Comparative judgments with numerical reference points

A model of subjective magnitude comparisons is explored, which assumes that subjects compare symbolic stimulus magnitudes with respect to a reference point. The reference point may be established implicitly by the question (e.g., “Which is larger?” vs “Which is smaller?”) or be presented explicitly (e.g., “Choose the stimulus closer to X.”). The model was tested in five experiments in which subjects judged which of two comparison digits was closer to (or further from) a reference digit. Regression analyses in three experiments revealed that reaction time depended on the ratio of the distances from the comparison items to the reference point. The other two experiments provided evidence that subjects can strategically vary the processes by which they compare stimuli to a reference point. The results indicated that subjects can perform various types of “analog arithmetic” using either the linear number scale or a nonlinear scale of subjective digit magnitude.     

Cognitive processes in stereotype formation: the role of correct contingency learning for biased group judgments

Three studies investigated contingency learning and stereotype formation in a scenario about group membership and behavior with a confounding context factor. The studies tested predictions from theoretical accounts of biased group judgments in terms of simplistic reasoning, parallel distributed memory, and pseudocontingencies. Study 1 revealed a positive correlation between erroneous stereotype formation and learning of the true contingencies with the confounding factor. Study 2 showed that a focus manipulation during encoding moderated the correlation between stereotype formation and contingency learning but not the strength of the erroneous stereotype. Study 3 used a quasiexperimental comparison between participants with biased versus unbiased group judgments and extended the findings of a positive relation between stereotype formation and contingency learning. The results support an explanation of biased group judgments by pseudocontingencies; that is, unwarranted inferences from accurately perceived bivariate correlations in complex environments.     

The ideologically objectionable premise model: Predicting biased political judgments on the left and right

This paper introduces the ideologically objectionable premise model (IOPM), which predicts that biased political judgments will emerge on both the political left and right, but only when the premise of a judgment is not ideologically objectionable to the perceiver. The IOPM generates three hypothesized patterns of bias: biases among both those on the left and right, bias only among those on the right, and bias only among those on the left. These hypotheses were tested within the context of the dual process motivational model of ideological attitudes (DPM; Duckitt, 2001), which posits that right-wing authoritarianism (RWA) and social dominance orientation (SDO) are related but distinct ideological attitudes. Across two studies, all three IOPM hypotheses were tested and supported on the RWA ideological attitude dimension, and two of the three IOPM hypotheses were tested and supported on the SDO dimension. These findings indicate that the context of the judgment is an important determinant of whether biases emerge in political judgment.     

Ordinal judgments of numerical symbols by macaques (Macaca mulatta)

Two rhesus monkeys (Macaca mulatta) learned that the arabic numerals 0 through 9 represented corresponding quantities of food pellets. By manipulating a joystick, the monkeys were able to make a selection of paired numerals presented on a computer screen. Although the monkeys received a corresponding number of pellets even if the lesser of the two numerals was selected, they learned generally to choose the numeral of greatest value even when pellet delivery was made arrhythmic. In subsequent tests, they chose the numerals of greater value when presented in novel combinations or in random arrays of up to five numerals. Thus, the monkeys made ordinal judgments of numerical symbols in accordance with their absolute or relative values.     

Directional verbal probabilities: inconsistencies between preferential judgments and numerical meanings

Verbal probability expressions (e.g., it is possible or doubtful) convey not only vague numerical meanings (i.e., probability) but also semantic functions, called directionality. We performed two experiments to examine whether preferential judgments are consistent with numerical meanings of verbal probabilities regardless of directionality. The results showed that because of the effects of directionality, perceived degrees of certainty for verbal probabilities differed between a binary choice and a numerical translation (Experiment 1), and decisions based on a verbal probability do not correspond to those based on a numerical translation for verbal probabilities (Experiment 2). These findings suggest that directionality of verbal probabilities is an independent feature from numerical meanings; hence numerical meanings of verbal probability alone remain insufficient to explain the effects of directionality on preferential judgments.     

Representations of numerical and non‐numerical magnitude both contribute to mathematical competence in children

A growing body of evidence suggests that non‐symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non‐numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non‐symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5‐ and 6‐year‐old children using comparison tasks of non‐symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non‐numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non‐symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non‐numerical magnitude.     

Children's numerical skill and judgments of confidence in estimation

This study investigated how children's fluent use of a basic numerical skill is related to their perceptions of confidence on a task of quantitative estimation. One hundred four 10th graders (age 15.7 years) were categorized according to level of numerical fluency as assessed by timed tasks of counting in various nonunity increments. They then made estimates of large numbers of dots and rated how confident they were on each trial. The estimation task varied according to two within-subject factors of task difficulty. It was found that counting skill was related not only to accuracy in estimating, but also to the appropriateness of the children's confidence in their responses. Skillful children made judgments that corresponded with actual task difficulty. A derived calibration score corroborated ANOVA findings of the interaction of skill level and task difficulty on confidence ratings. It also revealed that while girls were less confident than boys, they were actually more realistic in their judgments. Findings are discussed in terms of (a) the effect of stimulus knowledge on awareness of task difficulty and (b) sex differences in achievement-related expectancies.     

The role of subtractions and comparisons in comparative judgments involving numerical reference points

Experiments in which subjects are asked to decide which of two digits is closer in magnitude to a third raise problems for many theories of linear orders. Holyoak (1978), for example, performed a number of these reference point experiments and concluded that they posed serious difficulties for a number of leading models. In their place, he offered the distance ratio model in which the ease of the decision in a reference point task is a function of the ratio of the distances between each stimulus and the reference point. In the present article, three experiments are presented that bear on the adequacy of Holyoak's position. In the first two studies, we present evidence that an important assumption of the distance ratio model is incorrect. In the third experiment, we compare the empirical adequacy of the distance ratio model with our own subtraction model. This model treats the reference point task as a concatenation of two subtractions and a simple digit comparison. This comparison operation is equivalent to the magnitude comparison required in standard linear order experiments. Overall, the subtraction model gives a somewhat better account than the distance ratio.     

Grouping information for judgments

Models of cue weighting in judgment have typically focused on how decision-makers weight cues individually. Here, the authors propose that people might recognize and weight groups of cues. They examine how judgments change when decision-makers focus on cues individually or as parts of groups. Several experiments demonstrate that people can spontaneously pack information into cue groups. Moreover, group-level weighting depends on how people assess similarity or how they think of categorical hierarchies.     

SNARC effects with numerical and non-numerical symbolic comparative judgments: instructional and cultural dependencies

With English-language readers in an experiment requiring pairwise comparative judgments of the sizes of animals, the nature of the association between the magnitudes of the animal pairs and the left or right sides of response (i.e., the SNARC effect) was reversed depending on whether the participants had to choose either the smaller or the larger member of the pair. In contrast, such a dependence of the direction of the SNARC effect on the form of the comparative instructions was not evident for pairwise comparisons of numerical magnitude made by a similar group of participants. Furthermore, exactly the same configuration of findings was obtained for a single group of Israeli-Palestinian right-to-left reading and writing participants, except that the spatial direction of the SNARC effects for both the animal-size and number comparisons were completely reversed. In a final experiment with English readers, SNARC effects paralleling those for the animal-size comparisons were obtained for pairwise comparative judgments involving the just-learned height relations between 6 imaginary individuals. As will be discussed, such results serve to extend the generality of the SNARC effect far beyond the current modal view that it simply reflects culturally influenced, long-term learned associations between numerical magnitudes and the locations on a fixed mental number line. The implications that these results have for both the Proctor and Cho (2006) polarity correspondence view and the Gevers, Verguts, Reynvoet, Caessens, and Fias (2006) computational model of the SNARC effect will also discussed.     

Memory for psychophysical scaling judgments

In three experiments, we examined memory for responses and stimuli experienced in a single direct psychophysical scaling session in which subjects made absolute magnitude estimations of the loudnesses of pure tones. Recall of scaling responses was found to be accurate for the softest and loudest stimuli, but systematically greater than actual judgments for the intermediate stimulus amplitudes, yielding distorted psychophysical functions for the recall data which nonetheless had the same power function exponent as that for the judged stimuli. Also, memory for the range of stimulus amplitudes was fairly accurate, but subjects could not distinguish between judged and nonjudged amplitudes within that range. The results are consistent with the role of extreme stimuli as anchors for judgment, and indicate that memories for these stimuli and responses made to them can be expected to influence future scaling judgments. These results also are consistent with the uncertainty hypothesis of mnemophysics. The present research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada to L.M.W. Thanks to the reviewers for insightful critiques.     

Direct quantitative judgments of sums and a two-stage model for psychophysical judgments

Judged magnitudes of differences between stimuli have previously been shown to support a two-stage interpretation of magnitude estimation, in which input transformations and output transformations are each describable as power functions. In an effort to provide support for the model independent of the difference estimation procedure. the present investigation employed two additional judgment tasks. We obtained magnitude judgments and category judgments of the combined magnitudes (sums) of paired weights from two groups of Ss. Values of the inferred input exponent k calculated from the two sets of data were very similar and were also remarkably similar to the exponent previously calculated from magnitude estimations of differences between weights. The output exponent calculated from magnitude judgments of sums described a concave upward function; however. the similar function describing category judgments was essentially linear. These results show that the inferred input exponent is not the result of the difference estimation task, and in addition provides support for the contention that the interval scale may be a less biased sensory measure than the magnitude scale. The introduction of an additive constant to the model improved its fit to the data but the rule by which it was introduced made very little difference.     

Children's judgments of numerical quantity: A new view of early quantification

Three experiments investigated the role of length, density, and number dimensions in children's numerosity judgments of linear arrays of beads. The number dimension is physically defined by a length × density rule. Judgments based, even in part, on the number dimension would show signs of a length × density rule. Experiment 1 examined numerosity judgments of large arrays. Results showed that judgments by 3- and 4-year-olds obeyed a length + density rule, providing cogent evidence for a general adding strategy in young children's judgments of quantity. The physical length × density rule emerged gradually with increasing age. Experiment 2 showed that numerosity judgments of small arrays by 3- and 4-year-olds obeyed a length × density rule, indicating response to the number dimension. This result was expected and thereby validated the linearity of the response scale at the youngest ages. Experiment 3 verified the integration rules for individual 3- and 4-year-olds. The integration rules were interpreted in terms of Piaget's stages of the development of quantification. No evidence for Piaget's initial stage was found. This leads to a new view of early quantification which grants young children the ability to integrate stimulus dimensions.     

Reduced Memory Representations for Music

We address the problem of musical variation (identification of different musical sequences as variations) and its implications for mental representations of music. According to reductionist theories, listeners judge the structural importance of musical events while forming mental representations. These judgments may result from the production of reduced memory representations that retain only the musical gist. In a study of improvised music performance, pianists produced variations on melodies. Analyses of the musical events retained across variations provided support for the reductionist account of structural importance. A neural network trained to produce reduced memory representations for the same melodies represented structurally important events more efficiently than others. Agreement among the musicians' improvisations, the network model, and music-theoretic predictions suggest that perceived constancy across musical variation is a natural result of a reductionist mechanism for producing memory representations.     

Toward quality criteria for collective judgments

While most organizational and social decision making is done in a group or collective mode, there are few guides or evaluative criteria for judging when a high-quality outcome has been reached. Most past studies of group decision making have been conducted in laboratories using student subjects and factual problems with correct answers as means for judging outcome quality. Such proxies are rough approximations at best of real-world conditions where value differences can be intense and problems have no correct or best answer. Drawing upon the existing literature, evaluative criteria are proposed including process, content, and outcome concerns. An instrument based upon these criteria is then applied retrospectively to six cases of ad hoc collective decision making. Using dimensional analyses, a set of more detailed evaluative factors is derived from actual participant responses. The results provide some insights into the nature of high-quality collective judgments as well as the most effective procedures for their achievement.     

A plea for minimally biased naturalistic philosophy

Synthese - Naturalistic philosophers rely on literature search and review in a number of ways and for different purposes. Yet this article shows how processes of literature search and review are...     

Reminding as a basis for temporal judgments

In the normal course of events, some events bring to mind earlier events. This reminding or, in the context of list learning experiments, study-phase retrieval can serve as a basis for the accurate judgment of the relative recencies of the two events in question. In this article, evidence for this position is presented in three experiments. By manipulations of encoding using visual imagery instructions and word associations, appropriate conditions were arranged for reminding to occur. The results of all three studies support the position that reminding provides a direct basis for later judgments of the relative recency of events.     

Quantitative functions for size and distance judgments

A psychophysical approach was used to obtain judgments of visual extent under three conditions. In tuvo conditions a comparison stimulus at each of two distances was matched in size to a standard which varied in distance. Stimuli were presented on a well-lighted table and were judged by two observers under Objective instructions. Both the standard and comparison were located in either a frontal or longitudinal plane. In a third condition relative distance estimates were given of two stimuli which varied in their relative positions along the table. The mean results for all conditions were described as a power function of physical stimulus measures. The exponent was greater than 1.0 for frontal size and usually less than 1.0 for flat size and distance. The position of the comparison affected the magnitude of the exponents to a lesser degree. These findings have relevance for interpretations of size and distance judgments.