Abstract
The psychometric function relating stimulus intensity to response probability generally presents itself as a monotonically increasing sigmoid profile. Two summary parameters of the function are particularly important as measures of perceptual performance: the threshold parameter, which defines the location of the function over the stimulus axis (abscissa), and the slope parameter, which defines the (local) rate at which response probability increases with increasing stimulus intensity. In practice, the psychometric function may be modeled by a variety of mathematical structures, and the resulting algebraic expression describing the slope parameter may vary considerably between different functions fitted to the same experimental data. This variation often restricts comparisons between studies that select different functions and compromises the general interpretation of slope values. This article reviews the general characteristics of psychometric function models, discusses three strategies for resolving the issue of slope value differences, and presents mathematical expressions for implementing each strategy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acklam, P. J. (2004).An algorithm for computing the inverse normal cumulative distribution function. Retrieved November 12, 2004, from http://home.online.no/~pjacklam/notes/invnorm.
Berkson, J. (1951). Why I prefer logits to probits.Biometrics,7, 327–339.
Blackwell, H. R. (1953). Studies of the form of visual threshold data.Journal of the Optical Society of America,43, 456–463.
Chauhan, B. C., Tompkins, J. D., LeBlanc, R. P., &McCormick, T. A. (1993). Characteristics of frequency-of-seeing curves in normal subjects, patients with suspected glaucoma, and patients with glaucoma.Investigative Ophthalmology & Visual Science,34, 3534–3540.
Evans, M., Hastings, N., &Peacock, B. (1993).Statistical distributions (2nd ed.). New York: Wiley.
García-Pérez, M. A. (1998). Forced-choice staircases with fixed step sizes: Asymptotic and small sample properties.Vision Research,38, 1861–1881.
García-Pérez, M. A. (2001). Yes-no staircases with fixed step sizes: Psychometric properties and optimal setup.Optometry & Vision Science,78, 56–64.
Green, D. M., Richards, V. M., &Forrest, T. G. (1989). Stimulus step size and heterogeneous stimulus conditions in adaptive psychophysics.Journal of the Acoustical Society of America,86, 629–636.
Green, D. M., &Swets, J. A. (1966).Signal detection theory and psychophysics. Los Altos, CA: Peninsula.
Harvey, L. O., Jr. (1986). Efficient estimation of sensory thresholds.Behavior Research Methods, Instruments, & Computers,18, 623–632.
Kaernbach, C. (2001). Slope bias of psychometric functions derived from adaptive data.Perception & Psychophysics,63, 1389–1398.
King-Smith, P. E., &Rose, D. (1997). Principles of an adaptive method for measuring the slope of the psychometric function.Vision Research,37, 1595–1604.
Klein, S. A. (2001). Measuring, estimating, and understanding the psychometric function: A commentary.Perception & Psychophysics,63, 1421–1455.
Kontsevich, L. L., &Tyler, C. W. (1999). Bayesian adaptive estimation of psychometric slope and threshold.Vision Research,39, 2729–2737.
Laming, D. (1986).Sensory analysis. London: Academic Press.
Lipetz, L. E. (1969). The transfer functions of sensory intensity in the nervous system.Vision Research,9, 1205–1234.
Macmillan, N. A., &Creelman, C. D. (1991).Detection theory: A user’s guide. Cambridge: Cambridge University Press.
Nachmias, J. (1981). On the psychometric function for contrast detection.Vision Research,21, 215–223.
Naka, K. I., &Rushton, W. A. H. (1966). S-potentials from colour units in the retina of fish (Cyprinidae).Journal of Physiology,185, 536–555.
Patterson, V. H., Foster, D. H., &Heron, J. R. (1980). Variability of visual threshold in multiple sclerosis: Effect of background luminance on frequency of seeing.Brain,103, 139–147.
Pelli, D. G. (1987). On the relation between summation and facilitation.Vision Research,27, 119–123.
Quick, R. F. (1974). A vector magnitude model for contrast detection.Kybernetik,16, 65–67.
Snoeren, P. R., &Puts, M. J. H. (1997). Multiple parameter estimation in an adaptive psychometric method: MUEST, an extension of the QUEST method.Journal of Mathematical Psychology,41, 431–439.
Strasburger, H. (2001a). Converting between measures of slope of the psychometric function.Perception & Psychophysics,63, 1348–1355.
Strasburger, H. (2001b). Invariance of the psychometric function for character recognition across the visual field.Perception & Psychophysics,63, 1356–1376.
Treutwein, B. (1995). Adaptive psychophysical procedures.Vision Research,35, 2503–2522.
Treutwein, B., &Strasburger, H. (1999). Fitting the psychometric function.Perception & Psychophysics,61, 87–106.
Tyler, C. W. (1997). Why we need to pay attention to psychometric function slopes.OSA Technical Digest Series,1, 29–32.
Watson, A. B. (1979). Probability summation over time.Vision Research,19, 515–522.
Watson, A. B., &Pelli, D. G. (1983). QUEST: A Bayesian adaptive psychometric method.Perception & Psychophysics,33, 113–120.
Weibull, W. (1951). A statistical distribution function of wide applicability.Journal of Applied Mechanics,18, 292–297.
Wichmann, F. A., &Hill, N. J. (2001). The psychometric function: I. Fitting, sampling, and goodness of fit.Perception & Psychophysics,63, 1293–1313.
Wilson, H. R. (1980). A transducer function for threshold and suprathreshold human vision.Biological Cybernetics,38, 171–178.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gilchrist, J.M., Jerwood, D. & Ismaiel, H.S. Comparing and unifying slope estimates across psychometric function models. Perception & Psychophysics 67, 1289–1303 (2005). https://doi.org/10.3758/BF03193560
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03193560