r of overlapping, bell-shaped, probability (density) functions arrayed along a unidimensional variable, which is the weight of evidence derived from observation. In statistical theory, the probabilities are conditional on the null hypothesis or the alternative hypothesis, while in electronic detection theory they are observational probabilities conditional on noise-alone or signal-plus-noise. The greater the positive weight of evidence, the more likely a signal, or signiﬁcant experimental eﬀect, is present. A cutpoint must be set along the variable to identify those observed values above the cutpoint that would lead to rejection of the null hypothesis or acceptance of the presence of the designated signal. Errors are of commission or omission: type I and type II errors in statistics, and false alarms and misses in detection. Correct outcomes in detection are hits and correct rejections. Various decision rules specify optimal cutpoints for diﬀerent decision goals. The weight of evidence is unidimensional because just two decision alternatives are considered. Optimal weights of evidence are monotone increasing with the likelihood ratio, the ratio of the two overlapping bell-shaped probability functions. It is this ratio, and not the shapes of the functions, that is important.
3. Modern Signal Detection Theory
In the early 1950s, Peterson and Birdsall, then graduate students in electrical engineering and mathematics, respectively, at the University of Michigan, developed the general mathematical theory of signal detection that is still current. In the process they devised the ROC—a graphical technique permitting measurement of two independent aspects of detection performance: (a) the location of the observer’s cutpoint or decision criterion and (b) the observer’s sensitivity, or ability to discriminate between signal-plus-noise and noisealone, irrespective of any chosen cutpoint. The two measures characterize performance better than the single one often used, namely, the signal-to-noise ratio (SNR) expressed in energy 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 