QMPE: Estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation.¹ QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three “shifted” distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibull distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.     

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.     

Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package

Schwarz (2001, 2002) proposed the ex-Wald distribution, obtained from the convolution of Wald and exponential random variables, as a model of simple and go/no-go response time. This article provides functions for the S-PLUS package that produce maximum likelihood estimates of the parameters for the ex-Wald, as well as for the shifted Wald and ex-Gaussian, distributions. In a Monte Carlo study, the efficiency and bias of parameter estimates were examined. Results indicated that samples of at least 400 are necessary to obtain adequate estimates of the ex-Wald and that, for some parameter ranges, much larger samples may be required. For shifted Wald estimation, smaller samples of around 100 were adequate, at least when fits identified by the software as having ill-conditioned maximums were excluded. The use of all functions is illustrated using data from Schwarz (2001). The S-PLUS functions and Schwarz’s data may be downloaded from the Psychonomic Society’s Web archive, www. psychonomic.org/archive/.     

Intelligence, reaction time (RT) and a new ‘odd-man-out’ RT paradigm

A sample of adult Ss of reasonably normal intelligence were given an ‘IQ’ test, a series of RT tests using 0, 1, 2, 3 bits of information in a Hick paradigm and an RT task requiring choice of 1 of 3 lights as an ‘odd-man-out’ on the basis of its relative position. Negative correlations were found between both RT and measures of variation in RT and ‘IQ’ for both of the two tasks. Recent results showing no correlation between Hick slope and ‘IQ’ and no increase in correlation between ‘IQ’ and RT with increasing number of bits, are confirmed. An explanation for findings of Ss whose RT data do not conform to Hick's law is tested and found inadequate. The ‘odd-man-out’ task is found to show an effect of ‘learning’ across the period of the task, the size of the learning effect was found also to correlate with ‘IQ’, but no evidence for learning was found with the choice RT task.     

The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality

To assess the reliability of congeneric tests, specifically designed reliability measures have been proposed. This paper emphasizes that such measures rely on a unidimensionality hypothesis, which can neither be confirmed nor rejected when there are only three test parts, and will invariably be rejected when there are more than three test parts. Jackson and Agunwamba's (1977) greatest lower bound to reliability is proposed instead. Although this bound has a reputation for overestimating the population value when the sample size is small, this is no reason to prefer the unidimensionality-based reliability. Firstly, the sampling bias problem of the glb does not play a role when the number of test parts is small, as is often the case with congeneric measures. Secondly, glb and unidimensionality based reliability are often equal when there are three test parts, and when there are more test parts, their numerical values are still very similar. To the extent that the bias problem of the greatest lower bound does play a role, unidimensionality-based reliability is equally affected. Although unidimensionality and reliability are often thought of as unrelated, this paper shows that, from at least two perspectives, they act as antagonistic concepts. A measure, based on the same framework that led to the greatest lower bound, is discussed for assessing how close is a set of variables to unidimensionality. It is the percentage of common variance that can be explained by a single factor. An empirical example is given to demonstrate the main points of the paper. The authors are obliged to Henk Kiers for commenting on a previous version. Gregor Sočan is now at the University of Ljubljana.     

The asymptotic bias of minimum trace factor analysis, with applications to the greatest lower bound to reliability

In theory, the greatest lower bound (g.l.b.) to reliability is the best possible lower bound to the reliability based on single test administration. Yet the practical use of the g.l.b. has been severely hindered by sampling bias problems. It is well known that the g.l.b. based on small samples (even a sample of one thousand subjects is not generally enough) may severely overestimate the population value, and statistical treatment of the bias has been badly missing. The only results obtained so far are concerned with the asymptotic variance of the g.l.b. and of its numerator (the maximum possible error variance of a test), based on first order derivatives and the asumption of multivariate normality. The present paper extends these results by offering explicit expressions for the second order derivatives. This yields a closed form expression for the asymptotic bias of both the g.l.b. and its numerator, under the assumptions that the rank of the reduced covariance matrix is at or above the Ledermann bound, and that the nonnegativity constraints on the diagonal elements of the matrix of unique variances are inactive. It is also shown that, when the reduced rank is at its highest possible value (i.e., the number of variables minus one), the numerator of the g.l.b. is asymptotically unbiased, and the asymptotic bias of the g.l.b. is negative. The latter results are contrary to common belief, but apply only to cases where the number of variables is small. The asymptotic results are illustrated by numerical examples.This research was supported by grant DMI-9713878 from the National Science Foundation.     

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.     

Response time distributions and the Stroop Task: a test of the Cohen, Dunbar, and McClelland (1990) model.

Cohen, Dunbar, and McClelland's (1990) model was tested for Strooplike interference tasks by studying the shape of the distribution of response latencies produced by Ss and by the model. The model correctly anticipates changes in mean response latency (M(RT)) across congruent and incongruent conditions. It does not, however, correctly anticipate changes in the shape of the distributions, even though changes in the shape of the distributions underlie the changes in M(RT). Thus the model predicts M(RT) successfully but for the wrong reason. It is concluded that the model is not an adequate account of Ss' performance in the Stroop task.     

Morphological processing: a comparison between free and bound stem facilitation

Linguists distinguish between words formed from free stems (e.g., actor: act) and those formed from bound stems (e.g., spectator: spect). In a forward masked priming task, we observed significant morphological facilitation for prime-target pairs that shared either a free (e.g., deform-CONFORM) or a bound (e.g., revive-SURVIVE) stem. Relative to an unrelated baseline, magnitudes of facilitation for free (e.g., form) and bound (e.g., vive) stems were significant and comparable, but relative to an orthographic baseline free stem facilitation was greater than bound stem facilitation. In addition, the magnitude of bound (but not free) stem morphological facilitation correlated with the number of morphological relatives.     

Effects of alcohol, practice, and task complexity on reaction time distributions.

The effects of alcohol (1.0 ml/kg body weight) and practice (2 sessions) were investigated in 2-, 4-, and 8-choice reaction time (RT) tasks with 24 male subjects. The number of errors increased with alcohol, practice, and increasing task complexity (choice). Mean RT decreased with practice, but increased with alcohol and complexity. Both the alcohol and practice effects on mean RT increased as complexity increased. The effects of alcohol, practice, and complexity were all larger for the higher percentiles of the RT distributions than for the lower percentiles. RT distributions were further analysed at each level of choice by plotting percentiles (5th, 10th, ... , 95th) for alcohol conditions against corresponding percentiles for no-alcohol conditions, and percentiles obtained early in practice (Session 1) against those obtained later in practice (Session 2). These plots revealed that whereas at all levels of choice the effect of alcohol could be expressed as a simple linear transformation of all RTs, the effect of practice required a more complex curvilinear transformation. Thus, alcohol produces a general slowing of all RTs, whereas practice produces a disproportionate improvement at the slower end of the RT distribution.     

Gaze step distributions reflect fixations and saccades: a comment on

In three experimental tasks Stephen and Mirman (2010) measured gaze steps, the distance in pixels between gaze positions on successive samples from an eyetracker. They argued that the distribution of gaze steps is best fit by the lognormal distribution, and based on this analysis they concluded that interactive cognitive processes underlie eye movement control in these tasks. The present comment argues that the gaze step distribution is predictable based on the fact that the eyes alternate between a fixation state in which gaze is steady and a saccade state in which gaze position changes rapidly. By fitting a simple mixture model to Stephen and Mirman's gaze step data we reveal a fixation distribution and a saccade distribution. This mixture model captures the shape of the gaze step distribution in detail, unlike the lognormal model, and provides a better quantitative fit to the data. We conclude that the gaze step distribution does not directly suggest processing interaction, and we emphasize some important limits on the utility of fitting theoretical distributions to data.     

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: II: A search procedure to locate the greatest lower bound

Finding the greatest lower bound for the reliability of the total score on a test comprisingn non-homogenous items with dispersion matrix Σ _x is equivalent to maximizing the trace of a diagonal matrix Σ _E with elements θ _I , subject to Σ _E and Σ _T =Σ _x − Σ _E being non-negative definite. The casesn=2 andn=3 are solved explicity. A computer search in the space of the θ _i is developed for the general case. When Guttman's λ₄ (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ _i makes the rank of Σ _T less thann − 1. Numerical examples of various bounds are given. Present affiliation of the first author: St. Hild's College of Education, Durham City, England.     

Satisfaction with life in individuals with a lower limb amputation: The importance of active coping and acceptance

The aim of this study was to analyze the relationship between sociodemographic/clinical characteristics, coping strategies and satisfaction with life in individuals with lower limb amputation. Sixty‐three individuals with a lower limb amputation due to Diabetes and Peripheral Vascular Disease participated in the study and answered measures of coping strategies and satisfaction with life. Findings revealed high dissatisfaction with life. Acceptance and active coping were the most used coping strategies. Satisfaction with life was positively associated with active and planning coping, religion, acceptance and humour. There were differences in coping strategies according to gender, age, marital status, presence of residual limb pain, prosthesis use and mobility level. Results emphasize the differential role of coping strategies, for each individual. Psychosocial interventions need to take into consideration coping strategies during the process of rehabilitation and be specific regarding individuals` sociodemographic and clinical characteristics. This study may help design interventions that answer individuals with lower limb amputations given that coping strategies are a valuable resource in the promotion of satisfaction with life.     

Shapes of reaction-time distributions and shapes of learning curves: a test of the instance theory of automaticity.

The instance theory assumes that automatic performance is based on single-step direct-access retrieval from memory of prior solutions to present problems. The theory predicts that the shape of the learning curve depends on the shape of the distribution of retrieval times. One can deduce from the fundamental assumptions of the theory that (1) the entire distribution of reaction times, not just the mean, will decrease as a power function of practice; (2) asymptotically, the retrieval-time distribution must be a Weibull distribution; and (3) the exponent of the Weibull, which is the parameter that determines its shape, must be the reciprocal of the exponent of the power function. These predictions were tested and mostly confirmed in 12 data sets from 2 experiments. The ability of the instance theory to predict the power law is contrasted with the ability of other theories to account for it.     

Smiles as signals of lower status in football players and fashion models: Evidence that smiles are associated with lower dominance and lower prestige

Across four studies, the current paper demonstrates that smiles are associated with lower social status. Moreover, the association between smiles and lower status appears in the psychology of observers and generalizes across two forms of status: prestige and dominance. In the first study, faces of fashion models representing less prestigious apparel brands were found to be more similar to a canonical smile display than the faces of models representing more prestigious apparel brands.?In a second study, after being experimentally primed with either high or low prestige fashion narratives, participants in the low prestige condition were more likely to perceive smiles in a series of photographs depicting smiling and non-smiling faces. A third study of football player photographs revealed that the faces of less dominant (smaller) football players were more similar to the canonical smile display than the faces of their physically larger counterparts. Using the same football player photographs, a fourth study found that smiling was a more reliable indicator of perceived status-relevant personality traits than perceptions of the football players' physical sizes inferred from the photographs.     

VALCOR: A program for estimating standard error, confidence intervals, and probability of corrected validity

VALCOR is a Turbo-Basic program that corrects the observed (uncorrected) validity coefficients for criterion and predictor unreliability and range restriction in the predictor. Furthermore, using the formulas for the standard error of functions of correlations derived by Bobko and Rieck (1980), the program provides an estimation of the standard error, the confidence intervals, and the probability of the corrected validity coefficients. In this way, the probability and the boundaries of the corrected validity coefficients may be reported together with the probability of the uncorrected validity coefficients. The results are presented on the computer screen and may be saved in an external file.