gy terms that is necessary to provide a 50 percent hit (correct signal acceptance) probability at a ﬁxed decision cutpoint, say, one yielding a conditional false-alarm probability of 0.05. A curve showing conditional hit probability versus SNR is familiar in detection theory and is equivalent to the power function of statistical theory. These curves can be derived as a special case of the ROCs for the SNR values of interest.
4. The ROC
The ROC is a plot of the conditional hit (true–positive) proportion against the conditional false-alarm (false– positive) proportion for all possible locations of the decision cutpoint. Peterson’s and Birdsall’s SDT speciﬁes a form of ROC that begins at the lower left corner (where both proportions are zero) and rises with smoothly decreasing slope to the upper right corner (where both proportions are 1.0)—as the decision cutpoint varies from very strict (at the right end of the decision variable) to very lenient (at the left end). This is a ‘proper’ ROC, and speciﬁcally not a ‘singular’ ROC, or one intersecting axes at points between 0 and 1.0. The locus of the curve, or just the proportion of area beneath it, gives a measure of discrimination capacity. An index of a point along the curve, for example, the slope of the curve at the point, gives a measure of the decision cutpoint that yielded that point (the particular slope equals the criterial value of likelihood ratio). SDT speciﬁes the discrimination capacity attainable by an ‘ideal observer’ for any SNR for various practical combinations of signal and noise parameters (e.g., signal speciﬁed exactly and signal speciﬁed statistically) and hence the human observer’s eﬃciency can be calculated under various signal conditions. SDT also speciﬁes the optimal cutoﬀ as a function of the signal’s prior probability and the beneﬁts and costs of the decision outcomes.
5. Psychology’s Need For SDT And The ROC
Wilson P. Tanner, Jr., and John A. Swets, graduate students in psychology at Michigan in the early 1950s, became aware of Peterson’s and Birdsall’s work on the same campus as it began. Tanner became acquainted through his interest in mathematical and electronic concepts as models for psychological and neural processes and Swets was attracted because of his interest in quantifying the ‘instruction stimulus’ in psychophysical tasks, in order to control the observer’s response criterion. As they pursued studies in sensory psychology, these psychologists had become dissatisﬁed with psychophysical and psychological theories based on overlapping bell-shaped functions that assumed ﬁxed decision cutpoints and incorporated measures only of discrimination capacity, which were possibly confounded by attitudinal or decision processes. Prominent among such theories were those of Gustav Theodor Fechner (1860), Louis Leon Thurstone (1927), and H. Richard Blackwell (1963); see Swets (1996, Chap. 1).
Fechner and Thurstone conceived of a symmetrical cutpoint, where the two distributions cross, because they presented two stimuli on each trial with no particular valence; that is, both stimuli were ‘signals’ (lifted weights, samples of handwriting). Fechner also studied detection of single signals, but did not compare them to a noise-alone alternative and his theory was essentially one of variable representations of a signal compared to an invariant cutpoint. This cutpoint was viewed as a physiologically determined sensory threshold (akin to all-or-nothing nerve ﬁring) and hence the values of the sensory variable beneath it were thought to be indistinguishable from one another.
Blackwell’s task and model were explicitly of signals in noise, but the observer was thought to have a ﬁxed cutpoint near the top of the noise function, for which the false-alarm probability was negligible, and, again, the values below that cutpoint were deemed indistinguishable. For both threshold theorists, the single measure of performance was the signal strength required to yield 50 percent correct positive responses, like the eﬀective SNR of electronic detection theory, and it was take