The generalized matching law as a description of multiple-schedule responding

The literature was examined to determine how well the generalized matching law (Baum, 1974) describes multiple-schedule responding. In general, it describes the data well, accounting for a median of 91% of the variance. The median size of the undermatching parameter was 0.46; the median bias parameter was 1.00. The size of the undermatching parameter, and the proportion of the variance accounted for by the equation, varied inversely with the number of schedules conducted, with the number of sessions conducted per schedule, and with the time within a component. The undermatching parameter also varied with the operanda used to produce reinforcers and with the reinforcer used. The undermatching parameter did not vary consistently with component duration or with several other variables. Bias was greater when fewer rather than more schedules were conducted, when two rather than one operanda were used, and when White Carneaux rather than homing pigeons served as subjects. These results imply that the generalized matching law may describe both concurrent and multiple-schedule responding, but that the same variables do not always influence the bias and undermatching parameters in the same way for the two types of schedules.     

A latent trait theory via a stochastic learning theory for a knowledge space

To capture the cognitive organization of a set of questions or problems pertaining to a body of information, Doignon and Falmagne have proposed, and analyzed in a number of papers, the concept of aknowledge space, that is, a distinguished collection of subsets of questions, representing the possibleknowledge states. This collection of sets is assumed to satisfy a number of conditions. Since this concept is a deterministic one, the problem of empirical testing arises. A stochastic version of a knowledge space is developed in this paper, in which the knowledge states are considered as possible epochs in a subject's learning history. The knowledge space is decomposed as a union of a number of possible learning paths, calledgradations. The model specifies how a subject is channelled through and progresses along a gradation. A probabilistic axiom of the local indepencence type relates the knowledge states to the observable responses. The predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions. An application of the model to artificial data is described, based on maximum likelihood methods. The statistical analysis is shown to be capable of revealing the combinatoric core of the model.This work was supported by NSF grant IST-8418860 and ARI grant DAAG29-84-G-0083 to New York University. I am grateful to Jean-Paul Doignon, Mathieu Koppen, Geoff Iverson and Michael Landy for their reactions to previous drafts of this paper, to Michael Villano for carrying out the computer simulation and the analysis of the simulated data, and especially to one referee for his very useful comments.     

Assessing local influence for specific restricted likelihood: Application to factor analysis

In restricted statistical models, since the first derivatives of the likelihood displacement are often nonzero, the commonly adopted formulation for local influence analysis is not appropriate. However, there are two kinds of model restrictions in which the first derivatives of the likelihood displacement are still zero. General formulas for assessing local influence under these restrictions are derived and applied to factor analysis as the usually used restriction in factor analysis satisfies the conditions. Various influence schemes are introduced and a comparison to the influence function approach is discussed. It is also shown that local influence for factor analysis is invariant to the scale of the data and is independent of the rotation of the factor loadings. The authors are most grateful to the referees, the Associate Editor, and the Editor for helpful suggestions for improving the clarity of the paper.     

Multivariate normal integrals and contingency tables with ordered categories

Ak-dimensional multivariate normal distribution is made discrete by partitioning thek-dimensional Euclidean space with rectangular grids. The collection of probability integrals over the partitioned cubes is ak-dimensional contingency table with ordered categories. It is shown that loglinear model with main effects plus two-way interactions provides an accurate approximation for thek-dimensional table. The complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables. This approach imposes no restriction on the correlation matrix. Comparisons with other numerical integration algorithms are reported. The approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories.The contingency-table approach occurred to me while I was collaborating with Paul Holland of the Educational Testing Service in 1985 on bivariate dependence functions. Holland maintains a belief that the continuous can learn from the discrete. This work is a reassertion of his claim.This research was sponsored by the National Science Council, Republic of China.     

Key pecking in pigeons produced by pairing keylight with inaccessible grain

In Experiment I, keylight was paired with inaccessible grain delivery (under two conditions of keylight intensity) to determine if autoshaping would occur in the absence of primary reinforcement. In Experiment II, the procedure was repeated with accessible grain, for comparison. In Experiment III, the procedures were repeated with explicitly unpaired presentations of keylight and either inaccessible or accessible grain. The results indicated that key pecking occurred as quickly in the presence of keylight pairings with inaccessible grain as with accessible grain, though (except for one bird) key pecking was not maintained with inaccessible grain. Furthermore, compared to the dim keylight, the bright keylight greatly suppressed key pecking when paired with inaccessible grain, and reduced the rate of key pecking when paired with accessible grain. Little key pecking occurred in groups exposed to explicitly unpaired presentations of keylight (whether bright or dim) and grain (whether accessible or inaccessible). When the birds in Experiment III were retested with explicitly paired presentations of keylight and grain, little key pecking was observed, suggesting suppressive effects of prior explicitly unpaired presentations. It is suggested that the effects of key-brightness manipulation were produced by the association of grain with cues other than the response key, or by distraction produced by partial illumination of the grain hopper.     

Molecular contingencies: reinforcement probability

Pigeons obtained food by responding in a discrete-trials two-choice probability-learning experiment involving temporal stimuli. A given response alternative, a left- or right-key peck, had 11 associated reinforcement probabilities within each session. Reinforcement probability for a choice was an increasing or a decreasing function of the time interval immediately preceding the choice. The 11 equiprobable temporal stimuli ranged from 1 to 11 sec in 1-sec classes. Preference tended to deviate from probability matching in the direction of maximizing; i.e., the percentage of choices of the preferred response alternative tended to exceed the probability of reinforcement for that alternative. This result was qualitatively consistent with probability-learning experiments using visual stimuli. The result is consistent with a molecular analysis of operant behavior and poses a difficulty for molar theories holding that local variations in reinforcement probability may safely be disregarded in the analysis of behavior maintained by operant paradigms.     

Local contrast in multiple schedules: the effect of stimulus discriminability.

A three-ply multiple schedule assessed responding in a standard component as a function of the just-preceding schedule. The principal experimental condition was the difference among the wavelengths signaling the schedule components. Only the pigeons working in a narrow wavelength range showed persistent positive local contrast; that is, response rate during the standard component was higher when that component followed extinction than when it followed itself. Birds in both narrow- and medium-range groups showed persistent negative local contrast; that is, rate was lower following a relatively rich component. The dissipation of positive contrast appeared to be most clearly related to the establishment of differential responding. Negative contrast was inversely related to wavelength differences. Theories pertaining to contrast must account for the role of discrimination in both positive and negative types.     

Choice for aperiodic versus periodic ratio schedules: A comparison of concurrent and concurrent-chain procedures

Choice between mixed-ratio schedules, consisting of equiprobable ratios of 1 and 99 responses per reinforcement, and fixed-ratio schedules of food reinforcement was assessed by two commonly used procedures: concurrent schedules and concurrent-chains schedules. Rats were trained under concurrent fixed-ratio mixed-ratio schedules, in which both ratio schedules were simultaneously available, and under a concurrent-chains schedule, in which access to one of the mutually exclusive ratio schedules comprising the terminal links was contingent on a single “choice” response. The distribution of responses between the two ratio schedules was taken as the choice proportion under the concurrent procedure, and the distribution of “choice” responses was taken as the choice proportion under the concurrent-chains procedure. Seven of eight rats displayed systematic choice; of those, each displayed nearly exclusive choice for fixed-ratio 35 to the mixed-ratio schedule under the concurrent procedure, but each displayed nearly exclusive choice for the mixed-ratio schedule to fixed-ratio 35 under the concurrent-chains procedure. Thus, preference for a fixed or a mixed schedule of reinforcement depended on the procedure used to assess preference.     

Contrast and autoshaping in multiple schedules varying reinforcer rate and duration

Thirteen master pigeons were exposed to multiple schedules in which reinforcement frequency (Experiment I) or duration (Experiment II) was varied. In Phases 1 and 3 of Experiment I, the values of the first and second components' random-interval schedules were 33 and 99 seconds, respectively. In Phase 2, these values were 99 seconds for both components. In Experiment II, a random-interval 33-second schedule was associated with each component. During Phases 1 and 3, the first and second components had hopper durations of 7.5 and 2.5 seconds respectively. During Phase 2, both components' hopper durations were 2.5 seconds. In each experiment, positive contrast obtained for about half the master subjects. The rest showed a rate increase in both components (positive induction). Each master subject's key colors and reinforcers were synchronously presented on a response-independent basis to a yoked control. Richer component key-pecking occurred during each experiment's Phases 1 and 3 among half these subjects. However, none responded during the contrast condition (unchanged component of each experiment's Phase 2). From this it is inferred that autoshaping did not contribute to the contrast and induction findings among master birds. Little evidence of local contrast (highest rate at beginning of richer component) was found in any subject. These data show that (a) contrast can occur independently from autoshaping, (b) contrast assays during equal-valued components may produce induction, (c) local contrast in multiple schedules often does not occur, and (d) differential hopper durations can produce autoshaping and contrast.     

Motion as a natural category for pigeons: Generalization and a feature-positive effect

Three groups of pigeons were trained with a modified discriminative autoshaping procedure to discriminate video images of other pigeons on the basis of movement. Birds of all groups were shown the same video images of other pigeons, which were either moving or still. The group to whom food was presented only after moving images learned the discrimination very quickly. A second group, to whom food was given only after still images, and a pseudocategory group, to whom food was presented after arbitrarily chosen stimuli, showed no evidence of discrimination during acquisition training. Extinction conditions led to clear differences in peck rates to moving and still images in the second group but not in the pseudocategory group. The result is related to the feature-positive effect. Generalization tests showed that the discrimination performance was based on visual features of the stimuli but was invariant against changes of size, perspective, brightness, and color. Furthermore, discrimination was maintained when novel images of pigeons under different viewing angles and seven other types of motion categories were presented. It is argued that the discrimination is based not on a common motion feature but on motion concepts or high-order generalization across motion categories.