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.     

A comparison of generalization functions and frame of reference effects in different training paradigms

Six experiments were carried out to compare go/no-go and choice paradigms for studying the effects of intradimensional discrimination training on subsequent measures of stimulus generalization in human subjects. Specifically, the purpose was to compare the two paradigms as means of investigating generalization gradient forms and frame of reference effects. In Experiment 1, the stimulus dimension was visual intensity (brightness); in Experiment 2, it was line orientation (line-angle stimuli). After learning to respond (or to respond “right”) to stimulus value (SV) 4 and not to respond (or to respond “left”) to SV2 (in Experiment 1) or SV1 (in Experiment 2), the subjects were tested for generalization (recognition) with an asymmetrical set of values ranging from SV1 to SV11. Go/no-go training produced peaked gradients, whereas choice training produced sigmoid gradients. The asymmetrical testing resulted in a gradual shift of the peak of responding (go/no-go group) or in the point of subjective indifference (PSI; choice group) toward. the central value of the test series; thus, both paradigms revealed a frame of reference effect. The results were comparable for the quantitative (intensity) and the qualitative (line-angle) stimulus dimensions. Experiment 3 compared the go/no-go procedure with a yes/no procedure in which subjects responded “right” to SV4 and “left” to all other intensities and found no differences between these procedures. Thus the difference in gradient forms in go/no as opposed to (traditional) choice paradigms depends on whether one or two target stimuli are used in training. In Experiment 4, in which visual intensity was used, the shift in the PSI following choice training varied positively with the range of asymmetrical test stimuli employed. In Experiment 5, also with visual intensity, the magnitude of the peak shift following go/no-go training varied as a function of over representing a high or a low stimulus value during generalization testing. Experiment 6, with line angles, showed that the PSI following choice training varies in a similar way. The frame of reference effects obtained in these experiments are consistent with an adaptation-level model.     

Cognitive reference points in judgments of symbolic magnitude

Research on speeded symbolic magnitude comparisons indicates that decisions are made more quickly when the magnitudes of the stimuli being compared are relatively close to an explicit or implicit reference point. Alternative explanations of this phenomenon are tested by seeking similar effects in nonspeeded rating tasks. In accord with the predictions of discriminability models, rated magnitude differences between stimuli in the vicinity of a reference point are expanded relative to differences between stimuli far from it. The inferred locations of cities along a west-east axis varies systematically depending on which coast, Pacific or Atlantic, is specified as the reference point. Scales derived from the rating data are correlated with the pattern of reaction times obtained in a comparable speeded comparison task. In addition, the distance between the cities nearest the locale of our subjects is subjectively stretched. Reference point effects are also observed when the form of the comparative specifies an implicit reference point at either end of a continuum of subjective size; however, these effects are very small and do not clearly support a discriminability interpretation. Stronger evidence for discriminability effects is obtained when an explicit reference point is established at an arbitrary size value. An implicit scaling model, related to range-frequency theory, is proposed to account for the influence of reference points on relative discriminability of stimulus magnitudes. The implicit scaling model is used to develop an account of how symbolic magnitudes may be learned and of how habitual reference points can produce asymmetries in distance judgments.     

Chinese kindergartners’ automatic processing of numerical magnitude in Stroop-like tasks

Using Stroop-like tasks, this study examined whether Chinese kindergartners showed automatic processing of numerical magnitude. A total of 36 children (mean age 5 5 years 10 months) were asked to perform physical size comparison (i.e., “Which of two numbers is bigger in physical size?”) and numerical magnitude tasks (i.e., “Which of two numbers is bigger in numerical magnitude?”) on 216 number pairs. These number pairs varied in levels of congruence between numerical magnitude and physical size (for Stroop effect) and numerical distance (for distance effect). On the basis of analyses of response time and error rates, we found that Chinese kindergartners showed automatic processing of numerical magnitude. These results are significantly different from previous studies’ findings about the onset age (ranging from around the end of first grade to third grade) for automatic processing of numerical magnitude.     

The rate of “mental rotation” of images: A test of a holistic analogue hypothesis

This paper discusses the analogue-propositional distinction and argues that, given an appropriate understanding of this issue, the question of whether a particular cognitive function is analogue or not is an empirical one. As an example of how the question can be empirically investigated, the proposed analogue operation for mental rotation of images is considered. It is argued that the view that images are rotated in a holistic analogue manner should predict that rotation rate is independent of such factors as the conceptual complexity of the stimulus or of the comparison task. Two experiments are described that investigated the effects of several stimulus and task variables on the apparent rate of “mental rotation” of images in a Shepard-type task. Instead of comparing a stimulus and misoriented probe figure to determine whether they are identical (except for orientation) or mirror images, as was the case in most of previous studies, the present experiments required subjects to judge whether the misoriented probe was a subfigure of the target stimulus. The results showed that the “rotation rate” (i.e., the slope of the RT vs. angle of misorientation function) was influenced by practice, stimulus attributes, and the nature of the comparison task. In particular, when the probe was a “good” subfigure of the reference stimulus, apparent rotation rate was greater. These results are interpreted as indicating that the linear RT vs. angle relation is not due to a holistic analogue rotation of images, as had been supposed, but arises from a more articulated and piecemeal process in which analysis of the stimulus figure interacts with the comparison task.     

Familiarity-Matching: An Ecologically Rational Heuristic for the Relationships-Comparison Task

Previous studies have shown that people often use heuristics in making inferences and that subjective memory experiences, such as recognition or familiarity of objects, can be valid cues for inferences. So far, many researchers have used the binary choice task in which two objects are presented as alternatives (e.g., “Which city has the larger population, city A or city B?”). However, objects can be presented not only as alternatives but also in a question (e.g., “Which country is city X in, country A or country B?”). In such a situation, people can make inferences based on the relationship between the object in the question and each object given as an alternative. In the present study, we call this type of task a “relationships-comparison task.” We modeled the three inference strategies that people could apply to solve it (familiarity-matching [FM; a new heuristic we propose in this study], familiarity heuristic [FH], and knowledge-based inference [KI]) to examine people's inference processes. Through Studies 1, 2, and 3, we found that (a) people tended to rely on heuristics, and that FM (inferences based on similarity in familiarity between objects) well explained participants' inference patterns; (b) FM could work as an ecologically rational strategy for the relationships–comparison task since it could effectively reflect environmental structures, and that the use of FM could be highly replicable and robust; and (c) people could sometimes use a decision strategy like FM, even in their daily lives (consumer behaviors). The nature of the relationships–comparison task and human cognitive processes is discussed.     

Mental rotation under head tilt: Factors influencing the location of the subjective reference frame

We report five experiments on the effect of head tilt on the mental rotation of patterns to the “upright.” In Experiment 1, subjects rotated alphanumeric characters, displayed within a circular surround. Experiment 2 was similar except that the character was an unfamiliar letter-like symbol. In Experiment 3, subjects again rotated alphanumeric characters, but they were displayed within a rectangular frame tilted 60° to the right. Experiment 4 was similar, except that the subjects were instructed to rotate the characters to the “upright” defined by the tilted frame. In all four experiments, the subjects performed the task with their heads either upright or tilted 60°. In Experiment 5, subjects had their heads and bodies tilted 90°, and rotated alphanumeric characters displayed within a circular surround. In all except Experiment 4, analysis of response latencies revealed that the subjective vertical lay closer to the gravitational than to the retinal vertical, although it was somewhat displaced in the direction of the head tilt—more so in Experiments 2 and 3 than in Experiment 1, and more so still in Experiment 5. In Experiment 4, instructions to adopt the axes of the frame land thus of the retina) succeeded in bringing the subjective vertical closer to the retinal than to the gravitational vertical, although the subjective vertical was still some 20° on average from the gravitational vertical. The results show that the subjective reference frame is distinct from both gravitational and the retinal frames, and that the gravitational frame exerts the stronger influence. They also argue against the primacy of a “retinal factor” in the perception of orientation.     

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.     

Communicative functions of directional verbal probabilities: Speaker’s choice,listener’s inference,and reference points

Verbal probabilities have directional communicative functions, and most can be categorized as positive (e.g., “it is likely”) or negative (e.g., “it is doubtful”). We examined the communicative functions of verbal probabilities based on the reference point hypothesis According to this hypothesis, listeners are sensitive to and can infer a speaker’s reference points based on the speaker’s selected directionality. In four experiments (two of which examined speakers’ choice of directionality and two of which examined listeners’ inferences about a speaker’s reference point), we found that listeners could make inferences about speakers’ reference points based on the stated directionality of verbal probability. Thus, the directionality of verbal probabilities serves the communicative function of conveying information about a speaker’s reference point.     

Symbolic- and response-related contributions to blindness to compatible stimuli

Blindness to compatible stimuli refers to poorer target identification (e.g., right-pointing arrowhead) following compatible stimuli (e.g., right arrows; Stevanovski, Oriet, & Jolic?ur, 2003) and compatible responses (e.g., right key press; Müsseler & Hommel, 1997a and 1997b). To clarify the role of the response in this effect, we examined the impact of adding or removing an overt response. In three experiments, subjects saw an arrow cue (that sometimes required a response) followed by a brief, masked arrowhead, which was reported on all trials. In all experiments, making a response increased the magnitude of the blindness effect. Furthermore, “no response” performance was unaffected by whether subjects had previously responded to the cue. Results favour a two-factor symbolic- plus response-related activation model over a purely symbolic activation model.     

Decision latencies of “same” and “different” judgments

When a subject is asked to judge whether two stimuli are “same” or“different,” the time he takes to reach the decision same is frequently unequal to the time he takes to reach the decision different. We studied this discrepancy as a function of several variables, including stimulus modality, “codability” vs.“noncodability” of test stimuli, interstimulus interval, and discrimination difficulty. Results of four different experiments performed on a total of 111 subjects showed codability and discrimination difficulty to be the most important factors. Stimuli that are codable (i.e., which can be categorized by absolute judgment) yield a shorter latency for decision same, and noncodable stimuli (i.e., those requiring a reference stimulus for categorization) yield a longer latency for decision same. The modality of test stimuli, the prothetic or metathetic nature of the dimension to be judged, and simultaneous vs. successive presentation of the stimuli appear not to be crucial factors.     

Evidence against “absolute” scaling

Zwislocki and Goodman (1980) argue that there exists an “absolute coupling” of numbers to sensation magnitudes and conclude that, when subjects are left unconstrained by a designated stimulus-number pair, they may use this “absolute” scale. The purpose of the studies reported here was to test whether Zwislocki and Goodman’s (1980) absolute scaling procedure reduces contextual effects due to variations in the stimulus spacing. It was found that magnitude estimations vary as a function of the stimulus spacing, regardless of whether subjects are instructed to use a standard and modulus, and, furthermore, that category ratings yield effects of the stimulus spacing comparable to those obtained with magnitude estimations. It is argued that removing the so-called constraints of a standard and modulus does not yield an “absolute” scale of sensation. The absolute scaling procedure increases response variability and thereby lowers the power of a test for contextual effects.     

Effects of rank ordering stimuli on magnitude ratings of these and other stimuli

Five experiments investigated the effects of rank ordering stimuli on subsequent magnitude ratings of these and other stimuli. In Experiments 1 and 2, subjects first rank ordered environmental issues in terms of their importance. Ranking stimuli from “most” to “least” led to more extreme ratings than ranking them from “least” to “most,” regardless of whether the rating criterion was the same as or diametrically opposite to the ranking criterion. (For example, subjects who had previously ranked them beginning with the most important issue subsequently rated these issues not only as more important, but also as more trivial, than did subjects who had ranked them beginning with the least important.) These effects generalized to stimuli other than those that had previously been ranked, and generalized over stimulus domains. (For example, ratings of environmental issues were also affected by ranking the importance of attributes of a marriage partner.) Other experiments in the series circumscribed the conditions in which these effects occur. Results suggested that rank ordering stimuli leads subjects to adopt comparative standards, the use of which generalizes to subsequent magnitude rating tasks and produces an anchoring bias similar to that identified by A. Tversky and D. Kahneman (1974, Science (Washington, D.C.), 185, 1124–1131). Implications of these results for the cognitive processes that underlie social judgment are discussed.     

Examining the effects of group‐based instruction on emergent second‐language skills in young children

The present study evaluated the emergence of second‐language intraverbals in typically developing young children following a small‐group teaching intervention. Choral responding was employed with a group of 6 primary school children (5‐6 years old) to teach first‐language tacts (e.g., “What is this in English?” [“Hospital”]) and related second‐language tacts (e.g., “What is this in Welsh?” [“Ysbyty”]). A multiple‐probe design across stimulus sets was used to evaluate subsequent emergence of untrained first‐to‐second‐language derived intraverbals (e.g., “What is hospital in Welsh?” [“Ysbyty”]) and untrained second‐to‐first‐language intraverbals (e.g.,”What is ysbyty in English?” [“Hospital”]). Data indicated that the choral responding intervention produced robust increases in derived intraverbal relations for 3 of the 6 participants.     

Naming solids and nonsolids: Children's default construals

Four experiments used a free-naming task to examine children's and adults' default construals of solids and nonsolids. In Experiment 1,4-year-old children viewed entities presented in familiar geometric shapes (e.g., square, triangle), without touching them. One half saw solids (e.g., a square made of wood); the other half saw nonsol ids matched carefully in shape (e.g., a square with the same dimensions but made of peanut butter). To tap their default construals, children were simply asked, “What is that?” Answers varied sharply with the type of stimulus. If the entity was solid, children tended to provide an individual-related word (e.g., “a square”), even if they also knew a substance-related word (e.g., “wood”). But if the stimulus was nonsolid, children tended to give a substance-related word (e.g., “peanut butter”), even if they also knew an individual-related word (e.g., “a square”). These words were usually common nouns produced in appropriate sentential contexts, suggesting that 4-year-olds represented the words as naming kinds. The same pattern of results obtained in Experiments 2 and 3, which were modified replications of Experiment 1. The results of Experiment 4 replicated the main findings of Experiment 1 using adults as participants. The studies suggest that, as a default, 4-year-olds conceptualize solids and nonsolids in (a) fundamentally distinct, (b) kind-based (rather than perceptual property-based), and (c) adult-like ways.     

Identification and categorization in visual search

Seven experiments were addressed to the general question of whether the identification of letters and numbers is a more rapid process than the categorization of such stimuli. Subjects were required to make a single response if a target stimulus specified by name (e.g., “A,” “2”) or designated by category class alone (e.g., “letter,” “number”) was presented in a trial. The principal findings were: (1) identification reaction times (RTs) were faster than categorization RTs: (2) RTs for targets shown without a context were faster than RTs for targets shown in the context of other stimuli; (3) identification RTs for targets shown in the context of stimuli from a different conceptual-taxonomic category were faster than RTs for targets shown in the context of stimuli from the same category only when target-context stimulus discriminability differencet were optimized. The results were interpreted in terms of a two-stage processing model in which context face,ors affect the duration of an initial encoding-scanning stage and search instruction (effective memory size) factors affect the duration of the memory comparison stage.     

Individual differences in visual comparison processes

Two experiments are reported in which subjects compared the shape of two successively presented random visual forms. The first stimulus in the pair was one of five “standard” shapes, and the second stimulus was either the same as the standard or different by virtue of a perturbation in shape or an overall reflection. Marked individual differences were found in reaction time for the same-different comparison. For one type of subject “same” responses were faster than “different” responses, speed of “different” responses was unaffected by similarity of the test shape to the standard, and error rates and reaction times were not systematically related. For the other type of subject, “different” responses were generally faster than “same” responses, “different” reaction time decreased as the standard shape and the test shape became increasingly dissimilar, and error rates and reaction times were positively correlated. Implications of these individual differences for models of the same-different comparison process are discussed.     

Culture and perceptions of self—other similarity

In testing possible cultural effects of the use of the self as an habitual reference point to which others are compared, we expected that: (a) individualistic participants (i.e., those who give priority to personal goals) would rate self—other similarity higher when asked “How similar is X to you?” than when asked “How similar are you to X?”, whereas nondirectional similarity judgements (“How similar are these two people?”) would resemble the former directional comparison; (b) collectivistic participants (i.e., those who give priority to in‐group goals) would show a weaker or, possibly, reversed pattern, especially using in‐group comparison others. Neither hypothesis was upheld. However, the individualists perceived the in‐group to be relatively more similar to themselves as compared to the collectivists. This difference cannot be explained by response bias, status asymmetry, or role differentiation. We propose an explanation in terms of the differential relationship between self and other representations for people from collectivist versus individualist cultures.     

Binaural summation after learning psychophysical functions for loudness

Do response-related processes affect perceptual processes? Sometimes they may: Algom and Marks (1990) produced different loudness exponents by manipulating stimulus range, and thereby also modified the rules of loudness summation determined by magnitude scaling. The present study manipulated exponents by having a dozen subjects learn prescribed power functions with exponents of 0.3, 0.6, or 1.2 (re sound pressure). Subjects gave magnitude estimates of the loudness of binaural signals during training, and of monaural and binaural signals after training. During training, subjects’ responses followed the nominal functions reasonably well. Immediately following training, subjects applied the numeric response scales uniformly to binaural and monaural signals alike; the implicit monaural-binaural loudness matches, and thus the basic rules underlying binaural summation, were unaffected by the exponent learned. Comparison of these results with those of Algom and Marks leads us to conclude that changing stimulus range likely influences underlying perceptual events, whereas “calibrating” a loudness scale through pretraining leaves the perceptual processes unaffected.     

Contextual effects in difference judgments

Manipulating stimulus spacing, stimulus frequency, or stimulus range usually affects intensity judgments. In six experiments, I investigated the locus of analogues of these contextual effects in a “difference” estimation task. When all stimuli elicited the same taste quality, stimulus distribution affected the scale values only when water was included in the stimulus set (Experiments 1–3). When the subjective ranges of two taste qualities were manipulated, different scale values were obtained for the separate qualities in the two conditions (Experiment 4). Manipulation of the expected response distribution did not affect the scale values or the responses (Experiments 5–6). It is concluded that shifts in stimulus distributions or stimulus ranges result in shifts in subjective scale values. The contextual effects can be interpreted as relative shifts of a number of gustatory continua, with water lying on a separate continuum. Proposed is a model for context-dependent judgments, consisting of four stages: stimulus classification, stimulus placement, continuum placement, and continuum projection.