Confidence intervals for the parameters of psychometric functions LAURENCE

A Monte Carlo method for computing the bias and standard deviation of estimates of the parameters of a psychometric function such as the WeibulllQuick is described. The method, based on Efron's parametric bootstrap, can also be used to estimate confidence intervals for these parameters. The method's ability to predict bias, standard deviation, and confidence intervals is evaluated in two ways. First, its predictions are compared to the outcomes of Monte Carlo simulations ofpsychophysical experiments. Second, its predicted confidenceintervals were compared with the actual variability of human observers in a psychophysical task. Computer programs implementing the method are available from the author. The performance of an observer in a detection or discrimination task is typically summarized by fitting a psy-chometric function to the data. Examples of fitting methods include probit analysis (Finney, 1971) and maximum-likelihood fits using the Weibull/Quick psycho-These methods retain an estimate of threshold and a measure of variability (slope). Whatever fitting method is used, some measure of the variability of the estimated psychometric parameters is needed in order to compare the performances of an observer in two experimental situations, or to compare the performance of an observer to a theoretically motivated value. This article presents a Monte Carlo method for computing the bias, standard deviation, and confidence intervals for maximum-likelihood estimates of the location parameter rt and slope parameter f3 for the Weibull/ Section 2 contains a description and explanation of the method. Sections 3 and 4 report two evaluations of the method. The outcomes of both evaluations suggest that the parametric bootstrap provides useful estimates of bias, standard deviation, and confidence intervals for the location and slope parameters of the Weibull/Quick psychometric function. There are many different psychophysical procedures (staircase methods, method of constant stimuli, etc.) to select among in designing an experiment (see Levine & Shefner, 1981), and either forced-choice or yes-no tasks may be used. Furthermore, there are several fitting procedures , among which two, probit analysis and the maximum-likelihood fit to the Weibull/Quick psychomet-ric function, are most frequently employed. The paramet-ric bootstrap method can be adapted to each of the combinations of experimental design and data analysis that could arise. To ease the presentation, the method is first applied to maximum-likelihood fits for a two-parameter Weibull/Quick psychometric function using the method of constant stimuli and assuming a forced-choice task. The changes needed to use the method in other circumstances are then described. The programs …