Converting between measures of slope of the psychometric function

The psychometric function’s slope provides information about the reliability of psychophysical threshold estimates. Furthermore, knowing the slope allows one to compare, across studies, thresholds that were obtained at different performance criterion levels. Unfortunately, the empirical validation of psychometric function slope estimates is hindered by the bewildering variety of slope measures that are in use. The present article provides conversion formulas for the most popular cases, including the logistic, Weibull, Quick, cumulative normal, and hyperbolic tangent functions as analytic representations, in both linear and log coordinates and to different log bases, the practical decilog unit, the empirically based interquartile range measure of slope, and slope in a? representation of performance.     

Converting between measures of slope of the psychometric function.

The psychometric function's slope provides information about the reliability of psychophysical threshold estimates. Furthermore, knowing the slope allows one to compare, across studies, thresholds that were obtained at different performance criterion levels. Unfortunately, the empirical validation of psychometric function slope estimates is hindered by the bewildering variety of slope measures that are in use. The present article provides conversion formulas for the most popular cases, including the logistic, Weibull, Quick, cumulative normal, and hyperbolic tangent functions as analytic representations, in both linear and log coordinates and to different log bases, the practical decilog unit, the empirically based interquartile range measure of slope, and slope in a d' representation of performance.     

Invariance of the psychometric function for character recognition across the visual field

The psychometric function for recognition of singly presented digits as a function of digit contrast was measured at 2° steps across the horizontal meridian of the visual field, under monocular and binocular viewing conditions. A maximum-likelihood staircase procedure was used in a 10-alternative forcedchoice recognition paradigm to gather the data. Both the Weibull and the logistic psychometric functions provide excellent fits to the observed data. The slopes of these functions at their point of inflection ranged from 4.0 to 5.0 proportion-correct/log₁₀-unit contrast, for both monocular and binocular viewing and for all loci in the visual field. These slope values correspond to short-term measurements (around 30 trials, or 1 min) and do not include performance variations of longer duration; the latter are estimated to increase slope by a factor of about 1.5. A single psychometric function shape, centered around a threshold value, therefore describes recognition performance at all retinal loci and binocularity. An empirical comparison of slope results across the literature shows that the function’s slope is about twice that reported for a number of detection tasks. The comparison of recognition contrast thresholds, percentage correct values, and other performance measures across studies requires the knowledge of the psychometric function’s slope, and our results thus provide a firm basis for the study of low-contrast character recognition     

Shifts in the psychometric function and their implications for models of timing

This study examined how two models of timing, scalar expectancy theory (SET) and learning to time (LeT), conceptualize the learning process in temporal tasks, and then reports two experiments to test these conceptualizations. Pigeons responded on a two-alternative free-operant psychophysical procedure in which responses on the left key were reinforceable during the first two, but not the last two, quarters of a 60-s trial, and responses on the right key were reinforceable during the last two, but not the first two, quarters of the trial. In Experiment 1 three groups of birds experienced a difference in reinforcement rates between the two keys only at the end segments of the trial (i.e., between the first and fourth quarters), only around the middle segments of the trial (i.e., between the second and third quarters), or in both end and middle segments. In Condition 1 the difference in reinforcement rate favored the left key; in Condition 2 it favored the right key. When the reinforcement rates differed in the end segments of the trial, the psychometric function--the proportion of right responses across the trial--did not shift across conditions; when it occurred around the middle of the trial or in both end and middle segments, the psychometric function shifted across conditions. Experiment 2 showed that the psychometric function shifts even when the overall reinforcement rate for the two keys is equal, provided the rates differ around the middle of the trial. This pattern of shifts of the psychometric function is inconsistent with SET. In contrast, LeT provided a good quantitative fit to the 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.     

Invariance of the psychometric function for character recognition across the visual field.

The psychometric function for recognition of singly presented digits as a function of digit contrast was measured at 2 degrees steps across the horizontal meridian of the visual field, under monocular and binocular viewing conditions. A maximum-likelihood staircase procedure was used in a 10-alternative forced-choice recognition paradigm to gather the data Both the Weibull and the logistic psychometric functions provide excellent fits to the observed data. The slopes of these functions at their point of inflection ranged from 4.0 to 5.0 proportion-correct/log10-unit contrast, for both monocular and binocular viewing and for all loci in the visual field. These slope values correspond to short-term measurements (around 30 trials, or 1 min) and do not include performance variations of longer duration; the latter are estimated to increase slope by a factor of about 1.5. A single psychometric function shape, centered around a threshold value, therefore describes recognition performance at all retinal loci and binocularity. An empirical comparison of slope results across the literature shows that the function's slope is about twice that reported for a number of detection tasks. The comparison of recognition contrast thresholds, percentage correct values, and other performance measures across studies requires the knowledge of the psychometric function's slope, and our results thus provide a firm basis for the study of low-contrast character recognition.     

Fitting the psychometric function

A constrained generalized maximum likelihood routine for fitting psychometric functions is proposed, which determines optimum values for the complete parameter set--that is, threshold and slope--as well as for guessing and lapsing probability. The constraints are realized by Bayesian prior distributions for each of these parameters. The fit itself results from maximizing the posterior distribution of the parameter values by a multidimensional simplex method. We present results from extensive Monte Carlo simulations by which we can approximate bias and variability of the estimated parameters of simulated psychometric functions. Furthermore, we have tested the routine with data gathered in real sessions of psychophysical experimenting.     

Comparing categorization models

Four experiments are presented that competitively test rule- and exemplar-based models of human categorization behavior. Participants classified stimuli that varied on a unidimensional axis into 2 categories. The stimuli did not consistently belong to a category; instead, they were probabilistically assigned. By manipulating these assignment probabilities, it was possible to produce stimuli for which exemplar- and rule-based explanations made qualitatively different predictions. F. G. Ashby and J. T. Townsend's (1986) rule-based general recognition theory provided a better account of the data than R. M. Nosofsky's (1986) exemplar-based generalized context model in conditions in which the to-be-classified stimuli were relatively confusable. However, generalized context model provided a better account when the stimuli were relatively few and distinct. These findings are consistent with multiple process accounts of categorization and demonstrate that stimulus confusion is a determining factor as 10 which process mediates categorization.     

A test for psychometric function shift

A nonparametric, small-sample-size test for the homogeneity of two psychometric functions against the left- and right-shift alternatives has been developed. The test is designed to determine whether it is safe to amalgamate psychometric functions obtained in different experimental sessions. The sum of the lower and upper p-values of the exact (conditional) Fisher test for several 2 × 2 contingency tables (one for each point of the psychometric function) is employed as the test statistic. The probability distribution of the statistic under the null (homogeneity) hypothesis is evaluated to obtain corresponding p-values. Power functions of the test have been computed by randomly generating samples from Weibull psychometric functions. The test is free of any assumptions about the shape of the psychometric function; it requires only that all observations are statistically independent.     

Comparing learned predictiveness effects within and across compound discriminations

In four human learning experiments, we examined the extent to which learned predictiveness depends upon direct comparison between relatively good and poor predictors. Participants initially solved (a) linear compound discriminations in which one or both of the stimuli in each compound were predictive of the correct outcome, (b) biconditional discriminations where only the configurations of the stimuli were predictive of the correct outcome, or (c) pseudodiscriminations in which no stimulus features were predictive. In each experiment, subsequent learning and test stages were used to assay changes in the associability of each stimulus brought about by its role in the initial discriminations. Although learned predictiveness effects were observed in all experiments (i.e., previously predictive cues were more readily associated with a new outcome than previously nonpredictive cues), the same changes in associability were observed regardless of whether the stimulus was initially learned about in the presence of an equally predictive, more predictive, or less predictive stimulus. The results suggest that learned associability is not controlled by competitive allocation of attention, but rather by the absolute predictiveness of each individual cue.     

Sampling plans for fitting the psychometric function

Research on estimation of a psychometric function psi has usually focused on comparing alternative algorithms to apply to the data, rarely addressing how best to gather the data themselves (i.e., what sampling plan best deploys the affordable number of trials). Simulation methods were used here to assess the performance of several sampling plans in yes-no and forced-choice tasks, including the QUEST method and several variants of up-down staircases and of the method of constant stimuli (MOCS). We also assessed the efficacy of four parameter estimation methods. Performance comparisons were based on analyses of usability (i.e., the percentage of times that a plan yields usable data for the estimation of all the parameters of psi) and of the resultant distributions of parameter estimates. Maximum likelihood turned out to be the best parameter estimation method. As for sampling plans, QUEST never exceeded 80% usability even when 1000 trials were administered and rendered accurate estimates of threshold but misestimated the remaining parameters. MOCS and up-down staircases yielded similar and acceptable usability (above 95% with 400-500 trials) and, although neither type of plan allowed estimating all parameters with optimal precision, each type appeared well suited to estimating a distinct subset of parameters. An analysis of the causes of this differential suitability allowed designing alternative sampling plans (all based on up-down staircases) for yes-no and forced-choice tasks. These alternative plans rendered near optimal distributions of estimates for all parameters. The results just described apply when the fitted psi has the same mathematical form as the actual psi generating the data; in case of form mismatch, all parameters except threshold were generally misestimated but the relative performance of all the sampling plans remained identical. Detailed practical recommendations are given.     

Masking-level differences and the form of the psychometric function

Functions showing the relation between-the detectability and the energy of a signal were determined for various interaural phase conditions. The empirical relation between d′ and signal energy E is approximately d′=m(E/N₀)^k where m and k are constants for a particular function. The data show that k is fairly constant for a particular O and that m depends upon the interaural condition. That is, the various psychometrie functions for a given O all had the same form, independent of location. Therefore, the magnitude of the MLD=(10/k)log(m_i/m_r), where m_i is the constant for condition i, and m_r corresponds to a reference condition. Consequently, an MLD is relatively independent of the level of performance that is chosen for the determination of that MLD.     

Comparing hierarchical models of personality pathology

Two dimensional, hierarchical classification models of personality pathology have emerged as alternatives to traditional categorical systems: multi-tiered models with increasing numbers of factors and models that distinguish between a general factor of severity and specific factors reflecting style. Using a large sample (N = 840) with a range of psychopathology, we conducted exploratory factor analyses of individual personality disorder criteria to evaluate the validity of these conceptual structures. We estimated an oblique, “unfolding” hierarchy and a bifactor model, then examined correlations between these and multi-method functioning measures to enrich interpretation. Four-factor solutions for each model, reflecting rotations of each other, fit well and equivalently. The resulting structures are consistent with previous empirical work and provide support for each theoretical model.     

Comparing decision bound and exemplar models of categorization

The performance of a decision bound model of categorization (Ashby, J992a; Ashby & Maddox, in press) is compared with the performance of two exemplar models. The first is the generalized context model (e.g., Nosofsky, 1986, 1992) and the second is a recently proposed deterministic exemplar model (Ashby & Maddox, in press), which contains the generalized context model as a special case. When the exemplars from each category were normally distributed and the optimal decision bound was linear, the deterministic exemplar model and the decision bound model provided roughly equivalent accounts of the data. When the optimal decision bound was nonlinear, the decision bound model provided a more accurate account of the data than did either exemplar model. When applied to categorization data collected by Nosofsky (1986, 1989), in which the category exemplars are not normally distributed, the decision bound model provided excellent accounts of the data, in many cases significantly outperforming the exemplar models. The decision bound model was found to be especially successful when(1) single subject analyses were performed, (2) each subject was given relatively extensive training, and (3) the subject's performance was characterized by complex suboptimalities. These results support the hypothesis that the decision bound is of fundamental importance in predicting asymptotic categorization performance and that the decision bound models provide a viable alternative to the currently popular exemplar models of categorization.     

An antisymmetric psychometric function on a logarithmic scale

This very brief report introduces a psychometric function, very suitable for psychophysical data that displays Weber-like behaviour, because it is antisymmetric on a logarithmic scale.     

Personality and the slope of loudness function

The slope of the loudness function in direct magnitude estimation was studied at two frequencies in two groups of naive subjects. In the first group a limited scale was used and in the second group, an open ended scale. In all conditions the slope of the loudness function was found to be significantly positively correlated with test anxiety scores.     

The Job Insecurity Scale: A psychometric evaluation across five European countries

Multiple instruments have been developed and used to measure quantitative job insecurity (i.e., insecurity to lose the job as such), often without systematic evaluation of their psychometric characteristics across countries and language barriers. This may hamper consistent and reliable cross-study and cross-country comparisons. This study's aim was to introduce and validate the four-item Job Insecurity Scale (JIS) developed by De Witte across five European countries (i.e., Belgium, The Netherlands, Spain, Sweden, and the UK). Overall, the results demonstrated the construct validity (i.e., configural invariance and invariance of the measurement model parameters), the reliability (internal consistency of the items), and the criterion validity (with respect to affective organizational commitment, perceived general health, and self-reported performance) of the JIS. The different translations of the JIS can thus be considered as valid and reliable instruments to measure job insecurity and can be used to make meaningful comparisons across countries. Furthermore, the JIS translations may be utilized to assess how job insecurity is related to outcomes.     

Comparing exemplar-retrieval and decision-bound models of speeded perceptual classification

The authors compared the exemplar-based random-walk (EBRW) model of Nosofsky and Palmeri (1997) and the decision-bound model (DBM) of Ashby and Maddox (1994; Maddox & Ashby, 1996) on their ability to predict performance in Garner’s (1974) speeded classification tasks. A key question was the extent to which the models could predict facilitation in the correlated task and interference in the filtering task, in situations involving integral-dimension stimuli. To obtain rigorous constraints for model evaluation, the goal was to fit the detailed structure of the response time (RT) distribution data associated with each individual stimulus in each task. Both models yielded reasonably good global quantitative fits to the RT distribution and accuracy data. However, the DBM failed to properly characterize the interference effects in the filtering task. Apparently, a fundamental limitation of the DBM is that it predicts that the fastest RTs in the filtering task should be faster than the fastest RTs in the control task, whereas the opposite pattern was observed in our data.