Confidence regions for multidimensional scaling analysis

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Intertrial sources of stimulus control and delayed matching-to-sample performance in humans

Two experiments compared delayed matching-to-sample (DMTS) accuracy under 2 procedures in adults with mental retardation. In the trial-unique procedure, every trial in a session contained different stimuli. Thus, comparison stimuli that were correct on one trial were never incorrect on other trials in that session (or vice versa). In the 2-sample DMTS procedure, the same 2 comparison stimuli were presented on each trial, and their function changed quasi-randomly across trials conditional upon the sample stimulus. Across 2 experiments, 7 of 8 subjects showed the highest overall accuracy under the trial-unique procedure, and no subject showed consistently higher accuracy under the 2-sample procedure. Negative, exponential decay functions fit to logit p values showed that this difference was due largely to the steeper delay-mediated decline in sample control for the 2-sample procedure. Stimulus-control analyses indicated that, under the 2-sample procedure, the selection of the comparison stimulus on Trial N was often controlled by the comparison stimulus selection on Trial N-1 rather than the Trial-N sample stimulus. This source of competing stimulus control is not present in trial-unique procedures. Experiment 2 manipulated intertrial interval duration. There was a small but consistent increase in accuracy as a function of intertrial interval duration under the 2-sample procedure, but not under the trial-unique procedure.     

Emergent, untrained stimulus relations in many-to-one matching-to-sample discriminations in rats

The present experiment investigated whether rats formed emergent, untrained stimulus relations in many-to-one matching-to-sample discriminations. In Phase 1, rats were trained to match two samples (triangle and horizontal stripes) to a common comparison (horizontal stripes) and two additional samples (circle or vertical stripes) to another comparison (vertical stripes). Then, in Phase 2, the rats were trained to match the one sample (triangle) to a new comparison (black) and the other sample (circle) to another comparison (white). In the Phase 3 test, half the rats (consistent group) were given two new tasks in which the sample-correct comparison relation was consistent with any emergent stimulus relations that previously may have been learned. The remaining 6 rats (inconsistent group) were given two new tasks in which the sample-correct comparison relation was not consistent with any previously learned emergent stimulus relations. Rats in the consistent group showed more accurate performance at the start of Phase 3, and faster learning to criterion in this phase, as compared with rats in the inconsistent group. This finding suggests that rats may form emergent, untrained stimulus relations between the discriminative stimuli in many-to-one matching-to-sample discriminations.     

Simple and conditional visual discrimination with wheel running as reinforcement in rats.

Three experiments explored whether access to wheel running is sufficient as reinforcement to establish and maintain simple and conditional visual discriminations in nondeprived rats. In Experiment 1, 2 rats learned to press a lit key to produce access to running; responding was virtually absent when the key was dark, but latencies to respond were longer than for customary food and water reinforcers. Increases in the intertrial interval did not improve the discrimination performance. In Experiment 2, 3 rats acquired a go-left/go-right discrimination with a trial-initiating response and reached an accuracy that exceeded 80%; when two keys showed a steady light, pressing the left key produced access to running whereas pressing the right key produced access to running when both keys showed blinking light. Latencies to respond to the lights shortened when the trial-initiation response was introduced and became much shorter than in Experiment 1. In Experiment 3, 1 rat acquired a conditional discrimination task (matching to sample) with steady versus blinking lights at an accuracy exceeding 80%. A trial-initiation response allowed self-paced trials as in Experiment 2. When the rat was exposed to the task for 19 successive 24-hr periods with access to food and water, the discrimination performance settled in a typical circadian pattern and peak accuracy exceeded 90%. When the trial-initiation response was under extinction, without access to running, the circadian activity pattern determined the time of spontaneous recovery. The experiments demonstrate that wheel-running reinforcement can be used to establish and maintain simple and conditional visual discriminations in nondeprived rats.     

On the sampling interpretation of confidence intervals and hypothesis tests in the context of conditional maximum likelihood estimation

In the context ofconditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given theincidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that theconditional asymptotic confidence intervals are also valid under the other two sampling schemes. I am indebted to Theo Eggen, Norman Verhelst and one of Psychometrika's reviewers for their helpful comments.     

A parametric procedure for ultrametric tree estimation from conditional rank order proximity data

The psychometric and classification literatures have illustrated the fact that a wide class of discrete or network models (e.g., hierarchical or ultrametric trees) for the analysis of ordinal proximity data are plagued by potential degenerate solutions if estimated using traditional nonmetric procedures (i.e., procedures which optimize a STRESS-based criteria of fit and whose solutions are invariant under a monotone transformation of the input data). This paper proposes a new parametric, maximum likelihood based procedure for estimating ultrametric trees for the analysis of conditional rank order proximity data. We present the technical aspects of the model and the estimation algorithm. Some preliminary Monte Carlo results are discussed. A consumer psychology application is provided examining the similarity of fifteen types of snack/breakfast items. Finally, some directions for future research are provided.     

REVERSAL OF BASELINE RELATIONS AND STIMULUS EQUIVALENCE: II. CHILDREN

In a systematic replication of a study using college-student subjects (Pilgrim & Galizio, 1990), 5- to 7-year-old children learned two conditional discriminations (i.e., A1B1, A2B2, A1C1, and A2C2) in a two-choice arbitrary match-to-sample task and showed the emergence of two three-member equivalence classes (A1B1C1 and A2B2C2). Baseline conditional discrimination performances were quickly controlled by reversals of the AC reinforcement contingencies (i.e., choosing Comparison Stimulus C2 was reinforced given Sample A1, and choosing C1 was reinforced given Sample A2) when the reversals were introduced in restricted baselines. On reflexivity, symmetry, and transitivity/equivalence probes following the reversal, there was some limited indication of equivalence-class reorganization (i.e., A1B1C2 and A2B2C1) in keeping with the concurrently performed baseline relations for 2 of 5 subjects, but the predominant pattern across probe trials was one of inconsistent conditional control. These findings suggest that, given similar challenges, equivalence-class performances may be more easily disrupted in young children than in adults.     

Rasch models for item bundles

This paper discusses the application of a class of Rasch models to situations where test items are grouped into subsets and the common attributes of items within these subsets brings into question the usual assumption of conditional independence. The models are all expressed as particular cases of the random coefficients multinomial logit model developed by Adams and Wilson. This formulation allows a very flexible approach to the specification of alternative models, and makes model testing particularly straightforward. The use of the models is illustrated using item bundles constructed in the framework of the SOLO taxonomy of Biggs and Collis.The work of both authors was supported by fellowships from the National Academy of Education Spencer Fellowship.     

Development of conditional and equivalence relations without differential consequences

Two experiments were conducted to establish conditional stimulus relations without differential consequences and to test for the emergence of other relations. In Experiment 1, 3 adults responded to match-to-sample displays in which sample-comparison pairs were constant while the second comparison presented with each pair changed periodically across trials. No differential consequences followed any comparison selections. All subjects learned conditional relations between constant samples and comparisons, but results of tests for transitivity in those relations were equivocal. In Experiment 2, 4 children were given unreinforced training and testing similar to that provided to the adults in Experiment 1, with procedural refinements. One child learned conditional relations and demonstrated emergent relations that confirmed the development of two four-member equivalence classes. Another child learned the conditional relations but did not demonstrate any emergent relations reliably. A 3rd child, after reinforced training on two conditional relations, learned four new conditional relations without differential consequences and demonstrated symmetry but not equivalence in the trained relations. The 4th child did not learn the conditional relations. These findings emphasize the importance of careful construction of tests for stimulus equivalence and suggest a need for critical analyses of the apparent emergence of untrained stimulus relations on unreinforced tests that has been observed in several stimulus equivalence studies.     

Naming in conditional discrimination and stimulus equivalence.

Using a matching-to-sample procedure, McIntire, Cleary, and Thompson (1987) taught monkeys the conditional relations A1-R1-A1-R1, A2-R2-A2-R2, A1-R1-B1-R1, A2-R2-B2-R2, B1-R1-C1-R1, and B2-R2-C2-R2, where the first and third terms in each relation refer to the sample and comparison stimuli, respectively, and the second and last terms refer to the emission of a distinctive pattern of responding. The subjects were then tested for the emergent relations A-C, C-A, B-A, C-B, and B-B, with the differential response produced by a given stimulus during training also emitted on test trials (e.g., A1-R1-C1-R1). The performances of both subjects were as accurate on the tested relations as they had been on the trained relations. The new relations were characterized as demonstrations of stimulus equivalence. However, the conditional discrimination literature shows that such training procedures generate control of comparison selection by the differential response patterns. Therefore, no emergent relations were demonstrated because all of the trained response-stimulus relations were preserved on test trials. This paper suggests that these procedures do not provide an appropriate analogy for the kind of emergent stimulus-stimulus relations exhibited by human subjects in equivalence studies and outlines a paradigm for assessing the relative influence of stimulus-stimulus and response-stimulus relations.