A stochastic theory of matching processes

A theory of matching processes is developed within which serial models and parallel models based on within-stage independent intercompletion times are defined. These models are then specialized to a class of processes possessing exponential intercompletion time densities and equivalence properties of parallel and serial models within this class are investigated. Nonequivalence theorems are proved that designate possible differences in “same” (+) and different (?) matching rates as important in testing parallel and serial models. An experimental paradigm is then derived with the property that if + rates are different from — rates, and if processing is self-terminating, then the parallel and serial models are distinguishable at the level of mean reaction times. The serial class of models that is tested by this paradigm includes a large number of stochastic distributions whose central assumption is additivity of element processing times. The corresponding parallel class of models is currently limited to those assuming exponential intercompletion times. A numerical example and an example of non-parametric relations predicted by the serial or parallel models are given. Some advantages and limitations of the present treatment are discussed.     

Effects of stimulus order on discrimination processes in comparative and equality judgements: Data and models

In typical discrimination experiments, participants are presented with a constant standard and a variable comparison stimulus and their task is to judge which of these two stimuli is larger (comparative judgement). In these experiments, discrimination sensitivity depends on the temporal order of these stimuli (Type B effect) and is usually higher when the standard precedes rather than follows the comparison. Here, we outline how two models of stimulus discrimination can account for the Type B effect, namely the weighted difference model (or basic Sensation Weighting model) and the Internal Reference Model. For both models, the predicted psychometric functions for comparative judgements as well as for equality judgements, in which participants indicate whether they perceived the two stimuli to be equal or not equal, are derived and it is shown that the models also predict a Type B effect for equality judgements. In the empirical part, the models' predictions are evaluated. To this end, participants performed a duration discrimination task with comparative judgements and with equality judgements. In line with the models' predictions, a Type B effect was observed for both judgement types. In addition, a time-order error, as indicated by shifts of the psychometric functions, and differences in response times were observed only for the equality judgement. Since both models entail distinct additional predictions, it seems worthwhile for future research to unite the two models into one conceptual framework.     

Novelty as a reinforcer for position learning in children

The stimulus-familiarization-effect (SFE) paradigm, a reaction-time (RT) task based on a response to novelty procedure, was modified to assess response for novelty, i.e., a response-reinforcement sequence. Ninetysix kindergarten children were familiarized on one of two colored lights. Then Ss were confronted with a simple position discrimination task in which they could choose between two response buttons which produced either the previously familiarized light or a new (i.e., novel) light. Results showed that the choice response for the novel stimulus (NS) was facilitated by three variables: number of exposures to and duration per exposure of the familiar stimulus (FS), and degree of change provided by the NS relative to the FS. The parallel between the effective variables of this study and those which produce the SFE was noted, implicating an attention interpretation of the function of novelty as a reinforcer. The potential implications of attention for reinforcement theory and learning in general are discussed.     

Changes in threshold level produced by a signal preceding or following the threshold stimulus

In a previous experiment (Howarth and Treisman, 1958) it was shown that when a warning preceded a stimulus by a fixed interval, the threshold level for the stimulus was higher the longer the fixed interval. A model of this effect was proposed, assuming that the threshold criterion adopted by the subject at any moment was modified by his estimate of the probability of a stimulus at that moment.

Three predictions are derived and tested. It is found that: the standard deviation of the response is constant despite the shifts in threshold induced; the threshold continues to fall as the interval from warning to stimulus is decreased to zero, with the lowest threshold when the warning signal and the stimulus are simultaneous; and that a warning signal following the stimulus by an interval of less than 0 5 to 1 sec. also lowers the threshold.     

The Harvard Pigeon Lab under Herrnstein

The history of the Harvard Pigeon Lab is a history of two periods of remarkable productivity, the first under Skinner's leadership and the second under Herrnstein's. In each period, graduate students flocked to the leader and then began stimulating one another. Chance favored Herrnstein's leadership, too, because an unusually large number of graduate students were admitted in the fall of 1962. In each period, productivity declined as the leader lost interest in the laboratory and withdrew. Directly and indirectly, the laboratory finally died as a result of the cognitive "revolution." Skinner and his students saw the possibility of a natural science of behavior and set about establishing that science based on concepts such as response rate, stimulus control, and schedules of reinforcement. Herrnstein and his students saw that the science could be quantitative and set about making it so, with relative response rate, the matching law, and the psychophysics of choice (analogous to S. S. Stevens' psychophysics). The history might provide a golden research opportunity for someone interested in the impact of such self-organizing research groups on the progress of science.     

A sequential theory of psychological discrimination

A theory of discrimination which assumes that subjects compare psychological values evoked by a stimulus to a subjective referent is proposed. Momentary differences between psychological values for the stimulus and the referent are accumulated over time until one or the other of two response thresholds is first exceeded. The theory is analyzed as a random walk bounded between two absorbing barriers. A general solution to response conditioned expected response times is computed and the important role played by the moment generating function (mgf) for increments to the random walk is examined. From considerations of the mgf it is shown that unlike other random walk models [Stone, 1960; Laming, 1968] the present theory does not imply that response conditioned mean correct and error times must be equal. For two fixed stimuli and a fixed referent it is shown that by controlling values of response thresholds, subjects can produce Receiver Operating Characteristics similar or identical to those predicted by Signal Detection Theory, High Threshold Theory, or Low Threshold Theory.     

Limitations of exemplar models of multi-attribute probabilistic inference

Observers were presented with pairs of objects varying along binary-valued attributes and learned to predict which member of each pair had a greater value on a continuously varying criterion variable. The predictions from exemplar models of categorization were contrasted with classic alternative models, including generalized versions of a "take-the-best" model and a weighted-additive model, by testing structures in which interactions between attributes predicted the magnitude of the criterion variable. Under typical training conditions, observers showed little sensitivity to the attribute interactions, thereby challenging the predictions from the exemplar models. In a condition involving highly extended training, observers eventually learned the relations between the attribute interactions and the criterion variable. However, an analysis of the observers' response times for making their paired-comparison decisions also challenged the exemplar model predictions. Instead, it appeared that most observers recoded the interacting attributes into emergent configural cues. They then applied a set of hierarchically organized rules based on the priority of the cues to make their decisions.     

Is automatic imitation a specialized form of stimulus–response compatibility? Dissociating imitative and spatial compatibilities

In recent years research on automatic imitation has received considerable attention because it represents an experimental platform for investigating a number of interrelated theories suggesting that the perception of action automatically activates corresponding motor programs. A key debate within this research centers on whether automatic imitation is any different than other long-term S-R associations, such as spatial stimulus-response compatibility. One approach to resolving this issue is to examine whether automatic imitation shows similar response characteristics as other classes of stimulus-response compatibility. This hypothesis was tested by comparing imitative and spatial compatibility effects with a two alternative forced-choice stimulus-response compatibility paradigm. The stimulus on each trial was a left or right hand with either the index or middle finger tapping down. Speeded responses were performed with the index or middle finger of the right hand in response to the identity or the left-right spatial position of the stimulus finger. Two different tasks were administered: one that involved responding to the stimulus (S-R) and one that involved responding to the opposite stimulus (OS-R; i.e., the one not presented on that trial). Based on previous research and a connectionist model, we predicted standard compatibility effects for both spatial and imitative compatibility in the S-R task, and a reverse compatibility effect for spatial compatibility, but not for imitative compatibility, in the OS-R task. The results from the mean response times, mean percentage of errors, and response time distributions all converged to support these predictions. A second noteworthy result was that the recoding of the finger identity in the OS-R task required significantly more time than the recoding of the left-right spatial position, but the encoding time for the two stimuli in the S-R task was equivalent. In sum, this evidence suggests that the processing of spatial and imitative compatibility is dissociable with regard to two different processes in dual processing models of stimulus-response compatibility.     

Choice reaction time with a random foreperiod

Choice reaction latencies were measured at three different a priori probabilities for two stimulus alternatives. Unlike the results of some other studies, the mean latency of a given response was nearly the same whether the response was correct or incorrect. The discriminable stimuli were a 1000- or a 1700-Hz tone presented at 70 dB SPL. Latencies and standard deviations, based on about 17,000 observations, are reported for three observers. The data are compared with predictions of the optimum sequential model of Wald and Stone and two modifications of that random-walk model, one proposed by Link and Heath and the other proposed by Laming. Fast-guess analyses were also carried out. The three-parameter version of either the sequential or the modified random-walk models provided reasonably accurate predictions of the mean data for each observer. The parameters estimated by the fast-guess analysis were unrealistic. There are three obvious differences between this experiment and most previous choice reaction-time experiments. First is stimulus modality#x2014;we used an auditory signal, whereas most of the previous studies used a visual signal. Second, the observers practiced more in this experiment than in most previous experiments. Finally, there was a random foreperiod with a heavy penalty for anticipations. One or more of these factors is the probable reason for the discrepancy between our results and those of previous studies.     

Decision processes in temporal discrimination

The processing dynamics underlying temporal decisions and the response times they generate have received little attention in the study of interval timing. In contrast, models of other simple forms of decision making have been extensively investigated using response times, leading to a substantial disconnect between temporal and non-temporal decision theories. An overarching decision-theoretic framework that encompasses existing, non-temporal decision models may, however, account both for interval timing itself and for time-based decision-making. We sought evidence for this framework in the temporal discrimination performance of humans tested on the temporal bisection task. In this task, participants retrospectively categorized experienced stimulus durations as short or long based on their perceived similarity to two, remembered reference durations and were rewarded only for correct categorization of these references. Our analysis of choice proportions and response times suggests that a two-stage, sequential diffusion process, parameterized to maximize earned rewards, can account for salient patterns of bisection performance. The first diffusion stage times intervals by accumulating an endogenously noisy clock signal; the second stage makes decisions about the first-stage temporal representation by accumulating first-stage evidence corrupted by endogenous noise. Reward-maximization requires that the second-stage accumulation rate and starting point be based on the state of the first-stage timer at the end of the stimulus duration, and that estimates of non-decision-related delays should decrease as a function of stimulus duration. Results are in accord with these predictions and thus support an extension of the drift–diffusion model of static decision making to the domain of interval timing and temporal decisions.     

The frequentist implications of optional stopping on Bayesian hypothesis tests

Null hypothesis significance testing (NHST) is the most commonly used statistical methodology in psychology. The probability of achieving a value as extreme or more extreme than the statistic obtained from the data is evaluated, and if it is low enough, the null hypothesis is rejected. However, because common experimental practice often clashes with the assumptions underlying NHST, these calculated probabilities are often incorrect. Most commonly, experimenters use tests that assume that sample sizes are fixed in advance of data collection but then use the data to determine when to stop; in the limit, experimenters can use data monitoring to guarantee that the null hypothesis will be rejected. Bayesian hypothesis testing (BHT) provides a solution to these ills because the stopping rule used is irrelevant to the calculation of a Bayes factor. In addition, there are strong mathematical guarantees on the frequentist properties of BHT that are comforting for researchers concerned that stopping rules could influence the Bayes factors produced. Here, we show that these guaranteed bounds have limited scope and often do not apply in psychological research. Specifically, we quantitatively demonstrate the impact of optional stopping on the resulting Bayes factors in two common situations: (1) when the truth is a combination of the hypotheses, such as in a heterogeneous population, and (2) when a hypothesis is composite—taking multiple parameter values—such as the alternative hypothesis in a t-test. We found that, for these situations, while the Bayesian interpretation remains correct regardless of the stopping rule used, the choice of stopping rule can, in some situations, greatly increase the chance of experimenters finding evidence in the direction they desire. We suggest ways to control these frequentist implications of stopping rules on BHT.     

Discrete versus continuous stage models of human information processing: in search of partial output

This article introduces a new technique designed to study the flow of information through processing stages in choice reaction time tasks. The technique was designed to determine whether response preparation can begin before stimulus identification is complete ("continuous" models), or if a stimulus must be fully identified prior to any response activation ("discrete" models). To control the information available at various times during stimulus identification, some relevant stimulus characteristics were made easy to discriminate and some were made hard to discriminate. The experimental strategy was to look for effects of partial output based on information conveyed by characteristics that were easy to discriminate. The technique capitalized on the fact, demonstrated in Experiment 1, that preparation of two response fingers on the same hand is more effective than preparation of two response fingers on different hands. The usefulness of partial output was varied by manipulating the assignments of stimuli to responses. For some mappings partial information could contribute to effective response preparation because the responses consistent with partial information were assigned to fingers on the same hand. For other mappings partial information could not contribute to effective response preparation because the responses consistent with partial information were assigned to fingers of different hands. Performance differences between these mappings were considered evidence that partial information about a stimulus was transmitted to response activation processes before the stimulus was uniquely identified, and thus were considered evidence against discrete transmission of information about the stimulus as a whole. A variety of stimulus sets were studied; the results suggest that information is transmitted discretely with respect to stimulus codes, although distinct codes activated by a single stimulus may be transmitted at different times.     

Instance-based learning: integrating sampling and repeated decisions from experience

In decisions from experience, there are 2 experimental paradigms: sampling and repeated-choice. In the sampling paradigm, participants sample between 2 options as many times as they want (i.e., the stopping point is variable), observe the outcome with no real consequences each time, and finally select 1 of the 2 options that cause them to earn or lose money. In the repeated-choice paradigm, participants select 1 of the 2 options for a fixed number of times and receive immediate outcome feedback that affects their earnings. These 2 experimental paradigms have been studied independently, and different cognitive processes have often been assumed to take place in each, as represented in widely diverse computational models. We demonstrate that behavior in these 2 paradigms relies upon common cognitive processes proposed by the instance-based learning theory (IBLT; Gonzalez, Lerch, & Lebiere, 2003) and that the stopping point is the only difference between the 2 paradigms. A single cognitive model based on IBLT (with an added stopping point rule in the sampling paradigm) captures human choices and predicts the sequence of choice selections across both paradigms. We integrate the paradigms through quantitative model comparison, where IBLT outperforms the best models created for each paradigm separately. We discuss the implications for the psychology of decision making.     

Simultaneous temporal processing

Seven experiments assessed the ability of rats to process temporal information from two internal clocks simultaneously and independently. In the first six experiments a light stimulus signalled an overall interval between the beginning of a trial and the availability of food reinforcement (e.g., a 50-s fixed interval). During the overall interval a sound stimulus was used to signal shorter intervals that divided the overall interval into equal segments. When there was a fixed temporal relation between the final segment signal and the availability of reinforcement, there was a double-scallop pattern of responding throughout the segmented overall interval; the function relating response rate to time during segment intervals was similar to the function relating response rate to time in unsegmented overall intervals; a change in response rate occurred at the time that a normally presented segment signal was omitted. Taken together, the results indicate that rats timed the overall interval and the segment intervals simultaneously and independently without interference. In Experiment 7 a light stimulus was used on some trials, and a sound stimulus was used on other trials to signal a discrete-trial 50-s peak procedure. When these two signals were presented in compound, there was a leftward shift of the response function, which suggests that rats timed both signals simultaneously. For all of the experiments a scalar timing model with specific stimulus integration rules is used to explain the results. The stimulus integration rule used in the first six experiments, in which there were two signals for the same reinforcement, was to respond if both the segment and the overall interval had exceeded a response threshold. The stimulus integration rule used in Experiment 7, in which there were two signals for different reinforcements, was to respond if the response threshold for either interval had been exceeded.     

Restle's choice model: A reconceptualization for a special case

A reconceptualization of the choice model developed by Restle is proposed which involves a reformulation of the choice situation, consideration of absolute constraints, and a distinction between positively and negatively valued aspects of alternatives. Incorporation of these features results in four different models which are found to combine some characteristics of other choice models as well. It is felt that these models might be particularly appropriate for both general and specialized vocational choices.     

Second-order avoidance behavior in monkeys

Two monkeys were trained on a three-component multiple schedule using discrete trials. In one component (food), a response terminated a red stimulus and produced food and S(Delta). In the second component (avoidance), a response terminated a green stimulus and avoided shock. In the third component (optional), a response terminated a blue stimulus and produced S(Delta). The consequence of not responding in the blue stimulus, however, was the production of either a food or an avoidance trial. Manipulation of these consequences showed that when food trials were available only a small percent of the time after optional trials, the subjects tended not to respond in the blue, even though this led to an increase in the total number of avoidance trials per day. If only avoidance trials followed as a consequence of not responding in the optional component, the animals terminated the blue stimulus and avoided the avoidance trials. Throughout the experiment both subjects maintained consistently low shock rates (about 10 per day) and these were not affected by the various manipulations. The data suggest that a stimulus associated with avoidance can be a conditioned aversive stimulus and will maintain a more remote avoidance response under certain conditions.     

Optional stopping: No problem for Bayesians

Optional stopping refers to the practice of peeking at data and then, based on the results, deciding whether or not to continue an experiment. In the context of ordinary significance-testing analysis, optional stopping is discouraged, because it necessarily leads to increased type I error rates over nominal values. This article addresses whether optional stopping is problematic for Bayesian inference with Bayes factors. Statisticians who developed Bayesian methods thought not, but this wisdom has been challenged by recent simulation results of Yu, Sprenger, Thomas, and Dougherty (2013) and Sanborn and Hills (2013). In this article, I show through simulation that the interpretation of Bayesian quantities does not depend on the stopping rule. Researchers using Bayesian methods may employ optional stopping in their own research and may provide Bayesian analysis of secondary data regardless of the employed stopping rule. I emphasize here the proper interpretation of Bayesian quantities as measures of subjective belief on theoretical positions, the difference between frequentist and Bayesian interpretations, and the difficulty of using frequentist intuition to conceptualize the Bayesian approach.     

Unique transitions between stimuli and responses in SRT tasks: Evidence for the primacy of response predictions

The present experiments aimed at separating the impact stimulus and response predictions have on serial learning and performance in SRT tasks. In Experiment 1, a unique transition between two of four responses in an otherwise random response sequence was triggered by ambiguous stimulus transitions, allowing local response predictions but no stimulus predictions. The data indicated explicit transition knowledge and strong performance benefits. In Experiments 2 and 3, unique transitions between two of four stimuli in otherwise random stimulus sequences allowed local stimulus predictions under conditions of ambiguous response transitions. The data indicated fragmentary explicit transition knowledge but no performance effects. Experiments 4a and 4b reveal that the inefficacy of the unique stimulus transitions in Experiments 2 and 3 was presumably due to the fact that the stimuli differed with respect to conjunctions of response relevant and response irrelevant features which participants did not have to attend. However, although in Experiments 4a and 4b unique transitions between response relevant stimuli were applied, substantial explicit transition knowledge but only marginal performance effects resulted. It is argued i) that in SRT tasks learning mechanisms are addressed that primarily strive for reliable predictions of forthcoming responses and ii) that for these mechanisms to work the predictors have to be attended. Response transitions are easily learned and used because both criteria are fulfilled. In contrast, pure stimulus transitions are learned only if the predictive stimuli are attended, and learned stimulus transitions become effective only to the extent that the predicted stimuli specify the required responses.     

What response times tell of children's behavior on the balance scale task

Analysis of accuracy of responses to balance scale problems gives a global idea of the cognitive processes that underlie problem-solving behavior on this task. We show that response times (RTs) provide additional detailed information about the kind and duration of these processes. We derive predictions about the RTs from Siegler's (1981) model for the balance scale task, including the counterintuitive prediction that young adults are slower than children in solving particular balance scale problems. The predictions were tested in a study in which 191 6- to 22-year-old participants were presented with a computerized balance scale task. RTs were analyzed with regression models. In addition to qualitative differences between items, we also modeled quantitative differences between items in the regression models. Analyses supported the predictions and provided additional knowledge on the rules. Rule II was reformulated as a rule that always involves the encoding, but not always the correct application of the distance cue. RTs provided evidence for the use of a buggy-rule and not an addition-rule. Finally, a relation between rule inconsistency and increased RT was found.     

Developmental changes in the semantic processing of pictures and words

Second- and fifth-graders' semantic decision times for pictures and words were analyzed relative to the predictions derived from unitary- and dual-memory models. At both grade levels, word-word response latencies were greater than picture-word latencies which, in turn, were greater than picture-picture latencies. An interaction between Grade and Condition indicated that verbal access times decreased more than pictorial access times. The data fit the predictions of a memory model postulating category storage in a single memory system as opposed to simultaneous representation in verbal and nonverbal memory systems. It was concluded that with increasing experience verbal access to this single semantic system is more rapid.