On the form of psychometric functions for taste

Psychometric functions were determined for the discrimination of weak solutions of quinine hydrochloride or of hydrochloric acid from distilled water. The slopes of these functions were compared with those of some functions previously determined for sodium chloride and for sucrose. In general, the functions for QHCl were the least steep and those for NaCl were the most steep, but the differences were not great. Thus, it appears that the psychometric functions for these different taste qualities have substantially the same form, even though absolute sensitivity varies over several orders of magnitude.     

Confidence intervals for the parameters of psychometric functions

A Monte Carlo method for computing the bias and standard deviation of estimates of the parameters of a psychometric function such as the Weibull/Quick is described. The method, based on Efron's parametric bootstrap, can also be used to estimate confidence intervals for these parameters. The method's ability to predict bias, standard deviation, and confidence intervals is evaluated in two ways. First, its predictions are compared to the outcomes of Monte Carlo simulations of psychophysical experiments. Second, its predicted confidence intervals were compared with the actual variability of human observers in a psychophysical task. Computer programs implementing the method are available from the author.     

Adaptive psychophysical methods for nonmonotonic psychometric functions

Many psychophysical tasks in current use render nonmonotonic psychometric functions; these include the oddball task, the temporal generalization task, the binary synchrony judgment task, and other forms of the same–different task. Other tasks allow for ternary responses and render three psychometric functions, one of which is also nonmonotonic, like the ternary synchrony judgment task or the unforced choice task. In all of these cases, data are usually collected with the inefficient method of constant stimuli (MOCS), because extant adaptive methods are only applicable when the psychometric function is monotonic. This article develops stimulus placement criteria for adaptive methods designed for use with nonmonotonic psychometric functions or with ternary tasks. The methods are transformations of conventional up–down rules. Simulations under three alternative psychophysical tasks prove the validity of these methods, their superiority to MOCS, and the accuracy with which they recover direct estimates of the parameters determining the psychometric functions, as well as estimates of derived quantities such as the point of subjective equality or the difference limen. Practical recommendations and worked-out examples are provided to illustrate how to use these adaptive methods in empirical research.     

Nonparametric tests for equality of psychometric functions

Many empirical studies measure psychometric functions (curves describing how observers’ performance varies with stimulus magnitude) because these functions capture the effects of experimental conditions. To assess these effects, parametric curves are often fitted to the data and comparisons are carried out by testing for equality of mean parameter estimates across conditions. This approach is parametric and, thus, vulnerable to violations of the implied assumptions. Furthermore, testing for equality of means of parameters may be misleading: Psychometric functions may vary meaningfully across conditions on an observer-by-observer basis with no effect on the mean values of the estimated parameters. Alternative approaches to assess equality of psychometric functions per se are thus needed. This paper compares three nonparametric tests that are applicable in all situations of interest: The existing generalized Mantel–Haenszel test, a generalization of the Berry–Mielke test that was developed here, and a split variant of the generalized Mantel–Haenszel test also developed here. Their statistical properties (accuracy and power) are studied via simulation and the results show that all tests are indistinguishable as to accuracy but they differ non-uniformly as to power. Empirical use of the tests is illustrated via analyses of published data sets and practical recommendations are given. The computer code in matlab and R to conduct these tests is available as Electronic Supplemental Material.     

Modeling psychometric functions in R

We demonstrate some procedures in the statistical computing environment R for obtaining maximum likelihood estimates of the parameters of a psychometric function by fitting a generalized nonlinear regression model to the data. A feature for fitting a linear model to the threshold (or other) parameters of several psychometric functions simultaneously provides a powerful tool for testing hypotheses about the data and, potentially, for reducing the number of parameters necessary to describe them. Finally, we illustrate procedures for treating one parameter as a random effect that would permit a simplified approach to modeling stimulus-independent variability due to factors such as lapses or interobserver differences. These tools will facilitate a more comprehensive and explicit approach to the modeling of psychometric data.     

Comparing adaptive procedures for estimating the psychometric function for an auditory gap detection task

A subject’s sensitivity to a stimulus variation can be studied by estimating the psychometric function. Generally speaking, three parameters of the psychometric function are of interest: the performance threshold, the slope of the function, and the rate at which attention lapses occur. In the present study, three psychophysical procedures were used to estimate the three-parameter psychometric function for an auditory gap detection task. These were an up–down staircase (up–down) procedure, an entropy-based Bayesian (entropy) procedure, and an updated maximum-likelihood (UML) procedure. Data collected from four young, normal-hearing listeners showed that while all three procedures provided similar estimates of the threshold parameter, the up–down procedure performed slightly better in estimating the slope and lapse rate for 200 trials of data collection. When the lapse rate was increased by mixing in random responses for the three adaptive procedures, the larger lapse rate was especially detrimental to the efficiency of the up–down procedure, and the UML procedure provided better estimates of the threshold and slope than did the other two procedures.     

Measuring, estimating, and understanding the psychometric function: A commentary

The psychometric function, relating the subject’s response to the physical stimulus, is fundamental to psychophysics. This paper examines various psychometric function topics, many inspired by this special symposium issue ofPerception & Psychophysics: What are the relative merits of objective yes/no versus forced choice tasks (including threshold variance)? What are the relative merits of adaptive versus constant stimuli methods? What are the relative merits of likelihood versus up-down staircase adaptive methods? Is 2AFC free of substantial bias? Is there no efficient adaptive method for objective yes/no tasks? Should adaptive methods aim for 90% correct? Can adding more responses to forced choice and objective yes/no tasks reduce the threshold variance? What is the best way to deal with lapses? How is the Weibull function intimately related to thed’ function? What causes bias in the likelihood goodness-of-fit? What causes bias in slope estimates from adaptive methods? How good are nonparametric methods for estimating psychometric function parameters? Of what value is the psychometric function slope? How are various psychometric functions related to each other? The resolution of many of these issues is surprising.     

Spatial-frequency uncertainty and cuing effects on psychometric functions for contrast detection

In the present study, the effects of spatial-frequency uncertainty and cuing on psychometric functions for contrast detection of sinusoidal gratings are examined. For this purpose, psychometric functions were collected from 4 subjects under fixed-frequency, randomized-frequency, and cued-frequency conditions. The experiment was conducted with a temporal two-alternative forced-choice task, and five spatial frequencies in the range of 0.5 and 8.0 c/deg and seven contrast levels for each frequency were used. The results showed that the psychometric functions for the randomized-frequency condition were shallower than those for the fixed-frequency condition, supporting the single-band model for the uncertainty effects (Hübner, 1993a, 1993b). For the cued-frequency condition, the slopes of the functions were not clearly different from those for the randomized condition. These results clearly differ from those of Hübner (1996b), which showed, in the spatial two-alternative forced-choice task, steeper psychometric functions for the randomized-frequency condition than those for the fixed- and cued-frequency conditions, supporting the multiple-band model (Hübner, 1993a, 1993b). The difference suggests that the single-band model applies to the uncertainty effects in the temporal forced-choice task, whereas the multiple-band model does so in the spatial forced-choice task.     

Measuring, estimating, and understanding the psychometric function: a commentary.

The psychometric function, relating the subject's response to the physical stimulus, is fundamental to psychophysics. This paper examines various psychometric function topics, many inspired by this special symposium issue of Perception & Psychophysics: What are the relative merits of objective yes/no versus forced choice tasks (including threshold variance)? What are the relative merits of adaptive versus constant stimuli methods? What are the relative merits of likelihood versus up-down staircase adaptive methods? Is 2AFC free of substantial bias? Is there no efficient adaptive method for objective yes/no tasks? Should adaptive methods aim for 90% correct? Can adding more responses to forced choice and objective yes/no tasks reduce the threshold variance? What is the best way to deal with lapses? How is the Weibull function intimately related to the d' function? What causes bias in the likelihood goodness-of-fit? What causes bias in slope estimates from adaptive methods? How good are nonparametric methods for estimating psychometric function parameters? Of what value is the psychometric function slope? How are various psychometric functions related to each other? The resolution of many of these issues is surprising.     

Temporal order psychometric functions based on confidence-rating data

Psychophysical tasks involving confidence judgments allow the simultaneous generation of a family of psychometric functions. Sternberg, Knoll, and Mallows (1975) have demonstrated the power of the multiple-function approach in evaluating models concerned with specifying the source of errors in judgments of simultaneity and temporal order. In the present paper, data from a temporal order task requiring confidence ratings are examined, and a number of models for successiveness and order judgments evaluated.

     

Estimation of psychometric functions from adaptive tracking procedures

Because adaptive tracking procedures are designed to avoid stimulus levels far from a target threshold value, the psychometric function constructed from the trial-by-trial data in the track may be accurate near the target level but a poor reflection of performance at levels far removed from the target. A series of computer simulations was undertaken to assess the reliability and accuracy of psychometric functions generated from data collected in up-down adaptive tracking procedures. Estimates of psychometric function slopes were obtained from-trial-by-trial data in simulated adaptive tracks and compared with the true characteristics of the functions used to generate the tracks. Simulations were carried out for three psychophysical procedures and two target performance levels, with tracks generated by psychometric functions with three different slopes. The functions reconstructed from the tracking data were, for the most part, accurate reflections of the true generating functions when at least 200 trials were included in the tracks. However, for 50- and 100-trial tracks, slope estimates were biased high for all simulated experimental conditions. Correction factors for slope estimates from these tracks are presented. There was no difference in the accuracy and reliability of slope estimation due to -target-level-for the adaptive track, and only minor differences due to psychophysical procedure. It is recommended that, if both threshold and slope of psychometric functions are to be estimated-from the trial-by-trial tracking data, at least 100 trials should be included in the tracks, and a three- or four-alternative forced-choice procedure should be used. However, good estimates can also be obtained using the two-alternative forced-choice procedure or less than 100 trials if appropriate corrections for bias are applied.     

Estimation of psychometric functions from adaptive tracking procedures.

Because adaptive tracking procedures are designed to avoid stimulus levels far from a target threshold value, the psychometric function constructed from the trial-by-trial data in the track may be accurate near the target level but a poor reflection of performance at levels far removed from the target. A series of computer simulations was undertaken to assess the reliability and accuracy of psychometric functions generated from data collected in up-down adaptive tracking procedures. Estimates of psychometric function slopes were obtained from trial-by-trial data in simulated adaptive tracks and compared with the true characteristics of the functions used to generate the tracks. Simulations were carried out for three psychophysical procedures and two target performance levels, with tracks generated by psychometric functions with three different slopes. The functions reconstructed from the tracking data were, for the most part, accurate reflections of the true generating functions when at least 200 trials were included in the tracks. However, for 50- and 100-trial tracks, slope estimates were biased high for all simulated experimental conditions. Correction factors for slope estimates from these tracks are presented. There was no difference in the accuracy and reliability of slope estimation due to target level for the adaptive track, and only minor differences due to psychophysical procedure. It is recommended that, if both threshold and slope of psychometric functions are to be estimated from the trial-by-trial tracking data, at least 100 trials should be included in the tracks, and a three- or four-alternative forced-choice procedure should be used. However, good estimates can also be obtained using the two-alternative forced-choice procedure or less than 100 trials if appropriate corrections for bias are applied.     

Slope bias of psychometric functions derived from adaptive data

Several investigators have fit psychometric functions to data from adaptive procedures for threshold estimation. Although the threshold estimates are in general quite correct, one encounters a slope bias that has not been explained up to now. The present paper demonstrates slope bias for parametric and nonparametric maximum-likelihood fits and for Spearman-Kärber analysis of adaptive data. The examples include staircase and stochastic approximation procedures. The paper then presents an explanation of slope bias based on serial data dependency in adaptive procedures. Data dependency is first illustrated with simple two-trial examples and then extended to realistic adaptive procedures. Finally, the paper presents an adaptive staircase procedure designed to measure threshold and slope directly. In contrast to classical adaptive threshold-only procedures, this procedure varies both a threshold and a spread parameter in response to double trials.     

The Couples Emotion Rating Form: psychometric properties and theoretical associations

The Couples Emotion Rating Form assesses 3 types of negative emotion that are salient during times of relationship conflict. Hard emotion includes feeling angry and aggravated, soft emotion includes feeling hurt and sad, and flat emotion includes feeling bored and indifferent. In Study 1, scales measuring hard and soft emotion were validated by observation of 82 married couples in a series of conflict conversations. Self-report ratings for each emotion corresponded with observer ratings of the same emotion, and the emotion scales produced expected correlations with negative affect. In Study 2, a measure of flat emotion was added to the instrument, and 1,239 married people completed questionnaires. The rating form fit an expected 3-dimensional factor structure, and each scale correlated with a set of theoretically linked constructs. Hard emotion was associated with power assertion, pursuit of self-centered goals, and negative communication. Soft emotion was associated with expressions of vulnerability, pursuit of prosocial goals, and positive communication. Flat emotion was distinct from other emotions in being associated with withdrawal.     

Observers can voluntarily shift their psychometric functions without losing sensitivity

Psychometric sensory discrimination functions are usually modeled by cumulative Gaussian functions with just two parameters, their central tendency (μ) and their slope (1/σ). These correspond to Fechner's "constant" and "variable" errors, respectively. Fechner pointed out that even the constant error could vary over space and time and could masquerade as variable error. We wondered whether observers could deliberately introduce a constant error into their performance without loss of precision. In three-dot vernier and bisection tasks with the method of single stimuli, observers were instructed to favour one of the two responses when unsure of their answer. The slope of the resulting psychometric function was not significantly changed, despite a significant change in central tendency. Similar results were obtained when altered feedback was used to induce bias. We inferred that observers can adopt artificial response criteria without any significant increase in criterion fluctuation. These findings have implications for some studies that have measured perceptual "illusions" by shifts in the psychometric functions of sophisticated observers.     

Slope bias of psychometric functions derived from adaptive data.

Several investigators have fit psychometric functions to data from adaptive procedures for threshold estimation. Although the threshold estimates are in general quite correct, one encounters a slope bias that has not been explained up to now. The present paper demonstrates slope bias for parametric and nonparametric maximum-likelihood fits and for Spearman-K?rber analysis of adaptive data. The examples include staircase and stochastic approximation procedures. The paper then presents an explanation of slope bias based on serial data dependency in adaptive procedures. Data dependency is first illustrated with simple two-trial examples and then extended to realistic adaptive procedures. Finally, the paper presents an adaptive staircase procedure designed to measure threshold and slope directly. In contrast to classical adaptive threshold-only procedures, this procedure varies both a threshold and a spread parameter in response to double trials.     

On estimating processing variance: commentary and reanalysis of Kail's "Developmental functions for speeds of cognitive processes".

A recent paper by Kail (1988) in this journal appears to contain a significant error in the data analysis. The "goodness-of-fit" coefficients reported which suggest that overall about 94% of the variance can be accounted for by the model seem to be a substantial overestimation as a result of inappropriate procedures for statistical modeling. Using the data made available to us by Kail, we have reanalyzed these results. The corrected values range from 0.9 to 92.1% for the individual tasks with an overall average between 40 and 60%. We suggest that the support for the original conclusions is considerably weaker than reported.     

On the representation of error

Though he maintained a significant interest in theoretical aspects of measurement, Henry E. Kyburg, Jr. was critical of the representational theory that in many ways has come to dominate discussions concerning the foundations of measurement. In particular, Kyburg (in Savage and Ehrlich (eds) Philosophical and foundational issues in measurement theory, 1992) asserts that the representational theory of measurement, as introduced in (Scott and Suppes, Journal of Symbolic Logic, 23:113?C128, 1958) and developed in (Krantz et?al., Foundations of measurment: additive and polynomial representations. Academic Press, 1971), cannot account for the measurement of error. The present work examines and responds to this charge.