Abstract
Can accuracy and response bias in two-stimulus, two-response recognition or detection experiments be measured nonparametrically? Pollack and Norman (1964) answered this question affirmatively for sensitivity, Hodos (1970) for bias: Both proposed measures based on triangular areas in receiver-operating characteristic space. Their papers, and especially a paper by Grier (1971) that provided computing formulas for the measures, continue to be heavily cited in a wide range of content areas. In our sample of articles, most authors described triangle-based measures as making fewer assumptions than measures associated with detection theory. However, we show that statistics based on products or ratios of right triangle areas, including a recently proposed bias index and a not-yetproposed but apparently plausible sensitivity index, are consistent with a decision process based on logistic distributions. Even the Pollack and Norman measure, which is based on non-right triangles, is approximately logistic for low values of sensitivity. Simple geometric models for sensitivity and bias are not nonparametric, even if their implications are not acknowledged in the defining publications.
Article PDF
Similar content being viewed by others
References
Aaronson, D., &Watts, B. (1987). Extensions of Grier’s computational formulas forA′ andB′ to below-chance performance.Psychological Bulletin,102, 439–442.
Bradley, J. V. (1968).Distribution-free statistical tests. Englewood Cliffs, NJ: Prentice-Hall.
Donaldson, W. (1992). Measuring recognition memory.Journal of Experimental Psychology: General,121, 275–277.
Green, D. M. (1964). General prediction relating yes-no and forcedchoice results [Abstract].Journal of the Acoustical Society of America,36, 1042.
Green, D. M., &Swets, J. A. (1966).Signal detection theory and psychophysics. New York: Wiley.
Grier, J. B. (1971). Nonparametric indexes for sensitivity and bias: Computing formulas.Psychological Bulletin,75, 424–429.
Hodos, W. (1970). Nonparametric index of response bias for use in detection and recognition experiments.Psychological Bulletin,74, 351–354.
Luce, R. D. (1963). Detection and recognition. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.),Handbook of mathematical psychology (Vol. 1, pp. 103–189). New York: Wiley.
Macmillan, N. A., &Creelman, C. D. (1990). Response bias: Characteristics of detection theory, threshold theory, and “nonparametric” measures.Psychological Bulletin,107, 401–413.
Macmillan, N. A., &Creelman, C. D. (1991).Detection theory: A user’s guide. New York: Cambridge University Press.
Maloney, L. T., &Thomas, E. A. C. (1991). Distributional assumptions and observed conservatism in the theory of signal detectability.Journal of Mathematical Psychology,35, 443–470.
McNicol, D. (1972).A primer of signal detection theory. Sydney: Allen & Unwin.
Norman, D. A. (1964). A comparison of data obtained with different false-alarm rates.Psychological Review,71, 243–246.
Pollack, I., &Norman, D. A. (1964). A nonparametric analysis of recognition experiments.Psychonomic Science,1, 125–126.
Swets, J. A. (1986a). Form of empirical ROCs in discrimination and diagnostic tasks: Implications for theory and measurement of performance.Psychological Bulletin,99, 181–198.
Swets, J. A. (1986b). Indices of discrimination or diagnostic accuracy: Their ROCs and implied models.Psychological Bulletin,99, 100–117.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by National Science Foundation Grant DBS 92-12043 and a PSC-CUNY award to N.A.M. We thank R. Duncan Luce, Irwin Pollack, an anonymous reader, and Associate Editor Richard Schweickert for helpful comments during the review process.
Rights and permissions
About this article
Cite this article
Macmillan, N.A., Creelman, C.D. Triangles in ROC space: History and theory of “nonparametric” measures of sensitivity and response bias. Psychonomic Bulletin & Review 3, 164–170 (1996). https://doi.org/10.3758/BF03212415
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.3758/BF03212415