Psychometric functions for pure-tone frequency discrimination

Abstract

The form of the psychometric function (PF) for auditory frequency discrimination is of theoretical interest and practical importance. In this study, PFs for pure-tone frequency discrimination were measured for several standard frequencies (200-8000 Hz) and levels [35-85 dB sound pressure level (SPL)] in normal-hearing listeners. The proportion-correct data were fitted using a cumulative-Gaussian function of the sensitivity index, d', computed as a power transformation of the frequency difference, Δf. The exponent of the power function corresponded to the slope of the PF on log(d')-log(Δf) coordinates. The influence of attentional lapses on PF-slope estimates was investigated. When attentional lapses were not taken into account, the estimated PF slopes on log(d')-log(Δf) coordinates were found to be significantly lower than 1, suggesting a nonlinear relationship between d' and Δf. However, when lapse rate was included as a free parameter in the fits, PF slopes were found not to differ significantly from 1, consistent with a linear relationship between d' and Δf. This was the case across the wide ranges of frequencies and levels tested in this study. Therefore, spectral and temporal models of frequency discrimination must account for a linear relationship between d' and Δf across a wide range of frequencies and levels.

Figure 1

Example PFs constructed using data from adaptive tracks. The filled circles show proportions of correct responses estimated for each value of the tracking variable (Δf) visited by the adaptive track. The solid lines through the data points show maximum-likelihood fits (see text for details). The estimates of the PF parameters, threshold (α), slope (β), and lapse rate (λ), are indicated within each plot. The offset of the asymptote in PC of the PF from PC=1 is λ∕2.

Figure 2

Maximum-likelihood estimates of the slopes (β) of PFs. Panels L1– L5: data from individual listeners. Lower-right panel: mean across listeners for each sound level. Lower-left panel: grand mean obtained by averaging data across listeners and sound levels. The horizontal dashed lines indicate a slope of 1. Each error bar in the mean and grand-mean panels shows ± 1 standard error of the mean.

Figure 3

Thresholds (α): Panels L1–L5: data from individual listeners. Lower-right panel: mean across listeners for each sound level. Lower-left panel: grand mean obtained by averaging data across listeners and sound levels. The horizontal dashed lines indicate a slope of 1. Each error bar in the mean and grand-mean panels shows ± 1 standard error of the mean.

Figure 4

(Color online) Individual and overall distributions of the estimates of the parameter for the lapse rate (λ). Note that, for a given lapse rate of λ, the psychometric function has an upper asymptote of 1−λ∕2. Because λ∕2 is the offset of the asymptote PC of the psychometric function from a PC=1, it is directly visible on the fitted PF. For this reason, we use λ∕2 rather than λ in the abscissa.