1 Introduction Tanner and Swets (1954) Swets, Tanner, and Birdsall’s (1961) 1 1.1 Signal-detection theory Green & Swets, 1966 X (1) f X ( x ) = f ( x ; μ X , σ X ) = 1 2 π σ X exp - ( x - μ X ) 2 2 σ X 2 , x σ X σ X 1.2 Increasing variance Swets et al.’s (1961) (2) σ X = r μ X + 1 ,  μ X ⩾ 0 . 2 r σ X μ X r 1.3 Intrinsic uncertainty Solomon, 2007 Swets et al.’s (1961) M Pelli, 1985 1.4 Low-threshold theory Swets et al. (1961) Solomon, 2007 1.5 This study Pelli, 1985; Tanner, 1961 Swets et al.’s (1961) 2 Methods There were five observers: the author (JAS), another psychophysicist who understood the purposes of the experiment (MJM), two experienced psychophysical observers who were naïve to the purposes of this experiment (FV and MT) and one further observer who had no previous laboratory experience (NN). As described below, NN produced a very high proportion of “finger errors.” This suggested to us a general unreliability, and no further analyses were performed on his data. Watson & Solomon, 1997a http://vision.arc.nasa.gov/mathematica/psychophysica.html 2 2 λ σ Fig. 1 On each trial, three stimuli appeared with a pedestal contrast, which varied between blocks of 90 trials each. The contrast of the fourth stimulus was somewhat greater. After each 0.18-s stimulus exposure, observers gave two responses. The first response indicated which of the four positions the observer thought most likely to have contained the high-contrast target. The second responses from JAS and MJM indicated their second choices for the target position. Following their second responses, JAS and MJM received auditory feedback indicating which—if either—of their responses was correct. The naïve observers were not told that three of the four stimuli would have the same contrast. They were instructed merely to indicate their choices for the positions containing the two highest contrasts, in order. This encouraged them to fully consider their second responses, even when they felt confident about their first. The naïve observers received no feedback. Swets et al.’s (1961) Nachmias & Sainsbury, 1974 Fig. 2 Table 1 m x m x Watson & Pelli, 1983 c t c t c t c t c t 3 Results 3.1 Psychometric functions 3 (3) Ψ 1 ( c ) = 0.25 + ( 0.75 - δ ) ∫ - ∞ c f ( u ; c t , σ ) d u . c Ψ 1 f (1) c t σ δ δ 3.2 Threshold-vs.-contrast functions (first response only) Fig. 2 3.3 Second-vs.-first-response-accuracy functions: Detection Fig. 3 Swets et al.’s (1961) A Fig. 3 Swets et al. (1961) Ψ 1 Ψ 2 Ψ 2 Ψ 1 Ψ 2 Ψ 1 P 2 P 1 probability Fig. 3 B 3.4 Modeling finger errors ψ 1 ψ 1 δ δ ψ 1 δ ψ 1 ψ 1 ψ 1 ε ψ 1 ψ 1 ψ 1 (4) Ψ 2 ′ = ε ψ 1 + ( 1 - ε ) ( 2 + ψ 1 ) / 9 . δ ψ 2 δ εψ 1 ε ψ 1 ψ 2 ε ψ 1 (4) ε 3.5 Maximum-likelihood fits ε Figs. 3–6 Fig. 3 r (2) Swets et al., 1961 Solomon, 2007 C Nachmias (1972) Fig. 3 Table 2 3 Mood, Graybill, & Boes, 1974 Wichmann & Hill, 2001 Fig. 3 Swets et al. (1961) Swets et al.’s (1961) 3.6 r Kontsevich, Chen, & Tyler, 2002 Legge & Foley’s, 1980 Solomon, 2007 Swets et al., 1961 Katkov, Tsodyks, & Sagi, 2006a; Katkov, Tsodyks, & Sagi, 2006b Klein, 2006 r P 2 P 1 D Fig. 4 3.7 Binning accuracy Fig. 4 Fig. 3 δ (3) A 3.8 Second-vs.-first-response-accuracy functions: Suprathreshold discrimination Fig. 5 A Fig. 4 D r P 2 P 1 Fig. 5 r r Fig. 6 Swets et al. (1961) r r Swets et al.’s (1961) Kincaid and Hamilton’s (1959) Green & Swets, 1966 Blackwell, 1963 Eijkman and Vendrik (1963) r r 4 Discussion The most equitable summary of these results is that they are consistent with a performance-limiting source of noise, which increases slightly with suprathreshold contrast. Kontsevich et al., 2002 Solomon, 2007 4.1 Fitting contrast discrimination (5) σ = r μ q + σ 0 ,  0 < q < 1 . σ 0 σ 0 Kontsevich et al., 2002 Solomon (2007) r Foley (1994) Solomon, 2007 Swets et al., 1961 Foley’s (1994) Solomon, 2007 Table 3 M M q (5) M r M q r M Foley, 1994 4.2 Sensory thresholds for contrast discrimination Swets et al., 1961 Swets, 1961 Fig. 5 Swets et al.’s (1961) 4.3 Other models Foley (1994) Foley and Boynton (1994) Teo and Heeger (1994) Yu, Klein, and Levi (2004) Watson & Solomon, 1997b Henning and Wichmann (2007) Fig. 2 Blackwell (1998) Appendix A Tables A1–A3 Appendix B Ψ NP N a priori Ψ Ψ p G NP N p p p min Ψ p max Ψ p (6) h ( p ) = ∫ p min p g ( NP ; N , v ) ∫ p min p max g ( NP ; N , u ) d u d v , g G Ψ h −1 h −1 Appendix C Solomon (2007) Ψ ψ in the absence of finger errors ψ Ψ 3.4 (7) ψ 1 = ∫ - ∞ ∞ F N ( x ) 3 f S ( x ) d x (8) ψ 2 = 3 ∫ - ∞ ∞ F N ( x ) 2 f S ( x ) 1 - F N ( x ) d x 1 - ∫ - ∞ ∞ F N ( x ) 3 f S ( x ) d x . F N x f S x (9) f S ( x ) = F S ′ ( x ) . X N X S (10) F X ( x ) = [ F ( x ; 0 , 1 ) ] M - 1 F ( x ; μ X , r μ X q + 1 ) x ⩾ c [ F ( c ; 0 , 1 ) ] M - 1 F ( c ; μ X , r μ X q + 1 ) 0 ⩽ x < c , 0 otherwise M F ( x ; μ , σ ) = ∫ - ∞ x f ( u ; μ , σ ) d u (1) μ X r μ X q + 1 (5) c c M (11a) μ X = at X p , t X X N X S a p p Foley’s (1994) (11b) μ X = at p Z + t X q . a Z p q q (11b) (5) (10) (11b) q Appendix D μ N μ S μ S Ψ 1 r (1) and (2) r δ μ S r δ μ S Ψ 2 r r r Ψ 1 r Ψ 2 δ δ r Ψ 2 Ψ 1 r r r P 2 P 1 Table D1