An extension of the rasch model for ratings providing both location and dispersion parameters

An elaboration of a psychometric model for rated data, which belongs to the class of Rasch models, is shown to provide a model with two parameters, one characterising location and one characterising dispersion. The later parameter, derived from the idea of a unit of scale, is also shown to reflect the shape of rating distributions, ranging from unimodal, through uniform, and then to U-shaped distributions. A brief case is made that when a rating distribution is treated as a random error distribution, then the distribution should be unimodal.     

A note on the sampling properties of the Vincentizing (quantile averaging) procedure

To assess the effect of a manipulation on a response time distribution, psychologists often use Vincentizing or quantile averaging to construct group or “average” distributions. We provide a theorem characterizing the large sample properties of the averaged quantiles when the individual RT distributions all belong to the same location-scale family. We then apply the theorem to estimating parameters for the quantile-averaged distributions. From the theorem, it is shown that parameters of the group distribution can be estimated by generalized least squares. This method provides accurate estimates of standard errors of parameters and can therefore be used in formal inference. The method is benchmarked in a small simulation study against both a maximum likelihood method and an ordinary least-squares method. Generalized least squares essentially is the only method based on the averaged quantiles that is both unbiased and provides accurate estimates of parameter standard errors. It is also proved that for location-scale families, performing generalized least squares on quantile averages is formally equivalent to averaging parameter estimates from generalized least squares performed on individuals. A limitation on the method is that individual RT distributions must be members of the same location-scale family.     

Robust Bayesian growth curve modelling using conditional medians

Growth curve models have been widely used to analyse longitudinal data in social and behavioural sciences. Although growth curve models with normality assumptions are relatively easy to estimate, practical data are rarely normal. Failing to account for non-normal data may lead to unreliable model estimation and misleading statistical inference. In this work, we propose a robust approach for growth curve modelling using conditional medians that are less sensitive to outlying observations. Bayesian methods are applied for model estimation and inference. Based on the existing work on Bayesian quantile regression using asymmetric Laplace distributions, we use asymmetric Laplace distributions to convert the problem of estimating a median growth curve model into a problem of obtaining the maximum likelihood estimator for a transformed model. Monte Carlo simulation studies have been conducted to evaluate the numerical performance of the proposed approach with data containing outliers or leverage observations. The results show that the proposed approach yields more accurate and efficient parameter estimates than traditional growth curve modelling. We illustrate the application of our robust approach using conditional medians based on a real data set from the Virginia Cognitive Aging Project.     

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.     

When is the Wilcoxon–Mann–Whitney procedure a test of location? Implications for effect-size measures

The Wilcoxon–Mann–Whitney procedure is invariant under monotone transformations but its use as a test of location or shift is said not to be so. It tests location only under the shift model, the assumption of parallel cumulative distribution functions (cdfs). We show that infinitely many monotone transformations of the measured variable produce parallel cdfs, so long as the original cdfs intersect nowhere or everywhere. Thus there are infinitely many effect sizes measured as shifts of medians, invalidating the notion that there is one true shift parameter and thereby rendering any single estimate dubious. Measuring effect size using the probability of superiority alleviates this difficulty.     

Considerations in psychometric modeling of response time

Semiparametric models express a set of distributions of event times in terms of (a) a single parameter which varies across distributions and (b) a single function which does not vary across distributions and which has an unspecified form. These models appear to be attractive alternatives to parametric models of response times in psychometrics. However, our use of such models may require incorporating additional functions which do not vary across distributions and may require expressing the models in terms of the joint distribution of response class and response time.     

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. 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 Weibul 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.     

Reaction time distribution analysis of spatial correspondence effects

Since 1994, group reaction time (RT) distribution analyses of spatial correspondence effects have been used to evaluate the dynamics of the spatial Simon effect, a benefit of correspondence of stimulus location information with response location for tasks in which stimulus location is irrelevant. We review the history and justification for analyzing group RT distributions and clarify which conditions result in the Simon effect decreasing across the distribution and which lead to flat or increasing functions. Although the standard left-right Simon effect typically yields a function for which the effect decreases as RT increases, in most other task variations, the Simon effect remains stable or increases across the RT distribution. Studies that have used other means of evaluating the temporal dynamics of the Simon effect provide converging evidence that the changes in the Simon effect across the distribution are due mainly to temporal activation properties, an issue that has been a matter of some dispute.     

A new quantile estimator with weights based on a subsampling approach

Quantiles are widely used in both theoretical and applied statistics, and it is important to be able to deploy appropriate quantile estimators. To improve performance in the lower and upper quantiles, especially with small sample sizes, a new quantile estimator is introduced which is a weighted average of all order statistics. The new estimator, denoted NO, has desirable asymptotic properties. Moreover, it offers practical advantages over four estimators in terms of efficiency in most experimental settings. The Harrell–Davis quantile estimator, the default quantile estimator of the R programming language, the Sfakianakis–Verginis SV2 quantile estimator and a kernel quantile estimator. The NO quantile estimator is also utilized in comparing two independent groups with a percentile bootstrap method and, as expected, it is more successful than other estimators in controlling Type I error rates.     

QMLE: Fast,robust, and efficient estimation of distribution functions based on quantiles

Quantile maximum likelihood (QML) is an estimation technique, proposed by Heathcote, Brown, and Mewhort (2002), that provides robust and efficient estimates of distribution parameters, typically for response time data, in sample sizes as small as 40 observations. In view of the computational difficulty inherent in implementing QML, we provide open-source Fortran 90 code that calculates QML estimates for parameters of the ex-Gaussian distribution, as well as standard maximum likelihood estimates. We show that parameter estimates from QML are asymptotically unbiased and normally distributed. Our software provides asymptotically correct standard error and parameter intercorrelation estimates, as well as producing the outputs required for constructing quantile—quantile plots. The code is parallelizable and can easily be modified to estimate parameters from other distributions. Compiled binaries, as well as the source code, example analysis files, and a detailed manual, are available for free on the Internet.     

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.     

A comment on Heathcote,Brown, and Mewhort’s QMLE method for response time distributions

Heathcote, Brown, and Mewhort (2002) have introduced a new, robust method of estimating response time distributions. Their method may have practical advantages over conventional maximum likelihood estimation. The basic idea is that the likelihood of parameters is maximized given a few quantiles from the data. We show that Heathcote et al.’s likelihood function is not correct and provide the appropriate correction. However, although our correction stands on firmer theoretical ground than Heathcote et al.’s, it appears to yield worse parameter estimates. This result further indicates that, at least for some distributions and situations, quantile maximum likelihood estimation may have better nonasymptotic properties than a more theoretically justified approach.     

An evaluation of the Vincentizing method of forming group-level response time distributions

Vincentizing (quantile averaging) is a popular means of pooling response time distributions across individuals to produce a group average. The benefit of Vincentizing is that the resulting histogram “looks like” an average of the individuals. In this article, we competitively test Vincentizing against the more mundane approach of averaging parameter estimates from fits to individuals. We simulate data from three distributions: the ex-Gaussian, the Weibull, and the shifted-Wald. For the ex-Gaussian and the shifted-Wald, parameter averaging outperforms Vincentizing. There is only an advantage of Vincentizing for the Weibull and only when there are few observations per participant. Overall, we recommend that researchers use Vincentizing only in select circumstances and with the knowledge that Vincentized estimates are often inconsistent estimators of averaged parameters.     

Time Delay Embedding Increases Estimation Precision of Models of Intraindividual Variability

This paper investigates the precision of parameters estimated from local samples of time dependent functions. We find that time delay embedding, i.e., structuring data prior to analysis by constructing a data matrix of overlapping samples, increases the precision of parameter estimates and in turn statistical power compared to standard independent rows of panel data. We show that the reason for this effect is that the sign of estimation bias depends on the position of a misplaced data point if there is no a priori knowledge about initial conditions of the time dependent function. Hence, we reason that the advantage of time delayed embedding is likely to hold true for a wide variety of functions. We support these conclusions both by mathematical analysis and two simulations.     

What are the shapes of response time distributions in visual search?

Many visual search experiments measure response time (RT) as their primary dependent variable. Analyses typically focus on mean (or median) RT. However, given enough data, the RT distribution can be a rich source of information. For this paper, we collected about 500 trials per cell per observer for both target-present and target-absent displays in each of three classic search tasks: feature search, with the target defined by color; conjunction search, with the target defined by both color and orientation; and spatial configuration search for a 2 among distractor 5s. This large data set allows us to characterize the RT distributions in detail. We present the raw RT distributions and fit several psychologically motivated functions (ex-Gaussian, ex-Wald, Gamma, and Weibull) to the data. We analyze and interpret parameter trends from these four functions within the context of theories of visual search.     

The distribution of interresponse times in the pigeon during variable-interval reinforcement

Three pigeons' pecks were reinforced on 1- and 2-min variable-interval schedules, and frequency distributions of their interresponse times (IRTs) were recorded. The conditional probability that a response would fall into any IRT category was estimated by the interresponse-times-per-opportunity transformation (IRTs/op). The resulting functions were notable chiefly for the relatively low probability of IRTs in the 0.2- to 0.3-sec range; in other respects they varied within and between subjects. The overall level of the curves generally rose over the course of 32 experimental hours, but their shapes changed unsystematically. The shape of the IRT distribution was much the same for VI 1-min and VI 2-min. The variability of these distributions supports the notion that the VI schedule only loosely controls response rate, permitting wide latitude to adventitious effects. There was no systematic evidence that curves changed over sessions to conform to the distribution of reinforcements by IRT.     

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.     

Individual distributions under ordinal measurement

For ordinal measurement the concept of an individual propensity distribution is developed. For any given individual the mean of this distribution is his true score, for which estimation procedures are discussed. Two measures of individual dispersion are considered and their distributions derived in the null case. These measures are shown to be counterparts at the individual level of Kendall's tau and Spearman's rho. Estimation of the two dispersion measures from sample data is investigated, and the relation of these estimates to the variance of the individual propensity distribution is derived.