The psychometric function: I. Fitting, sampling, and goodness of fit

Abstract

The psychometric function relates an observer's performance to an independent variable, usually some physical quantity of a stimulus in a psychophysical task. This paper, together with its companion paper (Wichmann & Hill, 2001), describes an integrated approach to (1) fitting psychometric functions, (2) assessing the goodness of fit, and (3) providing confidence intervals for the function's parameters and other estimatesderived from them, for the purposes of hypothesis testing. The present paper deals with the first two topics, describing a constrained maximum-likelihood method of parameter estimation and developing severalgoodness-of-fit tests. Using Monte Carlo simulations, we deal with two specific difficulties that arise when fitting functions to psychophysical data. First, we note that human observers are prone to stimulus-independent errors (or lapses). We show that failure to account for this can lead to serious biases in estimates of the psychometric function's parameters and illustrate how the problem may be overcome. Second, we note that psychophysical data sets are usually rather small by the standards required by most of the commonly applied statistical tests. We demonstrate the potential errors of applying traditional c 2 methods to psychophysical data and advocate use of Monte Carlo resampling techniques that do not rely on asymptotic theory. We have made available the software to implement our methods.

Can measures of sound localization acuity be related to the precision of absolute location estimates?

“…Second, the cats determined that the sound source location was consonant with where they were already looking. In the first case, if the cats simply ignored a certain proportion of trials [consistent with a 'lapse-rate' in human psychophysical experiments (Wichmann and Hill, 2001)] then the proportion of non-responses would be approximately equal across all trials. Our experimental data are largely contrary to this, however.…”

Effects of stimulus duration on audio-visual synchrony perception

“…In this method, the response proportions for all three response categories were fitted with a psychometric function of the form c ? (1 -c -k)F(a, b), with F being the cumulative Gaussian distribution with mean a and standard deviation b. Fitting was done using the MATLAB psignifit toolbox, which implements the maximum-likelihood method described by Wichmann and Hill (2001). Stimulus-independent lapses (e.g., pressing the wrong key or blinking during stimulus presentation) were fitted by the c (lower horizontal asymptote) and k (higher horizontal asymptote) parameters.…”

Perception of across-frequency asynchrony and the role of cochlear delays

“…The fits were obtained using a maximum-likelihood procedure with a binomial distribution (Wichmann and Hill, 2001;Dai and Micheyl, 2011). The maximum-likelihood fits are shown by the curves in Fig.…”

“…(A1)-(A5) was used to fit the proportions of synchronous judgments measured in each listener using a maximum-likelihood procedure. The number of "synchronous" responses corresponding to a given P s (d) was assumed to be distributed according to a binomial distribution with the parameter n (number of trials) set to 100, the number of trials per delay condition per listener in experiment 1 (Wichmann and Hill, 2001;Dai and Micheyl, 2011). The negative logarithm of the likelihood of the data given the model was minimized using MATLAB's (The MathWorks, Natick, MA) fminsearch function, which implements the Nelder-Mead simplex algorithm.…”