Conditional discrimination and equivalence relations: A theoretical analysis of control by negative stimuli

A detailed analysis is presented of the ways in which control by the negative stimulus in two-comparison conditional discriminations may be expected to affect the outcome of tests for the properties of equivalence relations. Control by the negative stimulus should produce the following results: (a) no observable effect on symmetry tests; (b) reflexivity test results should look like “oddity” rather than “identity”; and (c) transitivity tests that involve an odd number of nodes should yield results that are 100% opposite to tests that involve an even number of nodes. The analysis also considers the effects of variation in the type of comparison-stimulus control between and within baseline conditional discriminations. Methods are suggested for experimentally regulating the type of control, and for verifying the predictions that the analysis generates. If suggested experiments continue to support the analysis, investigators who use two-comparison conditional discriminations to study equivalence relations will either have to control explicitly whether the positive or the negative comparison governs their subjects' choices, or they will have to abandon two comparisons and use three or more comparisons instead.     

Organic Unities

I defend the view that there are organic unities mainly by presenting examples of organic unities. I also defend the view against two objections. The first objection appeals to the notion of an evaluatively incomplete state of affairs. The second objection holds that the intrinsic value of a state of affairs can be different in different contexts. I argue that neither objection provides a compelling reason for rejecting these examples.     

Some remarks on Scheiblechner's treatment of isop models

Scheiblechner (1995) proposes a probabilistic axiomatization of measurement called ISOP (isotonic ordinal probabilistic models) that replaces Rasch's (1980) specific objectivity assumptions with two interesting ordinal assumptions. Special cases of Scheiblechner's model include standard unidimensional factor analysis models in which the loadings are held constant, and the Rasch model for binary item responses. Closely related are the doubly-monotone item response models of Mokken (1971; see also Mokken & Lewis, 1982; Sijtsma, 1988; Molenaar, 1991; Sijtsma & Junker, 1996; and Sijtsma & Hemker, in press). More generally, strictly unidimensional latent variable models have been considered in some detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg (1993), and Junker (1990, 1993). The purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from Scheiblechner's (1995) paper, and to point out important related work by Hemker, Sijtsma, Molenaar, & Junker (1996, 1997), Ellis and Junker (in press) and Junker and Ellis (1997). We also discuss counterexamples to three major theorems in the paper. By carrying out these three tasks, we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the ISOP approach the careful attention it deserves. This research was supported by the National Science Foundation, Grant DMS-94.04438. I thank the Editor and three anonymous referees for careful readings that improved the completeness of this note.     

Tail-measurability in monotone latent variable models

We consider latent variable models for an infinite sequence (or universe) of manifest (observable) variables that may be discrete, continuous or some combination of these. The main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality, monotonicity, conditional independence, andtail-measurability. Tail-measurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed. The characterizing,necessary and sufficient, conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence (as one conditions upon successively more other manifest variables). Our main theorem considerably generalizes and sharpens earlier results of Ellis and van den Wollenberg (1993), Holland and Rosenbaum (1986), and Junker (1993). It is also related to the work of Stout (1990).The main theorem is preceded by many results for latent variable modelsin general—not necessarily unidimensional and monotone. They pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of Suppes and Zanotti (1981). We discuss new definitions of the concepts of true-score and subpopulation, which generalize these notions from the stochastic subject, random sampling, and domain sampling formulations of latent variable models (e.g., Holland, 1990; Lord & Novick, 1968). These definitions do not require the a priori specification of a latent variable model.The authors made equivalent contributions to the results of this article. Ellis' research was supported by the Dutch Interuniversitary Graduate School of Psychometrics and Sociometrics. Junker's research was supported by ONR Grant N00014-87-K-0277, NIMH Grant MH15758, and a Carnegie Mellon University Faculty Development Grant. In addition Junker would like to acknowledge the hospitality of the Nijmegen Institute for Cognition and Information during his visit to the University of Nijmegen in August 5–10, 1993.     

Sample-specific ratio effects in matching to sample

In a symbolic matching-to-sample task, pigeons were trained using sample-specific, fixed-ratio “observing responses.” Subsequently, in a mixed condition, each sample was presented equally often with each ratio requirement, i.e., the ratios were no longer correlated with the samples. In a second experiment, pigeons were trained initially in the mixed condition and subsequently shifted to the sample-specific condition in which the required ratios were correlated with the samples. Results of both experiments suggested joint control of choices by ratio value and by the exteroceptive stimuli. The discriminative properties of the ratios appeared to outweigh absolute ratio-size effects.     

The role of physical identity of the sample and correct comparison stimulus in matching-to-sample paradigms

Pigeons were trained in a higher-order conditional discrimination paradigm to assess the role of physical identity in a within-subjects design. A line orientation which was super-imposed on all response keys signalled whether a response to the matching color or a response to the nonmatching color was correct. Following training under this paradigm, stimulus control gradients were obtained by varying the angularity of the lines. Orderly gradients of stimulus control were obtained and no bias toward or away from the physically identical comparison stimulus was observed. The data were interpreted as indicating that the pigeons acquired a discrimination for each specific stimulus configuration or a set of specific stimulus-response chains based on compound stimuli in which physical identity played no special role.     

Repeated acquisition of conditional discriminations

A new technique was developed to study the repeated acquisition of conditional discriminations. Using a discrete trial procedure, pigeons were required to learn during each session a different two-member chain of conditional discriminations. Key color and geometric forms were used as stimuli. After the pigeons had reached a steady state of relearning (40 to 60 sessions), the technique was used to investigate variables that have previously been shown to affect the repeated acquisition of response sequences. Various (0 to 90 seconds) durations of timeout for errors were investigated in Experiment I. The stimulus change associated with a timeout, rather than its duration, was found to be the critical variable in acquisition of the discrimination. Extended training on a single chain was found to reduce total errors across sessions in Experiment II. Extended training (three sessions) did not, however, change the pattern of within-session error reduction. In some cases, extended training facilitated acquisition of a partially reversed discrimination. In Experiment III, color rather than chain position was found to control behavior, for three of the four birds, as the second stimulus dimension in the conditional situation. The results of these experiments replicate and extend previous findings concerning some of the variables that affect the repeated acquisition of response sequences.     

A search for symmetry in the conditional discriminations of rhesus monkeys, baboons, and children.

Procedures for generating arbitrary matching-to-sample performances may generate only conditional discriminations. Rational grounds for this distinction are proposed, based on the properties that any equivalence relation must possess. Empirical tests are described for determining whether subjects trained on conditional discriminations are also engaged in true matching to sample. A series of studies than leads to the conclusion that proof of true matching to sample by monkeys, pigeons, or baboons is yet to be provided. Whether the absence of such proof reflects experiential factors or species-defined limitations is not presently clear.     

Food delivery as a conditional stimulus: Feature-learning and memory in pigeons

Three experiments investigated the learning and memory of discriminations based on presence versus absence of a pre-trial food delivery. In Experiment 1 half the illuminations of a response key were followed by food regardless of the subject's behavior. In one group an extra food delivery preceded only reinforced trials (feature-positive condition), whereas in a second group it preceded only nonreinforced trials (feature-negative condition). Key pecks and approaches revealed more rapid and superior discrimination learning in the first group. Experiment 2 replicated the results of Experiment 1 but yielded no evidence that greater “unexpectedness” of pretrial food conditions facilitates discriminative performance. In Experiment 3, individual pigeons trained on a conditional discrimination exhibited a within-subject feature-positive superiority. Delay between pretrial and trial stimuli interacted with feature-positive versus feature-negative training in both the between-group (Experiment 2) and within-subject (Experiment 3) procedures: performance was decremented at both short and long delays in the feature-positive condition but was decremented only at longer delays in the feature-negative condition. The feature-positive superiority obtained here is incompatible with explanations based on either the general concept of “perceptual organization” or on the conditional nature of feature-negative discriminations.     

The effects of number of sample stimuli and number of choices in a detection task on measures of discriminability

Six pigeons were trained on a conditional discrimination task involving the discrimination of various intensities of yellow light. The research asked whether stimulus—response discriminability measures between any pair of stimuli would remain constant when a third or fourth sample and reinforced response were added. The numbers of different sample stimuli presented and different responses reinforced were two (Part 1), three (Parts 2 and 4), and four (Part 3). Across conditions within parts, the ratios of reinforcers obtainable for correct responses were varied over at least five levels. In Part 5, the numbers of sample stimuli and reinforced responses were varied among two, three, and four, and the reinforcer ratio between consecutive remaining samples was constant at 2:1. It was found that once a particular response had been reinforced, subjects continued to emit that response when the conditional stimulus for that response was no longer presented. Data analysis using a generalization-based detection model indicated that this model was able to describe the data effectively. Four findings were in accord with the theory. First, estimates of stimulus—response discriminability usually decreased as the arranged physical disparity between the sample stimuli decreased. Second, stimulus—response discriminability measures were independent of response—reinforcer discriminability measures, preserving parameter invariance between these measures. Third, stimulus—response discriminability measures for constant pairs of conditional stimuli did not change systematically as conditional stimulus—response alternatives were added. Fourth, log stimulus—response discriminability values between physically adjacent conditional stimuli summed to values that were not significantly different from estimates of the discriminability values for conditional stimuli that were spaced further apart.