An analysis of visual detection by temporal probability summation

The probability of detection by temporal probability summation (TPS) is derived assuming a non-white Gaussian noise process, characterised by a finite second spectral moment λ₂; the assumption of stationary noise is discussed with respect to neurophysiological findings suggesting multiplicative noise for individual neurons. The usual white-noise assumption is shown to imply inconsistent detection models. d^′-estimates for 2AFC-experiments do not depend on the value of λ₂ and do not differ from those derived from the standard SDT-assumptions, in contrast to d^′-estimates for yes-no-experiments, which depend on the probability of false alarms. Estimates of properties of a visual channel depend on whether detection is by TPS or temporal peak detection (TPD); it is argued that TPD is not a special case of TPS, and that subjects chose a TPS- or a TPD-strategy depending on experimental conditions.     

Signal-detection models for multidimensional stimuli: Probability distributions and combination rules

Probabilistic independence among multiple random variables (e.g., among the outputs of multiple spatial-frequency channels) has been invoked to explain two effects found with many kinds of stimuli: increments in detection performance due to “probability summation” and decrements in detection and identification performance due to “extrinsic uncertainty.” Quantitative predictions of such effects, however, depend on the precise assumptions. Here we calculate predictions from multidimensional signal-detection theory assuming any of several different probability distributions characterizing the random variables (including two-state, Gaussian, exponential, and double-exponential distributions) and either of two rules for combining the multiple random variables into a single decision variable (taking the maximum or summing them). In general, the probability distributions predicting shallower ROC curves predict greater increments due to summation but smaller decrements due to extrinsic uncertainty. Some probability distributions yield steep-enough ROC curves to actually predict decrements due to summation in blocked-summation experiments. Probability distribution matters much less for intermixed-summation than for blocked-summation predictions. Of the two combination rules, the sum-of-outputs rule usually predicts both greater increments due to summation and greater decrements due to extrinsic uncertainty. Put another way, of the two combination rules, the sum-of-outputs rule usually predicts better performance on the compound stimulus under any condition but worse performance on simple stimuli under intermixed conditions.     

Facilitatory interactions between flashed lines

The detectability of pairs of flashed lines was investigated as a joint function of their separation in both space and time. In contrast to previous studies of contrast interactions as a function of either spatial or temporal separation alone, the predominant interaction is not lateral inhibition, but a delayed facilitation. A quantitative model based on probability summation between lateral-inhibitory detectors does not appear able to account for these results. Two related explanations are suggested: either a motion detection system or the nonlinear 'transient' subsystem is being tapped.     

The effects of duration,area, and background intensity on the visual intensity difference threshold given by the forced-choice procedure: Derivations from a statistical decision model for sensory discrimination

A model for visual discrimination is outlined. This assumes that detection is based on separating signaland noise by means of a statistical decision procedure, and that the noise arises from physical and biological sources. When the model is applied to such traditional detection procedures as the method of constant stimuli, it successfully predicts the laws of spatial and temporal summation and the form of the Weber function. The model is extended to the forced-choice procedures. It is found that it predicts departures from the classical laws for spatial and temporal summation and an increase in the slope of the Weber function when the pulsed forced-choice procedure is used, and data are presented that agree with the predictions. It is shown that data obtained with forced choice against a continuous background are consistent with the assumption that the Seither covertly detects or fails to detect an increment at each ofthe two alternative locations and this determines his final overt response, rather than that he directly compares the inputs at the two locations. A comparison between results obtained with the forced-choice procedure with a pedestal, or with the continuous forced-choice procedure, provides further evidence that Ss employ a double detection strategy in the latter case.     

The startle response and auditory temporal summation in neonates

The present study assessed temporal summation of transient and sustained stimuli in the startle eyeblink response system in neonates during quiet sleep. Subjects received 100-dB(A), fast-rising broadband noise bursts of two types: (a) single stimuli varying in duration from 20 to 100 ms and (b) pairs of 3-ms bursts presented at interpulse intervals corresponding to the single-stimulus durations. In addition, a single 3-ms pulse was used as an anchor point for both stimulus types. For startle amplitude, single stimuli were more effective than were paired stimuli, but the temporal summation functions were similar for the two types of stimuli. Response amplitude increased as stimulus duration/interval increased to 50 ms, but not beyond. For startle probability, temporal summation was similar for single and paired stimuli at 20 ms. Pairs of pulses were equally effective at 20, 35, and 50 ms, beyond which the second pulse was not effective. Increasing the duration of single stimuli from 20 to 35 ms resulted in increased probability, illustrating a contribution of sustained summation beyond that of transient summation. Response latency was generally greater for paired than for single stimuli. The results suggest that temporal summation of brief stimuli is deficient in the neonate. These data were compared to adult data from an analogous study, and suggest that the transient system is immature in infants, and that this immaturity is expressed differently by startle amplitude, probability, and latency.     

A simple derivation of Prelec's probability weighting function

Since Kahneman and Tversky [(1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291], it has been generally recognized that decision makers overweight low probabilities. Of the several weighting functions that have been proposed, that of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528] has the attractions that it is parsimonious, consistent with much of the available empirical evidence and has an axiomatic foundation. Luce [(2001). Reduction invariance and Prelec's weighting functions. Journal of Mathematical Psychology, 45, 167-179] provided a simpler derivation based on reduction invariance, rather than compound invariance of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528]. This note introduces a behavioral assumption that we call power invariance and provides a simple derivation of Prelec's function. Thus, we have three a priori different behavioral assumptions all leading to Prelec's function.     

Detection of changes in speed and direction of motion: Reaction time analysis

Observers reacted to the change in the movement of a random-dot field whose initial velocity,V _o, was constant for a random period and then switched abruptly to another value,V ₁. The two movements, both horizontally oriented, were either in the same direction (speed increments or decrements), or in the opposite direction but equal in speed (direction reversals). One of the two velocities,V ₀ orV ₁, could be zero (motion onset and offset, respectively). in the range of speeds used, 0–16 deg/sec (dps), the mean reaction time (MRT) for a given value ofV ₀ depended on |V ₁-V ₀| only: MRT ≈r +c(V ₀/|V ₁ ?V ₀|) ^β , where β=2/3,r is a velocity-independent component of MRT, andc(V ₀) is a parameter whose value is constant for low values ofV ₀ (0–4 dps), and increases beginning with some value ofV ₀ between 4 and 8 dps. These and other data reviewed in the paper are accounted for by a model in which the time-position function of a moving target is encoded by mass activation of a network of Reichardt-type encoders. Motion-onset detection (V ₀=0) is achieved by weighted temporal summation of the outputs of this network, the weights assigned to activated encoders being proportional to their squared spatial spans. By means of a “subtractive normalization,” the visual system effectively reduces the detection of velocity changes (a change fromV ₀ toV ₁) to the detection of motion onset (a change from 0 toV ₁-V ₀). Subtractive normalization operates by readjustment of weights: the weights of all encoders are amplified or attenuated depending on their spatial spans, temporal spans, and the initial velocityV ₀. Assignment of weights and weighted temporal summation are thought of as special-purpose computations performed on the dynamic array of activations in the motion-encoding network, without affecting the activations themselves.     

Is there low frequency vibrotactile temporal summation?

Reaction times of subjects to threshold vibratory stimuli of various durations and frequencies were measured simultaneously with the thresholds in order to test whether more time is needed for the detection of stimuli of long than short duration at threshold intensities. Vibratory stimuli of 20 or 150 Hz frequency and 50, 150, or 300 ms duration were applied to the back of the hand. Lower detection thresholds and longer reaction times were obtained for both 20 and 150 Hz vibration with increase of stimulus duration. The results suggest that temporal summation of high frequencies is due to energy integration , whereas at low frequencies probability summation explains better the threshold decrement.     

A programmable,multifunction dual-channel waveform generator for visual psychophysics

This paper describes a relatively low-cost, flexible means of presenting patterns on a CRT. The design allows for a number of options, such as presentation of gratings as well as nonstandard stimuli (white noise), the summation and differentiation of outputs to create complex luminance waveforms, and the switching of information from one channel to the other. In addition, the visual pattern generator permits a number of temporal options, such as phase reversals and the ramping on and/or off of a stimulus. This device incorporates a hardware means of scaling grating contrast. Finally, our procedure for producing synchronization signals, which uses internally generated interrupt signals, allows a number of software options, such as keyboard scanning and reaction timing, without significant time losses.     

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.     

The psychophysical inquiry into binocular summation

Experiments that compare monocular and binocular visual performance of human psychophysical Os on a variety of visual tasks are reviewed. The review attempts to include all experiments published in English in this century, excluding work on stereopsis, rivalry, and evoked potentials. The concept of probability summation as a baseline for assessing the presence of neural summation is discussed, and the assumptions of several models for estimating probability summation are considered. Experiments are classified in terms of visual task, major categories being increment detection, flicker fusion, brightness magnitude, and contour resolution. A major conclusion is that binocular performance is superior for essentially all task categories and in most cases by a magnitude greater than that predicted by appropriate probability summation models.     

Application of signal detection theory to perception of differences in line length

This paper is an attempt to specify the equinormal model for application to different tasks which are currently used in psychophysics (i.e. phenomenal report, detection, and 2AFC). The same set of stimuli (pairs of horizontal colinear lines) is presented in procedures which differ by the task, the probability of the signal, and the succession of the stimuli of the set. This approach leads to the specification of the sensorial and the decisional aspects of the model.The data indicate that (1) the sensitivity index d' to differences in line length is the same function of the physical stimulus independently of paradigm, as far as the same task is concerned. However, and according to the predictions, its value becomes larger by a factor of 2 in a two alternative forced choice task; (2) d' increases with stimulus exposure time; (3) latency of responses generally decreases with increasing d', and are dissimilar for correct and incorrect responses; (4) when the temporal succession of the stimuli is random the subject's criterion, x_c, is not chosen to optimize the percentage correct, which would require a dependance of criterion upon stimulus amplitude and prior probabilities.     

Vibrotactile temporal summation for threshold and suprathreshold levels of stimulation

Threshold measurement and matching procedures were used to determine the amount of temporal summation at threshold and suprathreshold levels of vibrotactile stimulation on the thenar eminence of the hand. The frequency of the stimulus was 25, 40, 80, or 200 Hz. At 25 Hz, temporal summation was absent at all intensity levels. Considerable amounts of temporal summation were observed for 80- and 200-Hz stimuli, although the effects decreased as a function of intensity. At 40 Hz, no temporal summation was observed at threshold, but above threshold, a small amount was observed at all intensity levels. The results support a duplex model of mechanoreception in which one of two receptor systems exhibits temporal summation.     

Knowledge is power? Outcome probability information impairs detection of deceptive intent

The benefits and costs of prior expectations that are (i.e., congruent) or are not in harmony (i.e., incongruent) with action outcomes appear to be balanced; however, researchers have yet to examine the influence on skilled detection of deception. In this study we investigated whether response bias resulting from probability information (a) is stronger for low-skilled than high-skilled participants, (b) is stronger for deceptive actions than genuine actions, and (c) impairs the discriminability of genuine and deceptive actions. High-skilled (n = 15) and low-skilled (n = 15) soccer players responded to life-sized projected video clips showing an oncoming opponent taking the ball to their left or right, with or without a deceptive ‘stepover’ action. Three probability conditions were used with respect to outcome direction: 50/50, 67/33, and 83/17. Participants responded by stepping on one of two corresponding pressure mats, as if attempting to intercept the player. Response accuracy for genuine and deceptive actions was used to generate measures of bias (c) and sensitivity (d’). The results confirmed stronger probability bias for deceptive actions than genuine ones, and for low-skilled than high-skilled participants. Congruence between high outcome probability and the direction of the fake significantly enhanced the effectiveness of the deceptive action. The study provides the first evidence that outcome probability information impairs skilled detection of deceptive intent.     

Binocular summation of equal-energy flashes of unequal duration

To determine if binocular summation occurs when increment flashes are of equal energy (Bloch’s law) but unequal in luminance-duration parameters, three Ss made temporal forced-choice judgments: (1) monocularly, (2) binocularly when the flashes to each eye were identical, (3) binocularly when the flashes to each eye were of equal energy but different in terms of their luminance and duration parameters, and (4) binocularly when flashes to each eye were separated by 100 msec. Binocular detection rates were consistently superior to monocular detection rates. Similarity in performance between Conditions 2 and 3 indicates that the binocular visual system responds only to the total energy of each monocular flash. The data from two Ss reveal that binocular performance was greater than that predicted on the basis of probability summation.     

Varieties of Binocular Interaction in Human Vision

Binocular processing was investigated using a quantitative, process-oriented metatheory of response times. The analyses are not confined to particular distributional assumptions or specific models. Upper and lower performance boundaries for probability summation in parallel processing are defined and compared with observed distributions of reaction times using a variety of dichoptic stimuli. Performance that exceeds the upper bound strongly suggests faciliatory convergence between the two eyes (binocular channel summation). Performance below the lower bound suggests that inputs to the two eyes are processed serially. The results indicate that binocular channel summation in subjects with normal stereo vision requires targets of the same luminance polarity (paired increments or decrements) in corresponding retinal locations. When corresponding retinal locations are stimulated with opposing luminance polarities (increment to one eye, decrement to the other), performance is consistent with probability summation, indicating that parallel ON and OFF pathways remain segregated at least to the level of binocular fusion. Further analyses of data from a stereo-blind observer suggests serial processing of binocular inputs.     

Binaural and temporal integration of the loudness of tones and noises

Subjects judged the loudness of tones (Experiment 1) and of bursts of noise (Experiment 2) that varied in intensity and duration as well as in mode of presentation (monaural vs. binaural). Both monaural and binaural loudness, for both types of signals, obeyed the bilinear-interaction prediction of the classic temporal integration model. The loudness of short tones grows as a power function of both intensity and duration with different exponents for the two factors (.2 and .3, respectively). The loudness of wide-band noises grows as a power function of duration (with an exponent of approximately .6) but not of sound pressure. For tones, binaural summation was constant but fell short of full additivity. For noises, summation changed across level and duration. Temporal summation followed the same course for monaural and binaural tonal stimuli but not for noise stimuli. Notwithstanding these differences between tone and noise, we concluded that binaural and temporal summation are independently operating integrative networks within the auditory system. The usefulness of establishing the underlying metric structure for temporal summation is emphasized.     

Estimation of careless error and lucky guess probabilities for dichotomous test items: A psychometric application of a biometric latent class model with random effects

Medical research has extensively dealt with the estimation of the accuracy (sensitivity and specificity) of a diagnostic test for screening individuals. In this paper we apply the biometric latent class model with random effects by Qu, Tan, and Kutner [(1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797-810] to estimate the response error (careless error and lucky guess) probabilities for dichotomous test items in the psychometric theory of knowledge spaces. The approach is illustrated with simulated data. In particular, we extend this approach to give a generalization of the basic local independence model in knowledge space theory. This allows for local dependence among the indicators given the knowledge state of an examinee and/or for the incorporation of covariates.     

Diagnostic causal reasoning with verbal information

In diagnostic causal reasoning, the goal is to infer the probability of causes from one or multiple observed effects. Typically, studies investigating such tasks provide subjects with precise quantitative information regarding the strength of the relations between causes and effects or sample data from which the relevant quantities can be learned. By contrast, we sought to examine people’s inferences when causal information is communicated through qualitative, rather vague verbal expressions (e.g., “X occasionally causes A”). We conducted three experiments using a sequential diagnostic inference task, where multiple pieces of evidence were obtained one after the other. Quantitative predictions of different probabilistic models were derived using the numerical equivalents of the verbal terms, taken from an unrelated study with different subjects. We present a novel Bayesian model that allows for incorporating the temporal weighting of information in sequential diagnostic reasoning, which can be used to model both primacy and recency effects. On the basis of 19,848 judgments from 292 subjects, we found a remarkably close correspondence between the diagnostic inferences made by subjects who received only verbal information and those of a matched control group to whom information was presented numerically. Whether information was conveyed through verbal terms or numerical estimates, diagnostic judgments closely resembled the posterior probabilities entailed by the causes’ prior probabilities and the effects’ likelihoods. We observed interindividual differences regarding the temporal weighting of evidence in sequential diagnostic reasoning. Our work provides pathways for investigating judgment and decision making with verbal information within a computational modeling framework.     

Binocular detection by normal and stereoblind observers

Binocular forced-choice detection performance was measured in three stereoblind observers and four observers with normal stereopsis. Detection rates of normal observers were greater than expected from probability summation, while those of the stereoblind observers were near or at a level expected from probability. It is concluded that binocular summation is reduced or absent in stereoblind persons.