Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
May, K. A.; Solomon, J. A. (2013)
Publisher: Public Library of Science (PLoS)
Journal: PLoS ONE
Languages: English
Types: Article
Subjects: Computational Biology, Applied Mathematics, Probability Theory, Research Article, BF, Signal Processing, Mathematics, Engineering and Technology, Computational Neuroscience, Statistical Distributions, Coding Mechanisms, Visual System, Physical Sciences, Psychology, Sensory Perception, Statistical Signal Processing, Biology and Life Sciences, Neuroscience, Psychometrics, Medicine, Psychophysics, Theoretical Biology, Q, R, Science, Sensory Systems
In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, Δx. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull "slope" parameter, β, can be approximated by [Formula: see text], where [Formula: see text] is the β of the Weibull function that fits best to the cumulative noise distribution, and [Formula: see text] depends on the transducer. We derive general expressions for [Formula: see text] and [Formula: see text], from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when [Formula: see text], [Formula: see text]. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4-0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull β reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of β for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 1. Foley JM, Legge GE (1981) Contrast detection and near-threshold discrimination in human vision. Vision Res 21: 1041-1053.
    • 2. Nachmias J (1981) On the psychometric function for contrast detection. Vision Res 21: 215-223.
    • 3. Mayer MJ, Tyler CW (1986) Invariance of the slope of the psychometric function with spatial summation. J Opt Soc Am A Opt Image Sci Vis 3: 1166- 1172.
    • 4. Meese T, Georgeson MA, Baker DH (2006) Binocular contrast vision at and above threshold. J Vis 6: 1224-1243.
    • 5. Wallis SA, Baker DH, Meese TS, Georgeson MA (2013) The slope of the psychometric function and non-stationarity of thresholds in spatiotemporal contrast vision. Vision Res 76: 1-10.
    • 6. Nachmias J, Sansbury RV (1974) Grating contrast: discrimination may be better than detection. Vision Res 14: 1039-1042.
    • 7. Wichmann FA, Hill NJ (2001) The psychometric function: I. Fitting, sampling, and goodness of fit. Percept Psychophys 63: 8/1293-1313.
    • 8. Weibull W (1951) A statistical distribution function of wide applicability. Journal of Applied Mechanics 18: 293-297.
    • 9. Quick RF (1974) A vector-magnitude model of contrast detection. Kybernetik 16: 65-67.
    • 10. Legge GE (1978) Sustained and transient mechanisms in human vision: Temporal and spatial properties. Vision Res 18: 69-81.
    • 11. Legge GE (1978) Space domain properties of a spatial frequency channel in human vision. Vision Res 18: 959-969.
    • 12. Watson AB (1979) Probability summation over time. Vision Res 19: 515-522.
    • 13. Robson JG, Graham N (1981) Probability summation and regional variation in contrast sensitivity across the visual field. Vision Res 21: 409-418.
    • 14. Tanner WP, Swets JA (1954) A decision-making theory of visual detection. Psychol Rev 61: 401-409.
    • 15. Swets JA, Tanner WP, Birdsall TG (1961) Decision processes in perception. Psychol Rev 68: 301-340.
    • 16. Laming D (2013) Probability summation-a critique. J Opt Soc Am A Opt Image Sci Vis 30: 300-315.
    • 17. Watson AB, Pelli DG (1983) QUEST: A Bayesian adaptive psychometric method. Percept Psychophys 33: 113-120.
    • 18. Neri P (2013) The statistical distribution of noisy transmission in human sensors. Journal of Neural Engineering 10: 016014.
    • 19. Pelli DG (1987) On the relation between summation and facilitation. Vision Res 27: 119-123.
    • 20. Tanner WP, Birdsall TG (1958) Definitions of d' and g as psychophysical measures. J Acoust Soc Am 30: 922-928.
    • 21. Pelli DG (1985) Uncertainty explains many aspects of visual contrast detection and discrimination. J Opt Soc Am A Opt Image Sci Vis 2: 1508-1532.
    • 22. O'Regan JK, Humbert R (1989) Estimating psychometric functions in forcedchoice situations: Significant biases found in threshold and slope estimations when small samples are used. Percept Psychophys 46: 434-442.
    • 23. Leek MR, Hanna TE, Marshall L (1992) Estimation of psychometric functions from adaptive tracking procedures. Percept Psychophys 51: 247-256.
    • 24. King-Smith PE, Rose D (1997) Principles of an adaptive method for measuring the slope of the psychometric function. Vision Res 37: 1595-1604.
    • 25. Treutwein B, Strasburger H (1999) Fitting the psychometric function. Percept Psychophys 61: 87-106.
    • 26. Kontsevich LL, Tyler CW (1999) Bayesian adaptive estimation of psychometric slope and threshold. Vision Res 39: 2729-2737.
    • 27. Kaernbach C (2001) Slope bias of psychometric functions derived from adaptive data. Percept Psychophys 63: 1389-1398.
    • 28. McIlhagga WH, May KA (2012) Optimal edge filters explain human blur detection. J Vis 12: 10/9/1-13.
    • 29. Strasburger H (2001) Converting between measures of slope of the psychometric function. Percept Psychophys 63: 1348-1355.
    • 30. Legge GE, Foley JM (1980) Contrast masking in human vision. Journal of the Optical Society of America 70: 1458-1471.
    • 31. Campbell FW, Kulikowski JJ (1966) Orientation selectivity of the human visual system. Journal of Physiology 187: 437-445.
    • 32. Swift DJ, Smith RA (1983) Spatial frequency masking and Weber's law. Vision Res 23: 495-505.
    • 33. Laming D (1989) Experimental evidence for Fechner's and Stevens's laws. Behav Brain Sci 12: 277-281.
    • 34. Bird CM, Henning GB, Wichmann FA (2002) Contrast discrimination with sinusoidal gratings of different spatial frequency. J Opt Soc Am A Opt Image Sci Vis 19: 1267-1273.
    • 35. Solomon JA (2009) The history of dipper functions. Attention, Perception, & Psychophysics 71: 435-443.
    • 36. Klein S (2001) Measuring, estimating, and understanding the psychometric function: A commentary. Percept Psychophys 63: 1421-1455.
    • 37. Stromeyer CF, Klein S (1974) Spatial frequency channels in human vision as asymmetric (edge) mechanisms. Vision Res 14: 1409-1420.
    • 38. Yu C, Klein SA, Levi DM (2003) Cross- and Iso- oriented surrounds modulate the contrast response function: The effect of surround contrast. J Vis 3: 527-540.
    • 39. Wilson HR (1980) A transducer function for threshold and suprathreshold human vision. Biol Cybern 38: 171-178.
    • 40. Henning GB, Wichmann FA (2007) Some observations on the pedestal effect. J Vis 7: 1/3/1-15.
    • 41. Hasselblatt B, Katok A (2003) A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge: Cambridge University Press.
    • 42. Legge GE (1981) A power law for contrast discrimination. Vision Res 21: 457- 467.
    • 43. Kontsevich LL, Chen C-C, Tyler CW (2002) Separating the effects of response nonlinearity and internal noise psychophysically. Vision Res 42: 1771-1784.
    • 44. Georgeson MA, Meese TS (2006) Fixed or variable noise in contrast discrimination? The jury's still out... Vision Res 46: 4294-4303.
    • 45. Solomon JA (2007) Intrinsic uncertainty explains second responses. Spat Vis 20: 45-60.
    • 46. Fechner GT (1912) In: Rand B, editor. The classical psychologists: Selections illustrating psychology from Anaxagoras to Wundt. Boston: Houghton Mifflin. (Original work published 1860).
    • 47. Henning GB, Bird CM, Wichmann FA (2002) Contrast discrimination with pulse trains in pink noise. J Opt Soc Am A Opt Image Sci Vis 19: 1259-1266.
  • Inferred research data

    The results below are discovered through our pilot algorithms. Let us know how we are doing!

    Title Trust
  • Discovered through pilot similarity algorithms. Send us your feedback.

    Title Year Similarity

    Four Theorems on the Psychometric Function


Share - Bookmark

Funded by projects

Cite this article