Threshold estimation in two-alternative forced-choice (2AFC) tasks: The Spearman-Kärber method

Abstract

The Spearman-Kärber method can be used to estimate the threshold value or difference limen in two-alternative forced-choice tasks. This method yields a simple estimator for the difference limen and its standard error, so that both can be calculated with a pocket calculator. In contrast to previous estimators, the present approach does not require any assumptions about the shape of the true underlying psychometric function. The performance of this new nonparametric estimator is compared with the standard technique of probit analysis. The Spearman-Kärber method appears to be a valuable addition to the toolbox of psychophysical methods, because it is most accurate for estimating the mean (i.e., absolute and difference thresholds) and dispersion of the psychometric function, although it is not optimal for estimating percentile-based parameters of this function.

Estimating the difference limen in 2AFC tasks: Pitfalls and improved estimators

“…This might bias results and increase sampling variability of threshold estimates. In addition and on the contrary to what has been claimed previously (e.g., Ulrich & Miller, 2004), P(c) corresponds to F 2 (c) only under certain assumptions, as will become evident in the text that follows.…”

“…It is often assumed in psychophysical work that G(c) can be scaled as G(c) .5 .5 P(c), where P(c) represents a cumulative probability function (e.g., Harvey, 1986;Simpson, 1989;Ulrich & Miller, 2004;Wichmann & Hill, 2001). Interestingly, Equation 2 does not advocate such a scaling assumption.…”

Reminder and 2AFC tasks provide similar estimates of the difference limen: A reanalysis of data from Lapid, Ulrich, and Rammsayer (2008) and a discussion of Ulrich and Vorberg (2009)

“…As sion" (p. 1399). They noted that the response r is typically "comparison longer," and they also correctly acknowledged that a psychometric function "must satisfy three conditions: (1) F(s) 0 as s , (2) F(s) 1 as s , and (3) F(s) is monotonically increasing with s." However, a subsequent article by Ulrich and Miller (2004) was entirely based on the mistake of considering that the psychometric function for discrimination must have a lower asymptote at .5, a mistake perhaps arising from the true fact that the chance level in a 2AFC task is .5. In reality, the latter fact implies only that the psychometric function for discrimination is nonmonotonic and has a discontinuity at the PSE when regarded as expressing the probability of a correct response in the strict physical sense (i.e., the probability of reporting the stimulus whose physical magnitude is actually larger; see Ulrich & Vorberg, 2009).…”

A comparison of adaptive psychometric procedures based on the theory of optimal experiments and Bayesian techniques: Implications for cochlear implant testing

“…Each point on the search path is an updated estimate of the parameters and results from independent steps in each of the two search space dimensions. The vertical dashed line identifies the values of the slope parameter tested to minimize the error after initializing of the mean parameter, using the Spearman-Kärber method (Ulrich & Miller, 2004). The value of the slope parameter that minimizes the error is chosen as the new value of the slope parameter.…”

“…To implement the iterative search, the initial value for the threshold parameter m 0 is calculated using the Spearman-Kärber method (Ulrich & Miller, 2004). The algorithm then iterates through five steps.…”