume a monotonic relationship. Consequently, the ratio and subtractive rules are tested with the raw and monotonically rescaled data, respectively. I evaluated these two approaches with managers and students from India. Experiment 1 varied merit and pay of two persons and obtained judgments of difference between unfairness to them. Experiments 2 and 3 manipulated two inputs of two persons and studied “fair” reward for them. I analyzed both the raw and rescaled data at the group and individual levels. The group analyses supported the ratio model; the individual analyses showed that majority was consistent with the subtractive model. Discrepant results from these analyses were due to individual differences in the models employed and use of the response scale. Implications of the findings are discussed for cross-cultural and developmental research in “fair” allocation.     

Representations for biased numerical judgments

Models are presented that introduce bias parameters into the numerical representations for ratio and interval judgments. An interval bias model is considered that has the capability of accounting for both interval and ratio judgments. A contrasting ratio bias model—a simple generalization of the classical model that accepts ratio judgments at face value—is also considered, and both models are tested with brightness data. The ratio estimation data lead to the conclusion that the constraints placed on the structure of the data by the classical model are too strong. In terms of goodness-of-fit and certain theoretical criteria, the interval bias model accounts for the ratio estimation data better than the ratio bias model. The interval estimation data are supportive of recent efforts directed toward the development of direct interval scaling theory.     

The "consistency" controversy and the accuracy of personality judgments

Areas of clear and acknowledged disagreement in the personality literature's “consistency controversy” are surprisingly difficult to pin down. The present essay suggests that one basic and real disagreement nowadays is the admissability of subjective judgments of personality as data: Personologists are willing to use them, and situationists are not. Situationists generally regard judgments as so influenced by error as to be essentially unusable, and instead prefer direct measurements of specific behaviors. The relative uses and limitations of global judgments and specific measures are considered. It is concluded that any position that judgments reflect “nothing but” error is untenable. It is further concluded that each sort of data has distinct uses and limitations, provides a check on the other, and is indispensable.     

Moral relativism and deontic logic

If a native of India asserts “Killing cattle is wrong” and a Nebraskan asserts “Killing cattle is not wrong”, and both judgments agree with their respective moralities and both moralities are internally consistent, then the moral relativist says both judgments are fully correct. At this point relativism bifurcates. One branch which we call content relativism denies that the two people are contradicting each other. The idea is that the content of a moral judgment is a function of the overall moral point of view from which it proceeds. The second branch which we call truth value relativism affirms that the two judgments are contradictory. Truth value relativism appears to be logically incoherent. How can contradictory judgments be fully correct? For though there will be a sense of correctness in which each judgment is correct — namely by that of being correct relative to the morality relative to which each was expressed — if contradictory, the judgments cannot both be true, and thus cannot both be correct in this most basic sense of correctness. We defend truth value relativism against this sort of charge of logical incoherence by showing it can be accommodated by the existing semantical metatheories of deontic logic. Having done this we go on to argue that truth value relativism is the best version of relativism.     

Scale convergence as a criterion for rescaling: Information integration with difference,ratio, and averaging tasks

Ss lifted pairs of weights simultaneously, one in each hand, and judged either the difference, ratio, or average heaviness of the two weights. Data for the difference and ratio tasks were in general agreement with subtractive and ratio models, but the averaging data showed discrepancies from the constant-weight averaging model similar to those reported in previous psychophysical research. Rescaling was ruled out for the averaging data, because responses to pairs of equal weight were a linear function of subtractive model scale values derived from the difference task data. Scale values for the ratio and difference task data were related exponentially, as were the responses to the pairs, consistent with Torgerson’s conjecture that Ss do not distinguish “differences” from “ratios.” They appear to use the same composition rule but different output functions, depending on the procedures for responding. The scale convergence criterion can thus prevent inappropriate rescaling when a model fails and can dictate rescaling even when a model fits.     

Metacognition in psychophysical judgment: An unfolding view of comparative judgments of mental workload

An experiment is reported in which it was found that when subjects were required to indicate which of two visual extents was more difficult to categorize as “long” or “short,” they executed these categorizations and then measured the distance of the representation of each stimulus from the long-short category boundary; the stimulus nearer the boundary was judged to be the more difficult. When they were requested to indicate which was easier to categorize, they selected the alternative that was farther. Coombs’s theory of data (1952, 1964) and his unfolding theory of preferential choice (1950, 1964) provided the conceptualization of metacognition in this psychophysical task context. Strong support for the probabilisitic version of unfolding theory was obtained from the observed selective effects of laterality on the levels of stochastic transitivity attained for various classes of triples and the reliably longer times for comparisons with bilateral pairs than with unilateral pairs. The semantic congruity effects obtained, together with the changes in the form of the relationship between probability and response time as a function of practice, can be best accounted for by an evidence accrual theory in which the distances from the active reference point are measured and compared with a criterion on each evidence accrual. No support is provided for the view that propositionally based semantic “ease”- “difficulty” codes serve as the basis for these metacognitive comparative judgments of ease and difficulty.     

A comparison of two models for performance under fixed interval schedules of reinforcement

Two models are proposed for responding under fixed-interval schedules of reinforcement. The first model is a Poisson model and seems suitable for situations in which responding produces a classical “FI scallop”. A second model is then developed to describe “break and run” performance, which is also known to occur under some Fixed Interval schedules. The models do not however give any indication of the circumstances under which a particular mode of responding should arise. A comparison of the models to a small set of data collected from rats performing under an FI 60 sec schedule indicates that for the data considered, the second model (a State model) produced by far the best fit.     

A model for realism of confidence judgments: Implications for underconfidence in sensory discrimination

In a recent issue of this journal, Björkman, Juslin, and Winman (1993) presented a model of the calibration of subjective confidence judgments for sensory discrimination which they called “subjective distance theory.” They proposed that there was a robust underconfidence bias in such judgments, that the model predicted such a bias, and that two different models were needed for the calibration of subjective confidence for cognitive judgments and for sensory ones. This paper addresses issues they raised. It points out that they have not presented a new model, but rather a portion of a more general one, the “decision-variable partition model” originally proposed in Ferrell and McGoey (1980). This paper explores properties of the model and shows, contrary to Björkman, Juslin, and Winman’s hypotheses, that the model does not predict under-confidence, that the “hard-easy effect” can be observed with sensory discriminations, and that the model fits not only sensory, but also cognitive judgments.     

Speed and accuracy of same and different responses in perceptual matching

Models of the visual perceptual matching process are evaluated in two experiments in which the relative speeds of “same” and “different” responses were manipulated. In the first experiment, subjects were tested under two bias conditions: they were instructed to respond “same” only when sure or to respond “different” only when sure. Such bias is found to have a very large effect on the speed and accuracy of “same” and “different” responses, changing “same” responses from being faster than “different” responses (the usual result) in the sure “different” condition to being slower in the sure “same” condition. In the second experiment, the relative speed of “same” responses was slowed significantly by the addition of difficult “different” judgments. These results are used to argue against models that contain an identity matcher and against models that have separate components to account for errors and reaction time. We also point out that, although it is well known that the relative speeds of positive and negative responses are subject to bias manipulations, this fact has been ignored by many researchers in developing models in which the reaction time difference between positive and negative responses is used as a measure of a stage of processing.     

“Ratios” and “differences” in perceived sweetness intensity

For a number of perceptual continua, it has been shown in previous studies that subjects use only one quantitative comparison between two sensory impressions of a pair of stimuli, irrespective of whether they are instructed to judge “ratios” or “differences”. This comparison can be described by algebraic subtraction. The present study was designed to investigate whether this one-operation theory for psychophysical judgment also applied to the sensory continuum of sweetness. Subjects were presented with pairs of fructose solutions, and judged “ratios” of, or “differences" in, perceived sweetness intensities. The pairs were constructed on the basis of a factorial judgment design. The results showed that the reported “differences” could be adequately described by a difference response model, and that the reported “ratios” could be adequately described by a ratio response model. However, the reported “ratios” and reported “differences” were monotonically related, and the marginal means of the log-transformed response matrix of “ratios” were a linear function of the marginal means of the response matrix of “differences”. These results are incompatible with the notion that subjects judged differences when instructed to judge “differences”, but ratios when instructed to judge “ratios”. The consistency of the ratio response model with “ratio” judgments is probably caused by a comparative operation based on “differences” in combination with an exponential response output function. It may be concluded that subjects judge only “differences”, and not “ratios”, between perceived sweetness intensities.     

Comparative judgments of distal size: A chronometric analysis

Observers were required to make comparative judgments of the distal sizes of squares at various perceived distances in a pictorial array. It was predicted that observers would normalize distance prior to judgment. Chronometric analyses indicated that the time to make “same” judgments increased systematically with the relative distance of the two stimuli. The time required to make “different” judgments depended on the nature of the difference. When the stimuli differed in proximal size, distal size, and distance, response time increased with distance ratio. However, when the stimuli differed in distal size and distance but not proximal size, response time decreased with distance ratio. In addition, when the stimuli differed in both distal and proximal size but not distance, RT decreased with size ratio. These results are consistent with a class of models that incorporate distance normalization into the comparative size-judging process. These and alternative models are discussed.     

Tests of product and additive scaling axioms

A product axiom for ratio judgments and an additive axiom for difference judgments were tested for six Ss judging brightness. The product axiom was rejected for all six Ss whereas the additive axiom was accepted for two Ss. Power functions did not fit the data well, although somewhat better for difference judgments than ratio judgments. A two-stage model fitted somewhat better, but failed to satisfy one important implication of that model, previously confirmed for lifted weights. The data gave stronger support for a theory of brightness estimation based on an additive axiom for difference judgments than a product axiom based on ratio judgments.     

Repeated magnitude estimations with a variable standard: Sequential effects and other properties

Twelve Ss made magnitude estimations of the loudness of each one of a sequence of pure tones according to the rule R(N) = R(N - 1) · [S(N)/S(N - 1)], where R(N) is the response on Trial N, R(N - 1) is the response on Trial N - 1, and S(N)/S(N - 1) is the judged ratio of the “loudness” of the pure tone presented on Trial N to that of the pure tone presented on Trial N - 1. It was found that these magnitude estimations were assimilated toward the immediately preceding stimuli as far as five trials back in the sequence of stimuli. In addition, ratio judgments were consistently asymmetric and the data displayed a form of “time order error.” In all cases, there are similar effects displayed in category judgment data. These and other data imply that at least some kinds of magnitude estimations may involve a judgment of the “difference” or “distance” between pairs of stimuli as a first step in the production of the response required by the judgment situation.     

Augmenting Data With Published Results in Bayesian Linear Regression

In most research, linear regression analyses are performed without taking into account published results (i.e., reported summary statistics) of similar previous studies. Although the prior density in Bayesian linear regression could accommodate such prior knowledge, formal models for doing so are absent from the literature. The goal of this article is therefore to develop a Bayesian model in which a linear regression analysis on current data is augmented with the reported regression coefficients (and standard errors) of previous studies. Two versions of this model are presented. The first version incorporates previous studies through the prior density and is applicable when the current and all previous studies are exchangeable. The second version models all studies in a hierarchical structure and is applicable when studies are not exchangeable. Both versions of the model are assessed using simulation studies. Performance for each in estimating the regression coefficients is consistently superior to using current data alone and is close to that of an equivalent model that uses the data from previous studies rather than reported regression coefficients. Overall the results show that augmenting data with results from previous studies is viable and yields significant improvements in the parameter estimation.     

Reaction times and error rates for judging nominal identity of letter strings

Most theories of “same”-“different” judgments predict that “same” responses should be at least as slow as “different” responses. However, the contrary has often been found. To explain this, a two-processor model has been proposed. In this model, a fast processor and a slow processor operate simultaneously. “Same” responses are initiated by whichever processor first indicates that the stimuli are “same,” whereas “different” responses are initiated only by the slow processor. In the experiment reported here, Ss judged whether two successively presented letter strings were nominally “same” or “different.” It was expected that the fast processor would be incapable of making nominal identity judgments. Thus, both “same” and “different” responses would be initiated by the slow processor. Consequently, “same” responses should have been slower than “different” responses. However, this did not occur. This finding casts doubt upon, but does not disprove, the two-processor model.