The microanalysis of fixed-interval responding

The fixed-interval schedule of reinforcement is one of the more widely studied schedules in the experimental analysis of behavior and is also a common baseline for behavior pharmacology. Despite many intensive studies, the controlling variables and the pattern of behavior engendered are not well understood. The present study examined the microstructure and superstructure of the behavior engendered by a fixed-interval 5- and a fixed-interval 15-minute schedule of food reinforcement in the pigeon. Analysis of performance typical of fixed-interval responding indicated that the scalloped pattern does not result from smooth acceleration in responding, but, rather, from renewed pausing early in the interval. Individual interresponse-time (IRT) analyses provided no evidence of acceleration. There was a strong indication of alternation in shorter-longer IRTs, but these shorter-longer IRTs did not occur at random, reflecting instead a sequential dependency in successive IRTs. Furthermore, early in the interval there was a high relative frequency of short IRTs. Such a pattern of early pauses and short IRTs does not suggest behavior typical of reinforced responding as exemplified by the pattern found near the end of the interval. Thus, behavior from clearly scalloped performance can be classified into three states: postreinforcement pause, interim behavior, and terminal behavior.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

Choice, changeover, and travel: A quantitative model

Six pigeons were trained on concurrent variable-interval schedules in which responding on fixed-interval schedules was required to give access to the alternate schedule. Responding on the concurrent schedules was not allowed, after changing over had commenced, until the changeover schedule had been completed. In Parts 1 to 3 of the experiment, the changeover fixed-interval schedules were equal and were 0 s, 10 s, and 20 s, respectively. In each part, the relative frequency of reinforcement obtained on the concurrent schedules was varied over at least five conditions. In Part 4, the concurrent schedules were equal, and one changeover fixed-interval schedule was twice the other. Under these conditions, the absolute sizes of the changeover schedules were varied. Increasing the changeover requirement from 0 s to 10 s (Parts 1 and 2) resulted in increases in the sensitivity of behavior allocation to reinforcers obtained, but no further increase was obtained when the changeover schedules were increased to 20 s (Part 3). In Part 4, performance was biased towards the concurrent schedule that took less time to enter. These results are consistent with a subtractive punishment model of travel in which the degree of punishment is measured by the number of reinforcers apparently lost from a schedule when the subject changes to that schedule. Absolute times spent on the main keys could be accurately described by a previous model of changeover performance.     

Minimum rank and minimum trace of covariance matrices

This paper considers some mathematical aspects of minimum trace factor analysis (MTFA). The uniqueness of an optimal point of MTFA is proved and necessary and sufficient conditions for a point x to be optimal are established. Finally, some results about the connection between MTFA and the classical minimum rank factor analysis will be presented.     

The Headache Symptom Questionnaire: Discriminant classificatory ability and headache syndromes suggested by a factor analysis

A brief Headache Symptom Questionnaire was administered to 129 chronic headache Sufferers. The questionnaire accurately classified 68.42% of headache subjects in their proper diagnostic category, comparable to, but statistically less accurate than, the 86.4% agreement between expert headache diagnosticians using clinical interviews. Results of a factor analysis of the Headache Symptom Questionnaire lend support for their being two commonly accepted global headache categories—vascular/migraine and muscle contraction—and one headache dimension concerned with duration of headache pain. Combined migraine-muscle contraction headache was found to be related more to migraine than to muscle contraction headache, and cluster headaches emerged as a separate clinical entity, not loading positively on any factor and loading negatively on all three.This research was supported in part by Grant NS-15235 from the National Institute of Neurological and Communicative Disorders and Stroke.     

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.     

On rotating to smooth functions

Tucker has outlined an application of principal components analysis to a set of learning curves, for the purpose of identifying meaningful dimensions of individual differences in learning tasks. Since the principal components are defined in terms of a statistical criterion (maximum variance accounted for) rather than a substantive one, it is typically desirable to rotate the components to a more interpretable orientation. Simple structure is not a particularly appealing consideration for such a rotation; it is more reasonable to believe that any meaningful factor should form a (locally) smooth curve when the component loadings are plotted against trial number. Accordingly, this paper develops a procedure for transforming an arbitrary set of component reference curves to a new set which are mutually orthogonal and, subject to orthogonality, are as smooth as possible in a well defined (least squares) sense. Potential applications to learning data, electrophysiological responses, and growth data are indicated.Portions of this research were supported by the National Research Council of Canada, Grant A8615 to the second author. We thank Jagdeth Sheth for supplying his raw data.     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Comments on “the measurement of factorial indeterminacy”

Guttman's index of indeterminacy (2² – 1) measures the potential amount of uncertainty in picking the right alternative interpretation for a factor. When alternative solutions for a factor are equally likely to be correct, then the squared multiple correlation ² for predicting the factor from the observed variables is the average correlation _AB between independently selected alternative solutionsA andB, while var ( _AB )=(1 – ²)²/s, wheres is the dimensionality of the space in which unpredicted components of alternative solutions are to be found. When alternative solutions for the factor are not equally likely to be chosen, ² is the lower bound for E( _AB ); however, E( _AB ) need not be a modal value in the distribution of _AB . Guttman's index and E( _AB ) measure different aspects of the same indeterminacy problem.