Some further tests of the constant-ratio rule

Five experiments sought to test the constant-ratio rule (CRR) with single dimensional ensembles composed of 2, 4, or 8 stimulus objects. Each S attempted to identify stimuli which varied in weight or in visual size or brightness. The results demonstrated: (a) The CRR predicts equally well the response proportions of single dimensional visual, kinesthetic, and auditory stimulus ensembles, but less well than those for multidimensional auditory stimuli. (b) Better predictions are obtained with four than with two stimulus objects.(c) The CRR is sensitive to variations in the spacing and range of the stimulus ensembles and to practice on the task. It is concluded that the rule tends to fail whenever stimulus conditions elicit differential amounts of stimulus and response confusion.     

Cognitive processing of multidimensional stimuli in schizophrenia: formal modeling of judgment speed and content

This article presents a formal, mathematical account of relations between response times on simple cognitive tasks and content of complex judgments involving multiple stimulus dimensions for people with schizophrenia. Changes in multidimensional judgments were viewed as the result of interference from increased stages of encoding with respect to the individual dimensions. Information on dimensional properties encoded earlier in a judgment trial was considered to be more susceptible to loss over the rest of the trial, because of a larger number of encoding stages applied to the remaining dimensional properties. Model predictions were tested with samples of paranoid and nonparanoid schizophrenic participants and controls. Unidimensional encoding speed was assessed by reaction times in an explicit similarity ratings task, and multidimensional judgment content was assessed by the relative importance of different stimulus dimensions to participants' ratings in an implicit similarity ratings task. Results support validity of the model.     

A Differential Geometric Description of the Relationships among Perceptions

We present a differential geometric method for measuring and characterizing the perceptions of an observer of a continuum of stimuli. Because the method is not based on a model of perceptual mechanisms, it can potentially be applied to a wide variety of observers and to many types of visual and auditory stimuli. The observer is asked to identify which small transformation of one stimulus is perceived to be equivalent to a small transformation of a second stimulus, differing from the first stimulus by a third small transformation. The observer's identification of a number of such transformations can be used to calculate an affine connection on the stimulus manifold. This quantity describes how the observer encodes an evolving stimulus as a perceived sequence of "reference" transformations. This type of encoding is a multidimensional generalization of Fechner's method and reduces to his psychophysical scale when the stimulus manifold is one dimensional and the reference transformation is chosen to be a just noticeable difference. The intrinsic aspects of the nature of the observer's perceptions can be characterized by the curvature and torsion tensors derived from the connection. The multidimensional analogues of psychophysical scale functions exist if and only if these quantities vanish. Differences between the affine connections of two observers characterize differences between their perceptions of the same evolving stimuli. The affine connections of two observers can also be used to map a stimulus perceived by one observer onto another stimulus, perceived in the same way by the other observer. Unlike multidimensional scaling techniques, this method does not assume that the observer has a sense of distance (a metric) or that he/she can otherwise compare stimulus pairs that are oriented along perceptually different directions in the manifold. The method was used to measure the affine connections of observers of a dot moving on different background graphics; e.g., a blank screen, a grid, or two converging lines similar to those used to create the Ponzo illusion. The results comprise quantitative measurements of the background graphic's influence on each observer's perceptions of straightness, parallelism, and distance. The measurements demonstrate differences between the perceptions of the two observers. Copyright 2000 Academic Press.     

Three case studies in the Bayesian analysis of cognitive models

Bayesian statistical inference offers a principled and comprehensive approach for relating psychological models to data. This article presents Bayesian analyses of three influential psychological models: multidimensional scaling models of stimulus representation, the generalized context model of category learning, and a signal detection theory model of decision making. In each case, the model is recast as a probabilistic graphical model and is evaluated in relation to a previously considered data set. In each case, it is shown that Bayesian inference is able to provide answers to important theoretical and empirical questions easily and coherently. The generality of the Bayesian approach and its potential for the understanding of models and data in psychology are discussed.     

A procedure for ordering object pairs consistent with the multidimensional unfolding model

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.The author wishes to thank Jack Hoadley, Larry Mayer, Sheldon Newhouse, Stuart Rabinowitz, Forrest Young, and three anonymous reviewers for their useful suggestions.     

Event-related brain potentials: report order and task switch on multidimensional stimuli identification

An experiment was conducted to evaluate the effect of order of reporting stimulus dimension in multidimensional stimulus identification using a switch-task paradigm. Participants were required to identify each two-dimensional symbol by pushing the corresponding buttons on the keypad on which there were two columns representing the two dimensions, part and shape. The two orders of report were Order Part/Shape and Order Shape/Part. There was a task cue prior to each presentation of a symbol indicating the particular report order the participants should perform. The cue changed randomly. Both behavioral data and event-related potentials were recorded from 12 college students. Analysis of behavioral data showed switch cost indicated by increase in response time was greater for Order Shape/Part, a less appropriate order of reporting dimensional values, than Order Part/Shape (88 msec. vs 9 msec. for response time for the first stimulus dimension; 96 msec. vs -1 msec. for response time total). Neural activities under Order Shape/Part and Order Part/Shape were different, and it seems plausible that participants put more effort into selection and showed more related semantic activation in Order Shape/Part than in Order Part/ Shape which fits the Chinese adjective-then-noun language habit.     

Assessing sensitivity in a multidimensional space: some problems and a definition of a general d'

This article provides a formal definition for a sensivity measure,d′ _g , between two multivariate stimuli. In recent attempts to assess perceptual representations using qualitative tests on response probabilities, the concept of ad′ between two multidimensional stimuli has played a central role. For example, Kadlec and Townsend (1992a, 1992b) proposed several tests based on multidimensional signal detection theory that allow conclusions concerning the perceptual and/or decisional interactions of stimulus dimensions. One proposition, referred to as thediagonal d′test, relies on specific stimulus subsets of a feature-complete factorial identification task to infer perceptual separability. Also, Ashby and Townsend (1986), in a similar manner, attempted to relate perceptual independence to dimensional orthogonality in Tanner’s (1956) model, which also involvesd′ between two multivariate signals. An analysis of the proposedd′ _g reveals shortcomings in the diagonald′ test and also demonstrates that the assumptions behind equating perceptual independence to dimensional orthogonality are too weak. Thisd′ _g can be related to a common measure of statistical distance, Mahalanobis distance, in the special case of equal covariance matrices.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.     

A COMPARATIVE STUDY OF SOME MULTIDIMENSIONAL VECTOR MODELS FOR SUBJECTIVE SIMILARITY

E kehammar B. A comparative study of some multidimensional vector models for subjective similarity. Scand. J. Psychol ., 1972, 1 3 , 190–197.—h this report, two theoretical vector models for subjective similarity were deduced from a "content" definition of subjective similarity. One of the derived similarity equations proved in empirical model testing to describe the similarity principle in perception of form better than earlier, similar models. Deviations between observed similarity estimates and values expected according to the model are discussed, among other things, in terms of the heterogeneity of the stimulus set.     

Dimensional and metric structures in multidimensional stimuli

The present experiments investigated two characteristics of subjects’ multidimensional representations: their dimensional organization and metric structure, for both analyzable and integral stimuli. In Experiment 1, subjects judged the dissimilarity between all pairs of stimuli differing in brightness and size (analyzable stimuli), while in Experiment 2, subjects made dissimilarity judgments for stimuli varying in width height, and area shape (integral stimuli). For the brightness size stimuli, the findings that (a) brightness judgments were independent of size (and vice versa) and (b) the best fitting scaling solution was one that depicted an orthogonal structure are strong evidence that subjects perceived brightness size as a dimensionally organized structure. In contrast, for the rectangle stimuli, neither width height nor area shape contributed additively to overall dissimilarity. The results of the metric fitting were more equivocal. For all stimulus sets, the Euclidean metric yielded scaling solutions with lower stress values than the city block metric. When bidimensional ratings were regressed on unidimensional ratings, the city block metric yielded a slightly higher correlation coefficient than the Euclidean metric for brightness size stimuli. The two rules of combination were equivalent for the width-height stimuli, but the Euclidean metric provided a better approximation for the area shape stimuli. The results were discussed in terms of how subjects integrate physical dimensions for the case of integral stimuli and the superiority of dimensional vs. metric structure as an indicator of stimulus analyzability.     

Perceptual primacy of dimensions: support for a model of dimensional interaction

Do Ss always process multidimensional stimuli according to psychologically primary dimensions? Our hypothesis is that they do: Primary dimensions provide one component of a new model of dimensional interaction, a model that distinguishes information processed at the level of attributes from information processed at the level of the stimulus. By using sound stimuli created from the dimensions pitch-loudness (Experiments 1 and 2), pitch-timbre (Experiment 3), and loudness-timbre (Experiment 4), we tested performance in selective- and divided-attention tasks at each of three orientations of axes: 0 degrees, 22.5 degrees, and 45 degrees. Each experiment revealed strong evidence of primacy: As axes rotated from 0 degrees to 45 degrees, selective attention deteriorated, but divided attention improved, producing a distinct pattern of convergence. Each experiment also revealed effects of congruity: Attributes from corresponding poles of a dimension (e.g., high pitch and loud) were classified faster than those from noncorresponding poles. The results fit well with our new conception but are inconsistent with other current models of dimensional interaction.     

Free classification: Element-level and subgroup-level similarity

Subjects classified sets of multidimensional stimuli into two groups in any way they wished. The sets were composed of 6 or 12 stimuli: 2 or 4 instances of 3 different stimuli (e.g., 2 blue circles, 2 green circles, 2 red circles). There were striking individual differences in the preferred classification. Some subjects maximized the similarity between subgroups by matching the composition of the subgroups--one instance of each stimulus was placed in each group. The other subjects maximized the similarity among stimuli within each subgroup by placing similar stimuli in each group (the blues and greens in one group, the reds in the other). The nature of the stimuli as well as the relationships among the three stimuli had little effect on classification. In this case, cognitive styles specific to individuals but general across diverse dimensions and stimulus sets determined classification.     

Multidimensional scale types and invariant multivariate statistics

This paper proposes an extension of the definition of one‐dimensional ordinal, interval, ratio, difference and absolute scales to the multidimensional case. The connection between measurement structures and statistical structures suggested in a previous paper is also extended to the multivariate case, and examples of invariance properties of multivariate functions and statistics are discussed.     

Discrete and continuous measures of dimensional stimulus control

In two sets of experiments, we examined dimensional stimulus control of pigeons' responses to a visual flicker-rate continuum. In the first experiment, responses to a single key were reinforced periodically during stimuli from one half of the stimulus continuum, and responses during other stimuli were extinguished. In the second experiment, two response keys were simultaneously available, with reinforcement for each response alternative associated with different halves of the stimulus continuum. Conditions of the second experiment involved either free-operant or discrete-trial stimulus presentations. Results from these experiments show that positive dimensional contrast appeared in discrimination tasks with one or two response alternatives, but only with free-operant procedures. In addition, discrimination between stimulus classes established by differential reinforcement was assessed as accurately by continuous rate measures as by discrete response choice in the two-alternative situation. The general implication of these experiments is that response rate measures, when properly applied, may reveal sources of variation within stimulus classes, such as dimensional contrast, that are not evident with discrete measures.     

The effects of stimulus type,training, and chronological age on children's identification and recoding of multidimensional stimuli

Kindergarten and second-grade children (75 and 99 months of age, respectively) were first taught that the two stimuli within each of four pairs of multidimentional stimuli differed with respect to their values on every one of three bivalued dimensions: color, size, and form. Each of the eight stimuli was then presented twice, one at a time, and the child was asked either to find its duplicate (eight encoding trials) or to find its complementary mate (eight recoding trials) among the complete set of eight stimuli. Next, each of the eight stimulus compounds was presented twice as a display stimulus for 5 sec and, after a delay of 10 to 15 sec, the child was asked to find the display stimulus (eight trials of encoding plus memory) or its complement (eight trials of recoding plus memory) among the complete set. The stimuli for half the children in each age group were standard unitary compounds (e.g., a large red circle, a small blue square, etc.) and for the other half they were partitioned stimuli with their dimensional values spatially separated (e.g., large arrow, circle outline, and patch of color). Half of the children in each of these subgroups were given a basic pretraining, and the other half received extended pretraining, in recoding the stimulus attributes. The results indicated that unitary stimuli were easier to encode but the partitioned stimuli were easier to recode, that recoding was much more difficult than encoding, that extended training improved recoding performance, and that second graders were slightly better at encoding and much better at recoding than the kindergarten children. The patterns of performance on the tasks involving memory were similar to those just described. The results were discussed in terms of children's abilities to analyze multidimensional stimuli into their dimensional components.     

Pictorial similarity judgments and the organization of visual memory in the rhesus monkey

The organization of visual memory for pictures was studied in the rhesus monkey. Two monkeys were tested in a same/different task in which sequentially presented pictures were compared to each other in a pair-wise fashion. The resulting confusion matrixes were analyzed using a multidimensional scaling procedure to obtain two- and three-dimensional graphic representations of the stimulus space. In Experiment 1, the monkeys' confusion errors caused pictures of human and rhesus monkey faces to fall in the same region of multidimensional space, which suggested that the monkeys categorized facial stimuli. A similar effect was found for pictures of different types of fruit. Experiment 2 replicated the categorization of faces with a more diverse collection of human and nonhuman primate faces. Experiment 3 explored the fruit category by varying stimulus attributes orthogonally. The results from this experiment showed that both monkeys encoded the pictures in this category by type of fruit (apples or grapes) and color (red or yellow). Taken together, these studies indicate that rhesus monkeys will treat some classes of pictorial stimuli categorically in visual memory.     

The prioritization of perceptual processing in categorization

In three experiments, the effects of selective attention on perceptual processes in a complex multidimensional object categorization task were investigated. In each experiment, participants completed a perceptual-matching task to gain estimates of the perceptual salience of each stimulus dimension, then a categorization task using the same stimuli. In Experiments 1 and 2, the perceptual processing of stimulus dimensions was faster when dimensions were more diagnostic of category membership, regardless of their perceptual salience. Experiment 3 demonstrated that this prioritization of perceptual processing was evident even when stimuli were presented in unpredictable locations during categorization, indicating that the physical characteristics of the stimulus guide selective attention to diagnostic stimulus dimensions.     

Fechnerian metrics in unidimensional and multidimensional stimulus spaces

A new theory is proposed for subjective (Fechnerian) distances among stimuli in a continuous stimulus space of arbitrary dimensionality. Each stimulus in such a space is associated with a psychometric function that determines probabilities with which it is discriminated from other stimuli, and a certain measure of its discriminability from its infinitesimally close neighboring stimuli is computed from the shape of this psychometric function in the vicinity of its minimum. This measure of discriminability can be integrated along any path connecting any two points in the stimulus space, yielding the psychometric length of this path. The Fechnerian distance between two stimuli is defined as the infimum of the psychometric lengths of all paths connecting the two stimuli. For a broad class of models defining the dichotomy of response bias versus discriminability, the Fechnerian distances are invariant under response bias changes. In the case in which physically multidimensional stimuli are discriminated along some unidimensional subjective attribute, a systematic construction of the Fechnerian metric leads to a resolution of the long-standing controversy related to the numbers of just-noticeable differences between isosensitivity curves. It is argued that for unidimensional stimulus continua, the proposed theory is close to the intended meaning of Fechner's original theory.