Effects of reinforcement context on choice

Two experiments investigated the effects of successive reinforcement contexts on choice. In the first, concurrent variable-interval schedules of primary reinforcement operated during the initial links of concurrent chains. The rate of this reinforcement arranged by the concurrent schedules was decreased across conditions: When it was higher than the terminal-link rate, preference for the higher frequency initial-link schedule increased relative to baseline. (During baseline, a standard concurrent-schedule procedure was in effect). When the initial-link reinforcement rate was lower than the terminal-link rate, preference converged toward indifference. In the second experiment, a chain schedule was available on a third key while a concurrent schedule was in effect on the side keys. When the terminal link of the chain schedule was produced, the side keys became inoperative. Availability of the chain schedule did not affect choice between the concurrent schedules. These results show that only when successive reinforcement contexts are produced by choice responding do those successive contexts affect choice in concurrent schedules.     

Aggregate item response analysis

A stochastic postulate is given for the multiple-item, successive-intervals scaling of populations. The logistic equivalent of this postulate provides an aggregate item response model in which a unidimensional submodel may be nested. This reduction provides a subtractive conjoint measurement of several items and stimuli on the same latent scale. Generalized-least-squares methods are used to estimate and test the multiple-item model, and its unidimensional reduction, on aggregate survey responses. The entire procedure is illustrated with an analysis of semantic-differential attitude data. This analysis exhibits an item selection procedure that is applicable to various social constructs.The authors dedicate this paper to the memory and contributions of Clyde Coombs.The programming and data analyses for the present paper were carried out by José Ventura of the Department of Industrial and Systems Engineering, and Jerry Meiten of the Department of Statistics, University of Florida.The study was also supported by the College of Business Administration, University of Florida, and the Faculty of Social Sciences, Hebrew University of Jerusalem.     

Canonical analysis of two convex polyhedral cones and applications

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples.     

Generalized approaches to the maxbet problem and the maxdiff problem,with applications to canonical correlations

Van de Geer has reviewed various criteria for transforming two or more matrices to maximal agreement, subject to orthogonality constraints. The criteria have applications in the context of matching factor or configuration matrices and in the context of canonical correlation analysis for two or more matrices. The present paper summarizes and gives a unified treatment of fully general computational solutions for two of these criteria, Maxbet and Maxdiff. These solutions will be shown to encompass various well-known methods as special cases. It will be argued that the Maxdiff solution should be preferred to the Maxbet solution whenever the two criteria coincide. Horst's Maxcor method will be shown to lack the property of monotone convergence. Finally, simultaneous and successive versions of the Maxbet and Maxdiff solutions will be treated as special cases of a fully flexible approach where the columns of the rotation matrices are obtained in successive blocks.The author is obliged to Henk Kiers for computational assistance and helpful comments.     

The autoshaping procedure as a residual block clock

In the first experiment, 4 pigeons were each presented with a recurring sequence of four key colors followed by the delivery of grain (block clock). Once the rate of pecking had stabilized, three of the colors were replaced, during different series of sessions, by a darkening of the key. The rate of pecking was reduced within those segments of the interval between deliveries of food during which the key was dark; when the key was dark during the final portion of the interval, rates were reduced throughout the entire interval. In the second experiment, 3 new pigeons were exposed to a different sequence of colors, and the final stimulus was replaced in successive conditions by a novel color, a darkened key, and a restoration of the original color. The data indicated that darkening the key had a more severe, more extensive, and more persistent effect than did a mere change in color. These results suggest that it may be fruitful to conceptualize the autoshaping procedure as a special version of the block clock in which pecking is suppressed throughout the greater part of the interval by darkening the key. In the final condition, the same stimulus appeared in each of the last three portions of the interval. The rate of pecking was lower during the last two portions than when distinctive colors were presented, with the peak rate now appearing in the fifth of seven equal temporal components.     

Constructing the exact significance level for a person fit statistic

This paper uses an extension of the network algorithm originally introduced by Mehta and Patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the Rasch model. By assuming that item difficulties are known, the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score. A simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less.     

Children's choice: Sensitivity to changes in reinforcer density

Two experiments were carried out in which children's sensitivity to changes in reinforcer density (number of reinforcers per session) was measured in a choice paradigm. In Experiment 1, 24 girls (ages 6, 9, and 12 years) performed on concurrent-chain schedules of reinforcement. The initial links were variable-interval 10-s schedules. One terminal link always gave three tokens after 30 s, but the parameters associated with the other were varied. Independent manipulations of reinforcer size (two tokens or four tokens) and prereinforcement delay (25 s or 65 s) led to equal changes in the relative density of tokens that could be earned on the schedules. Subjects at all ages were sensitive to changes in reinforcer density brought about by changes in reinforcer size, whereas only 3 12-year-olds showed sensitivity to the changes brought about by manipulation of prereinforcer delay. In Experiment 2, titration procedures were used to test the extent of this insensitivity to delay in 32 6- and 12-year-old children. In these procedures, a repeated choice of the large reinforcer increased the delay to its delivery, and a repeated choice of the small reinforcer reduced the delay to the delivery of the large reinforcer. Whereas 6-year-old boys and girls tended to maintain a strong preference for the large reinforcer, so increasing the delay to its delivery, 12-year-olds tended to distribute their responses to both alternatives, thus producing a stable level of delay to the large reinforcer. The results from the two experiments support the idea of two stages in the development of adaptive intertemporal choice.     

Estimation for structural equation models with missing data

A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.This research was supported in part by Research Grant DAD1070 from the U.S. Public Health Service. The author is indebted to anonymous reviewers for some very valuable suggestions. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Extending exploratory diagnostic classification models: Inferring the effect of covariates

Diagnostic models provide a statistical framework for designing formative assessments by classifying student knowledge profiles according to a collection of fine-grained attributes. The context and ecosystem in which students learn may play an important role in skill mastery, and it is therefore important to develop methods for incorporating student covariates into diagnostic models. Including covariates may provide researchers and practitioners with the ability to evaluate novel interventions or understand the role of background knowledge in attribute mastery. Existing research is designed to include covariates in confirmatory diagnostic models, which are also known as restricted latent class models. We propose new methods for including covariates in exploratory RLCMs that jointly infer the latent structure and evaluate the role of covariates on performance and skill mastery. We present a novel Bayesian formulation and report a Markov chain Monte Carlo algorithm using a Metropolis-within-Gibbs algorithm for approximating the model parameter posterior distribution. We report Monte Carlo simulation evidence regarding the accuracy of our new methods and present results from an application that examines the role of student background knowledge on the mastery of a probability data set.     

Momentary maximizing in concurrent schedules with a minimum interchangeover interval

Eight pigeons were trained on concurrent variable-interval variable-interval schedules with a minimum interchangeover time programmed as a consequence of changeovers. In Experiment 1 the reinforcement schedules remained constant while the minimum interchangeover time varied from 0 to 200 s. Relative response rates and relative time deviated from relative reinforcement rates toward indifference with long minimum interchangeover times. In Experiment 2 different reinforcement ratios were scheduled in successive experimental conditions with the minimum interchangeover time constant at 0, 2, 10, or 120 s. The exponent of the generalized matching equation was close to 1.0 when the minimum interchangeover time was 0 s (the typical procedure for concurrent schedules without a changeover delay) and decreased as that duration was increased. The data support the momentary maximizing theory and contradict molar maximizing theories and the melioration theory.