Two operations for "ratios" and "differences" of distances on the mental map.

Abstract

Ss judged "ratios of distances" and "differences of distances" between pairs of U.S. cities. Results fit the theory that Ss used two comparison processes as instructed. A ratio scale of distances between cities was constructed from the 2 rank orders. From this scale, an interval scale of the city locations on an east-west continuum was derived. This scale agrees with the subtractive model fit to "ratios" and "differences" of easterliness and westerliness, and it also agrees with multidimensional scaling of judged distances between cities. These findings are consistent with the theory that Ss use subtraction when instructed to judge either "ratios" or "differences," but that they can use both ratio and difference operations when the stimuli (in this case, distances) constitute a ratio scale on the subjective continuum. We thank Barbara A. Mellers, Allen Parducci, and anonymous reviewers for helpful suggestions.

Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production

“…The latter issue has been investigated extensively (for reviews, see Birnbaum, 1982Birnbaum, , 1990, but with mixed results. Whereas for most continua (such as loudness or heaviness), observers seem to employ a single operation whether instructed to judge differences or ratios (e.g., Mellers, Davis, & Birnbaum, 1984;Schneider, Parker, Farrell, & Kanow, 1976), there are occasional reports ofdifferent operations' being employed under these instructions (e.g., Birnbaum, Anderson, & Hynan, 1989;Popper, Parker, & Galanter, 1986). Narens's conceptualization, especially the inherent distinction between numerals and numbers, may shed new light on the issue of comparing direct ratio estimation with difference estima-tion both by providing a thorough theoretical basis (the foundations of which are developed in Narens, 1997) and by suggesting new methodologies to tackle the problem.…”

The role of part—whole information in reasoning about relative size

“…These judgments required participants to compare the distance between the members of one pair with the distance between the two members of a second pair. They were similar to judgments of triples because they cannot be solved by knowing the ordering of each pair but must be solved by determining the difference in the sizes of the two members of each pair (Birnbaum, Anderson, & Hynan, 1989;Petrusic, Baranski, & Kennedy, 1998). Comparison difficulty was manipulated by varying the differences in the size of the members ofone pair relative to the differences in the sizes of a second pair.…”

What are we estimating when we fit Stevens’ power law?

“…Indeed, in the approach the ratio model was often connected to a subtractive model for category scaling, relating overt rated differences to subjective differences. Various analyses comparing the two models demonstrated that while magnitude estimations and category ratings of the same stimuli are not linearly related, nevertheless they were monotonically related for a number of continua (e.g., Birnbaum, 1980Birnbaum, , 1982Birnbaum, Anderson, & Hynan, 1989;Elmasian, 1977 andVeit, 1974). This is quite interesting also from the perspective of axiomatic separable representations: on the one hand, it further shows the relevance of the behavioral properties underlying separable representations to address issues which go beyond Stevens' power law; 7 on the other hand, it suggests that the machinery used below for the empirical analysis of a ratio model may in principle be also applied to a subtractive model.…”