Temporal control of behavior and the power law

The performance of rats and pigeons under fixed-interval schedules was studied in two experiments. The duration of postreinforcement pause was a declining proportion of fixed-interval duration. For pigeons this was true both when the duration of the reinforcer was fixed and when it was increased in direct proportion to increases in fixed-interval duration; the longer reinforcer durations did, however, lengthen the postreinforcement pause at higher schedule values. A quantitative analysis of data from Experiments 1 and 2 and from other studies showed that fractional exponent power functions described the relationship between postreinforcement pause and fixed-interval value; similar functions have previously been observed in studies of temporal differentiation. It was concluded that power functions reflect a direct causal, rather than artifactual, relationship between performance and the temporal requirements of reinforcement schedules.     

Axiomatic and numerical conjoint measurement: An evaluation of diagnostic efficacy

Synthetic data are used to examine how well axiomatic and numerical conjoint measurement methods, individually and comparatively, recover simple polynomial generators in three dimensions. The study illustrates extensions of numerical conjoint measurement (NCM) to identify and model distributive and dual-distributive, in addition to the usual additive, data structures. It was found that while minimum STRESS was the criterion of fit, another statistic, predictive capability, provided a better diagnosis of the known generating model. That NCM methods were able to better identify generating models conflicts with Krantz and Tversky's assertion that, in general, the direct axiom tests provide a more powerful diagnostic test between alternative composition rules than does evaluation of numerical correspondence. For all methods, dual-distributive models are most difficult to recover, while consistent with past studies, the additive model is the most robust of the fitted models.Douglas Emery is now at the Krannert Graduate School of Management, Purdue University, West Lafayette, IN, on leave from the University of Calgary.     

Correctness of Kruskal's algorithms for monotone regression with ties

Kruskal has proposed two modifications of monotone regression that can be applied if there are ties in nonmetric scaling data. In this note we prove Kruskal's conjecture that his algorithms give the optimal least squares solution of these modified monotone regression problems. We also propose another (third) approach for dealing with ties.Comments by Dr. J. B. Kruskal have been most helpful.     

Transformation of a three-mode multidimensional scaling solution to indscal form

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.     

Fitting and testing carroll's weighted unfolding model for preferences

A quadratic programming algorithm is presented for fitting Carroll's weighted unfolding model for preferences to known multidimensional scale values. The algorithm can be applied directly to pairwise preferences; it permits nonnegativity constraints on subject weights; and it provides a means of testing various preference model hypotheses. While basically metric, it can be combined with Kruskal's monotone regression to fit ordinal data. Monte Carlo results show that (a) adequacy of true preference recovery depends on the number of data points and the amount of error, and (b) the proportion of data variance accounted for by the model sometimes only approximately reflects true recovery.This study is based on a doctoral dissertation submitted to the University of Illinois at Urbana-Champaign. The author wishes to thank the members of his dissertation committee—Lawrence E. Jones, Chairman, Charles Lewis, Stephen Golding, Ledyard Tucker, and Nancy Wiggins—for their helpful comments.     

Assessing depressive symptoms with the Borg centiMax Scale® in a Swedish sample of patients and students

The Borg centiMax Scale, is a psychophysically composed general intensity ratio scale, which could enable more precise inter- and intraindividual comparisons of the intensity of depressive symptoms. In the present study, the properties of the centiMax scale were examined in 38 patients with clinical depression and 109 students. Additionally, preliminary centiMax cut-off scores for mild, moderate and severe depression were estimated. The psychometric properties of the centiMax were found to be satisfactory regarding internal consistency, convergent, discriminative and predictive validity. Moreover, the centiMax was demonstrated to provide meaningful comparisons of symptom intensity, which makes it possible to evaluate the relative importance of individual symptoms in a profile and make more precise comparisons within and between individuals. With regard to intraindividual comparisons, patients rated , for example, the intensity of feeling “guilt” twice as strong as feelings of “being punished,” and the intensity of “loss of pleasure” almost three times as strong as “being punished.” With regard to interindividual comparisons, patients rated e.g., the intensity of “being punished” as 12 times stronger than controls, and the intensity of “worthlessness” about nine times stronger. In conclusion, the centiMax was shown to be reliable and valid for assessing depressive symptoms. The centiMax with level anchored ratio data, appears to be highly advantageous as it permits rather precise values of symptom intensity for intra- and interindividual comparisons that could be useful in the diagnostic process and in treatment planning.     

Pitch distribution,melodic contour or both? Modeling makam schema with multidimensional scaling and self-organizing maps

Experienced listeners internalize musical tonal knowledge via statistical learning of pitch distributions as a result of exposure to musical environment. Cross-cultural studies of music cognition offer new perspectives to investigate the acquisition of tonal schema. Makam music is a rich musical system characterized by modal structures defined by micro-tonal pitch sets, and melodic progression patterns (aka seyir features). Makam schema is possibly acquired by internalizing the seyir in addition to pitch features. In the current study, we examined whether an ideal model of makam schema is built with multidimensional scaling analysis and with self-organizing maps (SOMs). We were interested in whether statistical information about seyir features, in addition to pitch distributions, would form an acceptable makam schema model. We qualitatively analyzed topographical organizations in the models to understand whether they reflect complex relations between makams. Multidimensional scaling analyses did not produce an acceptable model for makam schema. The SOM trained with pitch distributions provided an adequate model for makam schema. However, the SOM trained with both pitch distributions and seyir features was better in capturing the complex relations between makams. Further behavioral research is necessary to understand whether melodic progression patterns are intrinsic features of the tonal knowledge of the experienced listeners of makam music.     

Getting the big picture: Development of spatial scaling abilities

Spatial scaling is an integral aspect of many spatial tasks that involve symbol-to-referent correspondences (e.g., map reading, drawing). In this study, we asked 3–6-year-olds and adults to locate objects in a two-dimensional spatial layout using information from a second spatial representation (map). We examined how scaling factor and reference features, such as the shape of the layout or the presence of landmarks, affect performance. Results showed that spatial scaling on this simple task undergoes considerable development, especially between 3 and 5 years of age. Furthermore, the youngest children showed large individual variability and profited from landmark information. Accuracy differed between scaled and un-scaled items, but not between items using different scaling factors (1:2 vs. 1:4), suggesting that participants encoded relative rather than absolute distances.     

Modeling the effect of reward amount on probability discounting

The present study with college students examined the effect of amount on the discounting of probabilistic monetary rewards. A hyperboloid function accurately described the discounting of hypothetical rewards ranging in amount from $20 to $10,000,000. The degree of discounting increased continuously with amount of probabilistic reward. This effect of amount was not due to changes in the rate parameter of the discounting function, but rather was due to increases in the exponent. These results stand in contrast to those observed with the discounting of delayed monetary rewards, in which the degree of discounting decreases with reward amount due to amount-dependent decreases in the rate parameter. Taken together, this pattern of results suggests that delay and probability discounting reflect different underlying mechanisms. That is, the fact that the exponent in the delay discounting function is independent of amount is consistent with a psychophysical scaling interpretation, whereas the finding that the exponent of the probability-discounting function is amount-dependent is inconsistent with such an interpretation. Instead, the present results are consistent with the idea that the probability-discounting function is itself the product of a value function and a weighting function. This idea was first suggested by Kahneman and Tversky (1979), although their prospect theory does not predict amount effects like those observed. The effect of amount on probability discounting was parsimoniously incorporated into our hyperboloid discounting function by assuming that the exponent was proportional to the amount raised to a power. The amount-dependent exponent of the probability-discounting function may be viewed as reflecting the effect of amount on the weighting of the probability with which the reward will be received.     

Profile Analysis of Psychological Symptoms Associated With Misophonia: A Community Sample

Misophonia is characterized by extreme aversive reactions to certain classes of sounds. It has recently been recognized as a condition associated with significant disability. Research has begun to evaluate psychopathological correlates of misophonia. This study sought to identify profiles of psychopathology that characterize misophonia in a large community sample. A total of N = 628 adult participants completed a battery of measures assessing anxiety and anxiety sensitivity, depression, stress responses, anger, dissociative experiences, obsessive-compulsive symptoms and beliefs, distress tolerance, bodily perceptions, as well as misophonia severity. Profile Analysis via Multidimensional Scaling (PAMS) was employed to evaluate profiles associated with elevated misophonia and those without symptoms. Three profiles were extracted. The first two accounted for 70% total variance and did not show distinctions between groups. The third profile accounted for 11% total variance, and showed that misophonia is associated with lower obsessive-compulsive symptoms for neutralizing, obsessions generally, and washing compared to those not endorsing misophonia, and higher levels of obsessive-compulsive symptoms associated with ordering and harm avoidance. This third profile extracted also showed significant differences between those with and without misophonia on the scale assessing physical concerns (that is, sensitivity to interoceptive sensations) as assessed with the ASI-3. Further research is called for involving diagnostic interviewing and experimental methods to clarify these putative mechanisms associated with misophonia.