The Psychometric Function I

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 2000), 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 estimates derived from them, for the purposes of hypothesis-testing. The current paper deals with the first two topics, describing a constrained maximum-likelihood method of parameter estimation, and developing several goodness-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 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. F.A. Wichmann and N.J. Hill The Psychometric Function I Outline The performance of an observer on a psychophysical task is typically summarized by reporting one or more response thresholds—stimulus intensities required to produce a given level of performance—and by characterization of the rate at which performance improves with increasing stimulus intensity. These measures are derived from a psychometric function, which describes the dependence of an observer’s performance on some physical aspect of the stimulus: one example might be the relation between the contrast of a visual stimulus and the observer’s ability to detect it. Fitting psychometric functions is a variant of the more general problem of modelling data. Modelling data is a three–step process. First, a model is chosen and the parameters are adjusted to minimize the appropriate error–metric or loss function. Second, error estimates of the parameters are derived and, third, the goodness–of–fit between model and data is assessed. This paper is concerned with the first and the third of these steps, parameter estimation and goodness–of–fit assessment. Our companion paper (Wichmann & Hill, 2000) deals with the second step and illustrates how to derive reliable error estimates on the fitted parameters. Together the two papers provide an integrated approach to fitting psychometric functions, evaluating goodness–of–fit, and obtaining confidence intervals for parameters, thresholds and slopes, avoiding the known sources of potential error. This paper is divided into two major subsections, fitting psychometric functions, and goodness–of–fit. Each subsection itself is again subdivided into two main parts, first an introduction to the issue and second a set of simulations addressing the issue raised in the respective introduction. Notation We adhere mainly to the typographic conventions frequently encountered in statistical texts (Collett, 1991; Dobson, 1990; Efron & Tibshirani, 1993). Variables are denoted by