Shifts of the psychometric function: distinguishing bias from perceptual effects.

Morgan, Dillenburger, Raphael, and Solomon have shown that observers can use different response strategies when unsure of their answer, and, thus, they can voluntarily shift the location of the psychometric function estimated with the method of single stimuli (MSS; sometimes also referred to as the single-interval, two-alternative method). They wondered whether MSS could distinguish response bias from a true perceptual effect that would also shift the location of the psychometric function. We demonstrate theoretically that the inability to distinguish response bias from perceptual effects is an inherent shortcoming of MSS, although a three-response format including also an "undecided" response option may solve the problem under restrictive assumptions whose validity cannot be tested with MSS data. We also show that a proper two-alternative forced-choice (2AFC) task with the three-response format is free of all these problems so that bias and perceptual effects can easily be separated out. The use of a three-response 2AFC format is essential to eliminate a confound (response bias) in studies of perceptual effects and, hence, to eliminate a threat to the internal validity of research in this area.

Bayesian adaptive estimation of arbitrary points on a psychometric function

Bayesian adaptive methods have been extensively used in psychophysics to estimate the point at which performance on a task attains arbitrary percentage levels, although the statistical properties of these estimators have never been assessed. We used simulation techniques to determine the small-sample properties of Bayesian estimators of arbitrary performance points, specifically addressing the issues of bias and precision as a function of the target percentage level. The study covered three major types of psychophysical task (yes-no detection, 2AFC discrimination and 2AFC detection) and explored the entire range of target performance levels allowed for by each task. Other factors included in the study were the form and parameters of the actual psychometric function Psi, the form and parameters of the model function M assumed in the Bayesian method, and the location of Psi within the parameter space. Our results indicate that Bayesian adaptive methods render unbiased estimators of any arbitrary point on psi only when M=Psi, and otherwise they yield bias whose magnitude can be considerable as the target level moves away from the midpoint of the range of Psi. The standard error of the estimator also increases as the target level approaches extreme values whether or not M=Psi. Contrary to widespread belief, neither the performance level at which bias is null nor that at which standard error is minimal can be predicted by the sweat factor. A closed-form expression nevertheless gives a reasonable fit to data describing the dependence of standard error on number of trials and target level, which allows determination of the number of trials that must be administered to obtain estimates with prescribed precision.

Psychometric functions for detection and discrimination with and without flankers.

Recent studies have reported that flanking stimuli broaden the psychometric function and lower detection thresholds. In the present study, we measured psychometric functions for detection and discrimination with and without flankers to investigate whether these effects occur throughout the contrast continuum. Our results confirm that lower detection thresholds with flankers are accompanied by broader psychometric functions. Psychometric functions for discrimination reveal that discrimination thresholds with and without flankers are similar across standard levels, and that the broadening of psychometric functions with flankers disappears as standard contrast increases, to the point that psychometric functions at high standard levels are virtually identical with or without flankers. Threshold-versus-contrast (TvC) curves with flankers only differ from TvC curves without flankers in occasional shallower dippers and lower branches on the left of the dipper, but they run virtually superimposed at high standard levels. We discuss differences between our results and other results in the literature, and how they are likely attributed to the differential vulnerability of alternative psychophysical procedures to the effects of presentation order. We show that different models of flanker facilitation can fit the data equally well, which stresses that succeeding at fitting a model does not validate it in any sense.

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test.

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.

The role of parametric assumptions in adaptive Bayesian estimation.

Variants of adaptive Bayesian procedures for estimating the 5% point on a psychometric function were studied by simulation. Bias and standard error were the criteria to evaluate performance. The results indicated a superiority of (a) uniform priors, (b) model likelihood functions that are odd symmetric about threshold and that have parameter values larger than their counterparts in the psychometric function, (c) stimulus placement at the prior mean, and (d) estimates defined as the posterior mean. Unbiasedness arises in only 10 trials, and 20 trials ensure constant standard errors. The standard error of the estimates equals 0.617 times the inverse of the square root of the number of trials. Other variants yielded bias and larger standard errors.

Stopping rules in Bayesian adaptive threshold estimation

Threshold estimation with sequential procedures is justifiable on the surmise that the index used in the so-called dynamic stopping rule has diagnostic value for identifying when an accurate estimate has been obtained. The performance of five types of Bayesian sequential procedure was compared here to that of an analogous fixed-length procedure. Indices for use in sequential procedures were: (1) the width of the Bayesian probability interval, (2) the posterior standard deviation, (3) the absolute change, (4) the average change, and (5) the number of sign fluctuations. A simulation study was carried out to evaluate which index renders estimates with less bias and smaller standard error at lower cost (i.e. lower average number of trials to completion), in both yes–no and two-alternative forced-choice (2AFC) tasks. We also considered the effect of the form and parameters of the psychometric function and its similarity with themodel function assumed in the procedure. Our results show that sequential procedures do not outperform fixed-length procedures in yes–no tasks. However, in 2AFC tasks, sequential procedures not based on sign fluctuations all yield minimally better estimates than fixed-length procedures, although most of the improvement occurs with short runs that render undependable estimates and the differences vanish when the procedures run for a number of trials (around 70) that ensures dependability. Thus, none of the indices considered here (some of which are widespread) has the diagnostic value that would justify its use. In addition, difficulties of implementation make sequential procedures unfit as alternatives to fixed-length procedures.

Does time ever fly or slow down? The difficult interpretation of psychophysical data on time perception.

Time perception is studied with subjective or semi-objective psychophysical methods. With subjective methods, observers provide quantitative estimates of duration and data depict the psychophysical function relating subjective duration to objective duration. With semi-objective methods, observers provide categorical or comparative judgments of duration and data depict the psychometric function relating the probability of a certain judgment to objective duration. Both approaches are used to study whether subjective and objective time run at the same pace or whether time flies or slows down under certain conditions. We analyze theoretical aspects affecting the interpretation of data gathered with the most widely used semi-objective methods, including single-presentation and paired-comparison methods. For this purpose, a formal model of psychophysical performance is used in which subjective duration is represented via a psychophysical function and the scalar property. This provides the timing component of the model, which is invariant across methods. A decisional component that varies across methods reflects how observers use subjective durations to make judgments and give the responses requested under each method. Application of the model shows that psychometric functions in single-presentation methods are uninterpretable because the various influences on observed performance are inextricably confounded in the data. In contrast, data gathered with paired-comparison methods permit separating out those influences. Prevalent approaches to fitting psychometric functions to data are also discussed and shown to be inconsistent with widely accepted principles of time perception, implicitly assuming instead that subjective time equals objective time and that observed differences across conditions do not reflect differences in perceived duration but criterion shifts. These analyses prompt evidence-based recommendations for best methodological practice in studies on time perception.

Fixed vs. variable noise in 2AFC contrast discrimination: lessons from psychometric functions.

Recent discussion regarding whether the noise that limits 2AFC discrimination performance is fixed or variable has focused either on describing experimental methods that presumably dissociate the effects of response mean and variance or on reanalyzing a published data set with the aim of determining how to solve the question through goodness-of-fit statistics. This paper illustrates that the question cannot be solved by fitting models to data and assessing goodness-of-fit because data on detection and discrimination performance can be indistinguishably fitted by models that assume either type of noise when each is coupled with a convenient form for the transducer function. Thus, success or failure at fitting a transducer model merely illustrates the capability (or lack thereof) of some particular combination of transducer function and variance function to account for the data, but it cannot disclose the nature of the noise. We also comment on some of the issues that have been raised in recent exchange on the topic, namely, the existence of additional constraints for the models, the presence of asymmetric asymptotes, the likelihood of history-dependent noise, and the potential of certain experimental methods to dissociate the effects of response mean and variance.