The components of latency of response in two models for auditory detection with deadlines

Two models are described for the case of auditory detection with response time deadlines. These are the “timing” model, modified from Luce and Green (1972), and the “interval of uncertainty” (IU) model, modified from the “counting” model of McGill (1967). The two models are expressed in the form of additive component latencies, including a component which varies inversely with probability of response and a component for a longer response on the “nondeadline” response condition. An experiment is described, incorporating deadlines on both s and n trials separately, and the additive component latencies of the models are calculated. The “interval of uncertainty” model is seen to represent the data better than the “timing” model, even though different versions of the latter model are considered. Parameters are then fitted to give a representation of the IU model. The data from one subject of another study not employing deadlines can also be fitted to the model, and an interesting problem then arises concerning the exact locus of the speed-accuracy tradeoff. It is concluded that the IU model needs extending if it is to describe the data without the use of a component of latency which varies inversely with response probability.     

Studies of letter identification using qualitative error analysis: effects of speed stress, tachistoscopic presentation, and word context

In Experiment 1 subjects named letters under a response deadline chosen so that an appreciable number of errors would be produced. The stimulus confusions were analyzed via the same mathematical models of stimulus recognition that have been applied to the confusion matrices generated in tachistoscopic experiments. Both the Luce choice model and the informed guessing model (a new model having a simple and elegant process interpretation) provided excellent fits to the data. The parameter values of the informed guessing model changed in logical and interpretable ways with changes in the response deadline. In Experiment 2 a direct comparison was made of the types of errors produced in the data-limited tachistoscopic situation and the resource-limited response deadline situation. It was found that, relative to the response deadline task, identification in the tachistoscopic task is much more likely to be based on partial information. In Experiments 3 and 4 the same research methodology was applied to the problem of the effect of a word context on letter perception. The methodology allowed this problem to be addressed in the context of both response deadline and tachistoscopic tasks. Several advantages of the methodology for investigating other issues of interest to cognitive psychologists are discussed.     

Pooling peripheral information: Averages versus extreme values

Suppose, as an idealization, that sensory intensity is coded in peripheral channels as identical Poisson pulse trains with intensity parameter a power function of signal intensity. Discrimination models based on either an average count computed over a fixed time or an average time computed for a fixed count per channel have difficulty in fitting the Weber function ($\text{ΔI}\text{I}$ versus I) if the free parameters are constrained to ranges determined from other experiments (magnitude estimation, reaction time). Here we study a different decision rule, namely, the most extreme observation in either the counting or timing mode. Our extremum-counting model, but not two timing ones, accounts very nicely for the Weber function. However, the ROC curves for these extremum models, which agree in shape with data of Green and Luce, yield estimates for the intensity parameter which are much larger than predicted by the power function growth used to calculate ΔI and about twice as large as those estimated from reaction time data collected in the same experiment.     

A biased random walk model for two choice reaction times

This paper generalizes Stone's (1960, Psychometrika 25, 251–260) random walk model of two choice response times (RTs) that is based on the sequential probability ratio test (SPRT) and that predicts equal correct and incorrect RTs to the same response. The generalized version allows a bias of the type found in signal detection theory to enter directly into the accumulation process so that ambivalent evidence may be seen as slightly favoring one alternative. The resulting biased SPRT model can predict any relation between correct and incorrect mean RTs. In particular, unlike the special symmetric case of Link and Heath's (1975, Psychometrika 40, 77–105) relative judgment theory (RJT), the biased SPRT model can predict that correct mean RTs are faster for one response but slower for the other. The biased SPRT model, the classical SPRT model, and the symmetric RJT model are all fit to the data of an RT deadline experiment reported by Green and Luce (1973, Attention and performance, New York: Academic Press) and it is shown that, of the three, the biased SPRT model provides the best account. Finally, a method for incorporating the same sort of bias into RJT models is sketched out.     

A theory of criterion setting: an alternative to the attention band and response ratio hypotheses in magnitude estimation and cross-modality matching

The attention band and response ratio hypotheses of Green and Luce (1974) and Luce and Green (1974) are discussed and some difficulties are noted. An alternative hypothesis is put forward. This is based on a Thurstonian model for magnitude estimation in which the presented stimulus intensities are subjected to a logarithmic transformation. Response criteria are then applied to the resulting quantities to select corresponding responses. The setting and maintenance of these response criteria are accounted for by a theory of criterion setting previously developed by the senior author (Treisman & Williams, 1984). A similar model is developed for cross-modality matching, and it is shown that these models can predict the V pattern for the coefficient of variation of response ratios, can predict the inverted V pattern for correlations between successive responses, and can account for some of the difficulties found in the literature.     

Second choices in a visual span of apprehension task

Four Ss were run in a visual span of apprehension experiment to determine whether second choices made following incorrect first responses are at the chance level, as implied by various high threshold models proposed for this situation. The relationships between response biases on first and second choices, and between first choice biases on trials with two or three possible responses, were also examined in terms of Luce’s (1959) choice theory. The results were: (a) second choice performance in this task appears to be determined by response bias alone, i.e., second choices were at the chance level; (b)first and second choice response biases were not related according to Luce’s choice axiom; and (c) the choice axiom predicted with reasonable accuracy the relationships between first choice response biases corresponding to trials with different numbers of possible response alternatives.     

The response ratio hypothesis for magnitude estimation

Assume that each presentation of a signal produces two independent random variable representations and that the ratio of responses on successive trials of a magnitude estimation experiment are proportional to the ratio of a representation from the present trial, which representation is then lost, to the remaining one from the previous trial. The mean response to a particular signal depends on the mean of the representation used, but in general exhibits drift over trials and sequential effects due to the preceeding trial; the mean response ratio does not exhibit drift, but it has a simple form only when there are no sequential effects; however, a modified mean ratio function has a simple form. A model suggested by D. V. Cross is a special case of this one. Simple timing and counting models for the representations fail to exhibit sequential effects, contrary to considerable data. However, data of the authors have suggested a version of the timing model in which the sample size of the representation varies by an order of magnitude depending on how close the signal is to the preceding one; this hypothesis accounts for the observed sequential effects and other aspects of the data.     

Two classes of models for magnitude estimation

One class of models assumes that presentation of a signal results in an internal representation as a random variable. Depending on whether the signal is close to or far from the preceding signal, the variance of the representation is smaller or larger. Responses are determined largely by this random variable; however, when the signal is close to the preceding one, the response is generated by modifying the representation multiplicatively by some function of the ratio of the previous response to its representation. Power and linear functions are explored. The form of the random variable is assumed to be that arising from either the timing or the counting model operating on a Poisson process. Detailed analyses are carried out successfully only for the timing model with neural sample sizes independent of intensity; however, the data require the sample to increase with intensity. The linear response function coupled with the constant sample size counting model appears somewhat viable, but detailed calculations are very difficult to carry out. The second class of models postulates a power function relation between magnitude estimates and signals intensity for which the exponent is a Gaussian distributed random variable and the unit is the product of two log normal random variables. Again we assume an attention band such that succesive stimuli that are widely separated in intensity lead to independent samples of the random variables while a variety of assumptions is explored for successive stimuli that are near each other in intensity. Although they each give rise to the qualitative features of the data, estimates of parameters are sufficiently inconsistent that we are led to reject all of the submodels studied.     

A model for detection and recognition with signal uncertainty

A model for signal detectability suggested by Luce is extended to situations in which the observer is uncertain of some important characteristic of the signal, such as frequency. By making a single assumption concerning the observer's covert response behavior, two solutions are obtained corresponding to qualitatively different behavior. Decrements in detectability and in recognition with uncertainty are shown to be particular functions of discriminability and detectability of the stimuli in other situations. Relevant experimental data are considered.This work was supported by grant NSF-G5544 from the National Science Foundation. The author is indebted to R. Duncan Luce for critical discussion of the content and helpful suggestions concerning the form of this paper. Useful comments have been made by David M. Green, Francis W. Irwin, Roger M. Shepard, and W. P. Tanner, Jr.     

An analysis of confusion errors in naming letters under speed stress

Three subjects named six visually presented letters under two levels of speed stress. The obtained confusion matrices for each stress condition were fit by Luce’s choice theory, which provided measures of stimulus similarity and response bias. Speed stress produced proportional increases in pairwise similarity measures but had no systematic effect on response biases. In Experiment 2, the same three subjects named the same letters under two levels of stimulus quality and a constant response-time deadline. As with speed stress, degrading the stimulus produced proportional increases in pairwise similarity measures but had no systematic effect on response biases. In Experiment 3, two of the subjects participating in Experiments 1 and 2 named the six letters under conditions where the probabilities of the letters were unequal. The letters toward which the subject had been most biased in Experiments 1 and 2 were assigned low probabilities, and the letters toward which he was least biased were assigned high probabilities. The result of this manipulation was to completely reverse the ordering of the response bias parameters of the Luce choice model. It is suggested that the present methodology provides a means of validating as psychological constructs the parameters of various mathematical models of stimulus recognition.     

Pigeons flexibly time or count on cue

In Experiment 1, pigeons were presented with a sequence of light flashes and cued to peck a key for reward either after a fixed time or after a fixed number of flashes. Curves that showed the rate of key pecking over time within trials indicated that peak rates of response were reached near the fixed time on timing-cued trials and near the fixed number of flashes on counting-cued trials. In Experiment 2, the key cue was shifted from timing to counting or from counting to timing midway through a trial. The peak times reached after the cue change indicated that pigeons kept track of time while cued to count but did not count while cued to time. These findings suggest a basic asymmetry in the dual-mode model of timing and counting.     

Response deadline and subjective awareness in recognition memory

Level of processing and generation effects were replicated in separate experiments in which recognition memory was tested using either short (500 ms) or long (1500 ms) response deadlines. These effects were similar at each deadline. Moreover, at each deadline these effects were associated with subsequent reports of remembering, not of knowing. And reports of both knowing and remembering increased following the longer deadline. These results imply that knowing does not index an automatic familiarity process, as conceived in some dual-process models of recognition, and that both remembering and knowing increase with the slower, more controlled processing permitted by the longer response time.     

Comparing single‐ and dual‐process models of memory development

This experiment examined single‐process and dual‐process accounts of the development of visual recognition memory. The participants, 6–7‐year‐olds, 9–10‐year‐olds and adults, were presented with a list of pictures which they encoded under shallow or deep conditions. They then made recognition and confidence judgments about a list containing old and new items. We replicated the main trends reported by Ghetti and Angelini ( 2008 ) in that recognition hit rates increased from 6 to 9 years of age, with larger age changes following deep than shallow encoding. Formal versions of the dual‐process high threshold signal detection model and several single‐process models (equal variance signal detection, unequal variance signal detection, mixture signal detection) were fit to the developmental data. The unequal variance and mixture signal detection models gave a better account of the data than either of the other models. A state‐trace analysis found evidence for only one underlying memory process across the age range tested. These results suggest that single‐process memory models based on memory strength are a viable alternative to dual‐process models for explaining memory development.     

The latencies of correct and incorrect responses in discrimination and detection tasks: Their interpretation in terms of a model based on simple counting

A model for two-choice discrimination based on a process of simple counting is described, and two experiments are performed to test the predictions of the model concerning the graph of latency as a function of response proportion. Two main forms of this graph are identified and predicted to arise in different circumstances. The experimental results support the model, and its possible extension to other psychophysical situations, especially signal detection, are then discussed. It is compared with a model derived directly from the detection situation, and the usefulness of testing these models is pointed out.     

Some exact results for random walk models with applications

This article presents a random walk model that can be analyzed without recourse to Wald's (1947) approximation, which neglects the excess over the absorbing barriers. Hence, the model yields exact predictions for the absorption probabilities and all mean conditional absorption times. We derive these predictions in some detail and fit them to the extensive data of an identification experiment published by Green et al. (1983). The fit of the model seems satisfactory. The relationship of the model to existing classes of random walk models (SPRT and SSR; see Luce, 1986) is discussed; for certain combinations of its parameters, the model belongs either to the SPRT or to the SSR class, or to both. We stress the theoretical significance of the knowledge of exact results for the evaluation of Wald's approximation and general properties of the several models proposed derived from this approximation.     

Context Shift and Protein Synthesis Inhibition Disrupt Long-Term Habituation after Spaced,but Not Massed,Training in the CrabChasmagnathus

An opaque screen moving overhead elicits an escape response in the crabChasmagnathusthat after a few presentations habituates for a long period (long-term habituation, LTH). Previous results suggested that spaced (15 trials separated by 171 s) and massed training (300 trials without rest interval) were correlated with two different memory components of LTH. The present experiments were aimed at further studying the mechanisms subserving these components. Results indicate that LTH acquired by spaced but not by massed training is blocked either by a training-to-testing context shift or by cycloheximide (15–25 μg) pre- or posttraining injection and that LTH after spaced training persists for longer time (5 days) than after massed training (2 days). A model based on these results that distinguishes two LTH-memory components is proposed: a (context-signal) LTH yielded by spaced training, dependent of context, sensitive to cycloheximide (CYX), and long lasting; and a (signal) LTH yielded by massed training, dependent only on the signal invariance, insensitive to CYX, and shorter lasting.     

The integration of familiarity and recollection information in short-term recognition: modeling speed-accuracy trade-off functions

Dual process models postulate familiarity and recollection as the basis of the recognition process. We investigated the time-course of integration of the two information sources to one recognition judgment in a working memory task. We tested 24 subjects with a response signal variant of the modified Sternberg recognition task (Oberauer, 2001) to isolate the time course of three different probe types indicating different combinations of familiarity and source information. We compared two mathematical models implementing different ways of integrating familiarity and recollection. Within each model, we tested three assumptions about the nature of the familiarity signal, with familiarity having (a) only positive values, indicating similarity of the probe with the memory list, (b) only negative values, indicating novelty, or (c) both positive and negative values. Both models provided good fits to the data. A model combining the outputs of both processes additively (Integration Model) gave an overall better fit to the data than a model based on a continuous familiarity signal and a probabilistic all-or-none recollection process (Dominance Model).     

A model of response time effects in symbolic comparison

A cognitive process model is developed that predicts the 3 major symbolic comparison response time effects (distance, end, and semantic congruity) found in the results of the linear syllogistic reasoning task. The model includes a simple connectionist learning component and dual evidence accumulation decision-making components. It assumes that responses can be based either on information concerning the positional difference between the presented stimulus items or on information concerning the endpoint status of each of these items. The model provides an excellent quantitative account of the mean correct response times obtained from 16 participants who performed paired comparisons of 6 ordered symbolic stimuli (3-letter names).     

Effects of response time deadlines on adults' strategy choices for simple addition

In this investigation of adults' solution strategies for simple arithmetic, participants solved addition problems (e.g., 2 + 3, 8 + 7) under fast and slow response deadlines: The participants were instructed either to respond before a 750-msec warning beep, or to wait for a 2,500-msec beep before responding. After each trial, they indicated whether they had solved the problem by direct memory retrieval or by using a procedural strategy (e.g., counting, transformation). It was predicted that the fast deadline condition should curtail the use of procedural strategies, which generally are slower than direct retrieval. Furthermore, this deadline effect should be exaggerated for numerically larger problems because procedural strategies are especially slow for the larger problems. As predicted, we observed a deadline x size interaction whereby the fast deadline increased reported use of retrieval, especially for large problems. The results confirm that reported use of direct retrieval decreases systematically with elapsed time, and they provide additional evidence that young, educated adults rely substantially on procedural strategies even for simple addition.     

Memory retrieval given two independent cues: cue selection or parallel access?

A basic but unresolved issue in the study of memory retrieval is whether multiple independent cues can be used concurrently (i.e., in parallel) to recall a single, common response. A number of empirical results, as well as potentially applicable theories, suggest that retrieval can proceed in parallel, though set forth a model that denies that possibility. In this paper, five quantitative models are developed to test broad candidate principles. In multiple experiments, subjects were trained to retrieve a vocal digit response for each member of a set of letter or color cues. In subsequent test and transfer phases, single cue trials were randomly mixed with dual cue trials on which the two cues always required the same response. For the first few repetitions of each new set of dual cue items, there was no evidence of parallel retrieval over any part of the RT distribution. After more repetitions, dual cue trials were performed faster than single cue trials, but only under conditions that were favorable to development of a "chunked" dual cue representation. These results indicate that associative independence is an important modulating variable that must be heeded in any general model of attention and memory retrieval. Further, the results are most consistent with a model that places the performance bottleneck prior to the retrieval stage of processing.