Thresholds from psychometric functions: Superiority of bootstrap to incremental and probit variance estimators.

Abstract

The bootstrap method provides a powerful, general procedure for estimating the variance of a parameter ofa function. The parametric version ofthe method was used to estimate the standard deviation of a threshold from a psychometric function and the standard deviation of its slope. Bootstrap standard deviations were compared with those obtained by a classical incremental method and by the asymptotic method of probit analysis. Twelve representative experimental conditions were tested in Monte Carlo studies, each of 1,000 data sets. All methods performed equally well with large data sets, but with small data sets the bootstrap was superior in both percentage bias and relative efficiency. There are many occasions in which it is desirable to measure the strength ofa stimulus in terms of its response in an organ­ ism. Typically, different levels ofa known treatment are applied to subjects and the effects ofthat treatment are recorded at each level. Thus, in psychophysics, one mightconstruct a psychomet­ ric ./imction, which describes the relationship between the level ofa stimulus and the probability ofa subject making a particu­ lar response at that level (Falmagne, 1982). In a biological or medical assay, one might determine a stimulus-responsecurve or dose-response curve, which relates the dosage of a drug or poison and the proportion of subjects that on average are af­ fected at that dosage (Finney, 1978). In practice, the potency ofa stimulus may need to be charac­ terized by a single number that corresponds to a particular criterion level of efficacy. For a psychometric function, this stimulus level is the threshold value of the stimulus, for that particular criterion. In a simple yes-no detection task, per­ centage of successes might be recorded at a number of testing levels and a theoretical function in the form, for example, ofa normal probability integral function fitted to those data. The situation is illustrated in Figure la. Threshold would be defined for a criterion performance level of 50%. For a two-alternative forced-choice task, where theoretical performance ranges from 50% to 100%, the criterion level could be 75%. For a dose-re­ sponse curve, the situation is similar. The criterion level ofeffi­ cacy would be the median (or mean) effective dose, symbolized by ED50, which on average produces a response in 50% ofsub­ jects. Similarly, ED75 is the dose that produces a response in 75% ofsubjects.