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.     

Generalized bilinear models

Generalized bilinear models are presented for the statistical analysis of two-way arrays. These models combine bilinear models and generalized linear modeling, and yield a family of models that includes many existing models, as well as suggest other potentially useful ones. This approach both unifies and extends models for two-way arrays, including the ability to treat response and explanatory variables differently in the models, and the incorporation of external information about the variables directly into the analysis. A unifying framework for the generalized bilinear models is provided by considering four particular cases which have been proposed and used in the existing statistical literature. A three-step procedure is proposed to analyze data sets by generalized bilinear models. Two data sets of different nature are analyzed.     

Discriminative properties of briefly presented stimuli

In Experiment I, pigeons' responses produced food according to a fixed-interval schedule while responses on the key also produced brief stimuli according to a variable-interval schedule. Each brief stimulus reset the fixed interval. Thus, a brief stimulus occurred irregularly but a fixed minimum time separated the occurrence of food from a brief stimulus. Pauses followed brief stimuli and were followed by an accelerated response rate until another brief stimulus or food occurred. In Experiment II, four control procedures were examined. (1) Brief-stimulus presentations were omitted, producing a loss of response patterning. (2) A second-order schedule was studied with fixed-interval components. This schedule produced patterning following brief stimuli similar in kind and degree to that found in Experiment I. (3) A conjoint schedule was arranged in which food was no longer separated from the stimulus by a fixed time; pauses following the stimulus no longer resulted. (4) A brief food reinforcer replaced the brief visual stimulus, resulting in a constant response rate with no pausing following the brief food stimulus. The results suggest that the brief-stimulus effects were due to discriminative functions produced by the fixed temporal relation separating the stimulus from food.     

A COMPARISON OF THE EFFECTS OF FIXED AND VARIABLE RATIO SCHEDULES OF REINFORCEMENT ON THE BEHAVIOR OF DEAF CHILDREN

The performance of all five students in an adjustment class for deaf children was compared under fixed and variable ratio schedules of reinforcement. During the fixed ratio (FR) condition, students earned checks if they were attentive and did not engage in disruptive behavior. These checks could be exchanged for the opportunity to draw a prize from a grab bag. During the variable ratio (VR) condition, they earned a draw from the grab bag according to a variable ratio schedule with a mean ratio equal to the value of the preceding FR schedule. During the VR condition, students visually attended more and engaged in disruptive behavior less often than during the FR condition. The number of math problems completed per minute was also higher during the VR condition although no specific consequences were applied to math performance.     

Relationship between response rate and reinforcement frequency in variable-interval schedules: the effect of the concentration of sucrose reinforcement

Four rats were exposed to variable-interval schedules specifying a range of different reinforcement frequencies, using sucrose of two different concentrations and distilled water as the reinforcer. With sucrose, the rates of responding of all four rats were increasing negatively accelerated functions of reinforcement frequency, the data conforming closely to Herrnstein's equation; this was also true of the data from three of the four rats when distilled water was used as the reinforcer. The values of both constants in Herrnstein's equation were related to the sucrose concentration: the asymptotic response rate decreased, and the reinforcement frequency corresponding to the half-maximal response rate increased, with decreasing sucrose concentration.     

Schedule-induced defecation.

Excessive defecation, typically considered to be a concomitant of stress, was experimentally induced or eliminated under specific schedules of positive reinforcement of lever pressing by rats. The schedules were, by and large, those under which polydipsia is typically induced. In the first of three experiments, rats under fixed-interval 32-second schedules and variable interval 32-second schedules for food and water reinforcers defecated profusely, but not under fixed-interval one-second schedules or other small interval schedules. Somewhat higher rates of defecation were observed on variable interval 32-second schedules than on fixed-interval 32-second schedules. In a second experiment, fixed-ratio schedules were used, some of which resulted in responding such that reinforcement densities were similar to those on the interval schedules that induced defecation. Defecation was not systematically induced by these ratio schedules. In a third experiment, fixed-time schedules of food presentations were utilized. High rates of defecation were induced comparable to those induced by interval schedules of the same time parameter. No other behavior commonly termed "emotional" was observed in any of these experiments.     

The effect of punishment on free-operant choice behavior in humans

During Phase I, three female human subjects pressed a button for monetary reinforcement in five variable-interval schedules specifying different frequencies of reinforcement. On alternate days, responding was also punished (by subtracting money) according to a variable-ratio 34 schedule. In the absence of punishment, response rates conformed to Herrnstein's equation for single variable-interval schedules. Punishment suppressed responding at all frequencies of reinforcement. This was reflected in a change in the values of both constants in Herrnstein's equation: the value of the theoretical maximum response-rate parameter was reduced, and the parameter describing the reinforcement frequency corresponding to the half-maximal response rate was elevated. During Phase II, the same five schedules (A) were in operation (without punishment), but in addition, a concurrent variable-interval schedule (B) of standard reinforcement frequency was introduced. On alternate days, responding in Component B was punished according to a variable-ratio 34 schedule. In the absence of punishment, absolute response rates conformed to equations proposed by Herrnstein to describe performance in concurrent schedules; the ratios of the response rates in the two components and the ratios of the times spent in the two components conformed to the Matching Law. When responding in Component B was punished, response rates in Component B were reduced and those in Component A were elevated, these changes being reflected in distortions of the matching relationship.     

Some Effects Of Intertrial Interval Duration On Discrete-trial Choice

Pigeons were trained in Experiment 1 on a discrete-trial concurrent variable-interval (VI) 1-min VI 3-min schedule, and in Experiment 2 on a discrete-trial concurrent VI 1.5-min VI 1.5-min schedule. In each experiment, the intertrial-interval durations (ITIs) were 0 s, 6 s, 22 s, and 120 s, and the schedules were both independent and interdependent. The purpose of the research was to determine whether lengthening the ITI would disrupt any local control that existed, measured with respect to relative response rate and changeover probability. In Experiment 1, with the independent schedules, both preference and obtained relative reinforcement rate approximated .75 at short ITIs, but then decreased toward .50 with longer ITIs. With interdependent schedules, both preference and obtained relative reinforcement rate approximated .75 at all ITIs. In both experiments, with both independent and interdependent schedules, changeover probabilities for each response in a sequence of up to five successive responses to a given schedule were variable for individual birds. The average changeover probabilities for all birds suggested perseveration rather than a systematic increase within a given ITI or a systematic trend toward chance responding as ITI duration increased. Finally, the changeover functions did not differ when a sequence of responses was calculated to begin anew after reinforcement rather than with the first response on a schedule. Taken together, the data were inconsistent with a theory holding that only local processes underlie choice in discrete-trial procedures.     

Optimality And Concurrent Variable-interval Variable-ratio Schedules

Despite claims to the contrary, all leading theories about operant choice may be seen as models of optimality. Although melioration is often contrasted with global maximization, both make the same core assumptions as other versions of optimality theory, including momentary maximizing, hill climbing, and the various versions of optimal foraging theory. The present experiment aimed to test melioration against more global optimality and to apply the visit-by-visit analysis suggested by foraging theory. Rats were exposed to concurrent schedules in which one alternative was always variable-ratio 10 and the other alternative was a variable-interval schedule. Although choice relations varied from rat to rat, the overall results roughly confirmed the matching law, a result often taken to support melioration. Pooling the data across sessions and across rats, however, resulted in no increment in unsystematic variance, lending support to the contention by Ziriax and Silberberg (1984) that the choice relation is partly constrained. When the data were analyzed at the level of visits, the results either disconfirmed predictions of melioration or showed regularities about which melioration is silent. Instead, performance tended toward a rough optimization, in which responding favored the variable ratio, but with relatively brief visits to the variable interval. There were no asymmetries in travel or variability that would indicate that different processes were involved in generating visits at the two different schedules. The findings point toward a more global optimality model than melioration and demonstrate the value of per-visit analysis in the study of concurrent performances.     

Bull's Theorem by the Method of Diagrams

We show how to use diagrams in order to obtain straightforward completeness theorems for extensions of K4.3 and a very simple and constructive proof of Bull's theorem: every normal extension of S4.3 has the finite model property.