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.     

Comparing and unifying slope estimates across psychometric function models

The psychometric function relating stimulus intensity to response probability generally presents itself as a monotonically increasing sigmoid profile. Two summary parameters of the function are particularly important as measures of perceptual performance: the threshold parameter, which defines the location of the function over the stimulus axis (abscissa), and the slope parameter, which defines the (local) rate at which response probability increases with increasing stimulus intensity. In practice, the psychometric function may be modeled by a variety of mathematical structures, and the resulting algebraic expression describing the slope parameter may vary considerably between different functions fitted to the same experimental data. This variation often restricts comparisons between studies that select different functions and compromises the general interpretation of slope values. This article reviews the general characteristics of psychometric function models, discusses three strategies for resolving the issue of slope value differences, and presents mathematical expressions for implementing each strategy.     

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.     

A psychometric comparison of anxiety-relevant attention measures

Background and Objectives: Preferential attention to threat, emotional response inhibition, and attentional control each purportedly play a key role in anxiety disorders. Divergent psychometric properties among attention measures may produce differential detection of anxiety-related associations and treatment-related changes. However, no studies have directly compared the psychometric properties of these attention measures in the same sample.

Design: Eighty-five young adults (M?=?19.41 years, SD?=?1.47, 48 Females) completed a cognitive task battery and a subset of 60 participants (M?=?19.42 years, SD?=?1.48, 33 Females) completed the task battery again approximately two weeks later.

Method: To assess preferential attention to threat, emotional response inhibition, and attentional control, the cognitive task battery included a dot-probe task, emotion and gender Stroop tasks, and a flanker task. Tasks varied in how attention was directed and if emotional stimuli were included. Internal consistency and test-retest reliability were compared across measures.

Results: Within the same sample, internal consistency and reliability differed across attention measures. Explicit attention measures (emotional Stroop and flanker) exhibited stronger internal consistency and greater test-retest reliability compared to implicit measures (dot-probe and gender Stroop).

Conclusions: These results inform clinical research using attention measures to assess anxiety-related differences and treatment response.     

Effects of age and the relationship between response time measures and psychometric intelligence in younger adults

The major goal of the present study was to investigate the potential influence of age on the relationship among performance measures derived from a Hick reaction time task and general intelligence (psychometric g). Participants were 130 male and 130 female younger adults (mean age: 24.6 years) covering a wide range of individual levels of intelligence. Results from structural-equation modelling analyses clearly indicate that there is no evidence for the notion of a general speed factor underlying psychometric g. A much more sufficient model is provided by the assumption that median reaction time (RTmd) and intraindividual variability of reaction time (RTSD), though highly correlated, reflect two distinct sources of variance for psychometric g. While age was negatively associated with RTmd, no such relationship could be established for RTSD.     

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.     

The psychometric properties of several measures of body image

Despite the recent upsurge of interest in the construct of body image, there is relatively little information on the psychometric properties of the instruments used to measure it. This study investigated the reliability and validity of several measures of body image and compared bulimics and normals on these measures. One hundred ten normal weight females, half of whom were diagnosed as bulimic, were administered two measures of affect toward one's body, two measures of perceptions of one's entire body, and three measures of perceptions of the size of specific body sites (face, shoulders, waist, and hips). In themain, the measures provided reliable indices of body image. Examination of the correlation matrix for the measures disclosed convergence for the affective measures of body image and for all but one of the perceptual measures of body image. There was also significant covariation between the affective and the perceptual measures. The multitrait-multimethod technique was used to investigate the construct validity of the measures concerned with perceptions of the size of body sites. The multitrait-multimethod matrices disclosed substantial convergence between perceptions of face, shoulder, waist, and hip size across the three measures. However, the measure which used kinesthetic estimates of body-site size produced low reliabilities and all three of the measures showed substantial method variance. Bulimics and normals differed significantly on both the affective and the perceptual components of body image.This study is based on the first author's masters thesis. Portions of this study were represented at the 1986 meeting of the Southeastern Psychological Association. A grant to the third author from the Anclote Psychiatric Center provided support for this research project.     

The non-monotonicity of the psychometric function in recognition memory

The technique of using the signal detectability theory parameter d’ as a measure of trace strength in recognition memory is extended in a decision-making model that predicts that a S’s psychometric function lp(Y/s) plotted against stimulus intensity / will have a peak for intermediate values of trace strength when the S has a bias towards a “Yes” response. An experiment is reported where Ss were required to recognize previously presented nonsense syllables under a number of pay-off conditions designed to give different groups of Ss different response biases. The predictions of the model were confirmed. It is suggested that a S’s discriminability and response bias parameters may not be as stable as previous studies, that pool large samples of data, have indicated, and the results also provide support for a two-stage model of recognition, where a preliminary estimate of the trace strength of an item is obtained before the location of the response criterion is decided.     

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.     

The impact of social desirability on psychometric measures of aggression

Although many studies have focused on the effects of social desirability in personality measures, few have analysed its effects on such highly undesirable behaviour as aggressiveness. The present study analyzes the impact of social desirability on measures of direct and indirect aggression and on the relationships between both kinds of aggression with impulsivity, using a method that enables the content factors of the measures to be isolated from social desirability. Results showed that aggression measures are highly affected by social desirability and that the relationships between the two forms of aggression and impulsivity are due to the content measured by the tests and not to a common social desirability factor.     

Factor analysis to examine psychometric properties of family functioning measures

To assess the relationship of family functioning to problem behaviors and alcohol and drug use among youth, researchers must test the effects of interventions using suitably constructed and psychometrically sound scales. This study evaluated whether originally calculated coefficients alpha underestimate the reliability of the family functioning measures given. Through exploratory factor analysis, estimates of alternative internal consistency reliability which might improve the estimate of reliability were examined. Responses of 755 adults from Strengthening Multi-ethnic Families and Communities were analyzed. Coefficients alpha for two scales were modest (alpha = .68 and alpha = .75), and factor analysis indicated that the scales were multidimensional. After exploratory factor analysis, the reassessment of reliability based on the extracted factors indicated an overall increase in the coefficients alpha.     

A psychometric evaluation of behavioral inhibition and approach self-report measures

Despite the putative applicability and unquestioned heuristic value of capturing individual variation in behavioral inhibition (BIS) and approach system (BAS) sensitivities, the field has yet to achieve widespread agreement regarding a self-report instrument of choice. The current study evaluates perhaps the two strongest candidates, the BIS/BAS scales (Carver & White, 1994) and the Sensitivity to Punishment and Sensitivity to Reward Questionnaire (SPSRQ; Torrubia, Avila, Molto, & Caseras, 2001). Using both confirmatory and exploratory factor analytic techniques in two university samples, we determined that neither measure achieved adequate fit to our data set, and both contain multiple items we deemed to be problematic. Models trimmed of the poor items achieved better fit than the full models. However, even after trimming the data, model fit was marginal at best. Caution is urged in the continued use of both measures on conceptual and psychometric grounds.     

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.     

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 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.     

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.