Contrast discrimination: second responses reveal the relationship between the mean and variance of visual signals

Abstract

To explain the relationship between first- and second-response accuracies in a detection experiment, Swets, Tanner, and Birdsall [Swets, J., Tanner, W. P., Jr., & Birdsall, T. G. (1961). Decision processes in perception. Psychological Review, 68, 301-340] proposed that the variance of visual signals increased with their means. However, both a low threshold and intrinsic uncertainty produce similar relationships. I measured the relationship between first- and second-response accuracies for suprathreshold contrast discrimination, which is thought to be unaffected by sensory thresholds and intrinsic uncertainty. The results are consistent with a slowly increasing variance.

Fig. 1

Example stimulus. One Gabor has more contrast than the others. When those others have sufficient “pedestal” contrast for essentially perfect detection, neither intrinsic uncertainty nor a sensory threshold can contaminate an observer’s decision as to which of the four is most intense. For JAS and MJM, all of the black spots disappeared during each 0.18-s stimulus exposure. For the other observers, only the central fixation spot disappeared.

Fig. 2

Threshold-vs.-contrast functions for first responses in four observers. Error bars contain 95% confidence intervals.

Fig. 3

Two-response, four-alternative-forced-choice (2R4AFC), detection results for three observers. Each box represents a unique intensity. Error bars contain 95% confidence intervals. The solid black curves show simple signal-detection theory (SDT). Dashed lines show high-threshold theory. Dotted lines are mathematical and theoretical upper bounds for second-response accuracies. The green, blue and red curves show, respectively, the maximum-likelihood fits of SDT modified with increasing noise, intrinsic uncertainty and a low threshold.

Fig. 4

Cleaned-up 2R4AFC detection results (see text). Green curves show the best fit of SDT with increasing variance. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

2R4AFC contrast-discrimination results. Green curves show the best fit of SDT with increasing variance. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

How the sigma-to-mean ratio varies with signal intensity. Dashed lines indicate Swets et al.’s (1961) estimate, based on detection data.