Psychometric Function from Rayleigh-Rayleigh ROC Curves

Abstract

While a visiting professor at the University of Texas, Egan (1967) showed us how power-law ROC curves can be derived from overlapping Rayleigh distributions. Such distributions suggest immediately that detection performance for narrow-band noise signals should differ markedly from that for sinusoidal signals. Accordingly, the psychometric function for Rayleigh signals was derived. The equations governing the probability of being correct in a 2AFC experiment for sinusoidal and noise signals are: P(c) = 1 − 12(exp− σs2/2σn2), which is another way of expressing Marill's famous equation, and P(c) = (σn2+σs2)/(2σn2+σs2), respectively; σn and σs are the rms voltages of the effective masking noise and the signal. For these signals, both rating-scale and 2AFC experiments, appropriate to the tenets of detection theory, show that the shapes of human psychometric functions are predicted in detail by the mathematical models. [Work supported by National Institute of Mental Health, Naval Ship Systems Command, and National Aeronautics and Space Administration.]

Nonparametric Relationships between Single-Interval and Two-Interval Forced-Choice Tasks in the Theory of Signal Detectability

“…Experiment II provided some evidence that the nonparametric relationships hold for a continuous random variable, but the results were not as strong. These results are more robust than the results of previous experimental research (Emmerich, 1968a;Green & Moses, 1966;Whitmore et al, 1968), however, because error due to observer inconsistency was reduced using GOC analysis.…”

“…In Experiment II, observers were asked to detect narrow-band short-duration Gaussian noise with a bandwidthduration product of unity, masked by wide-band Gaussian noise. Three ideal observers have been suggested for this signal-known-statistically observer: the energy detector (Green & McGill, 1970), the envelope detector (Drga, 1988;Whitmore et al, 1968), and the full-linear detector (Lapsley Miller, 1999). The theoretical evidence distributions are unknown but are suggested to be q 2 (energy) or Rayleigh (envelope) for the SIFC task, although Lapsley Miller (1999) found evidence that neither distribution was appropriate.…”

“…On average, across all observers and conditions, A SI approximately equaled P(C) 2I , but the difference between the two measures for individual observers ranged from −0.111 to 0.062, where the difference could range between −0.5 and 0.5 (assuming A SI and P(C) 2I are both \ 0.5). Whitmore, Williams, and Ermey (1968), in a study on envelope detection, considered both tone-in-noise and noise-in-noise detection using a rating-scale SIFC task and a binary-decision 2IFC task. Because different signal-to-noise ratios were used in the two conditions, A SI and P(C) 2I could not be directly compared.…”

Amplitude discrimination of sinusoids and narrow-band noise with Rayleigh properties

“…However, as gratifying as this may be, it leaves unanswered the origin of the difference between Ronken's (1969) data and those of Whitmore et al (1968). In the present study, both the shapes and the locations of the psychometric functions fitted to each type of stimulus were quite similar (Experi- …”

“…This is much more than the 0.7 dB reported by Ronken (1969), but within the range found by Whitmore et al (1968). Some of the difference between our result and Ronken's is attributable to the difference in the dimension he used, 10 10g(a.…”