The law of categorical judgment (Corrected) and the interpretation of changes in psychophysical performance

Signal detection theory (SDT) makes the frequently challenged assumption that decision criteria have no variance. An extended model, the Law of Categorical Judgment, relaxes this assumption. The long accepted equation for the law, however, is flawed: It can generate negative probabilities. The correct equation, the Law of Categorical Judgment (Corrected), is derived; the SDT rating model is a special case. An example shows how to invert the Law of Categorical Judgment (Corrected) numerically, thereby extracting estimates of signal and criterion density parameters and their confidence limits from rating data. The SDT rating model predicts linear Zeta-transformed operating characteristics (ZetaROCs), whereas the new equation can produce nonlinear ZetaROCs. For single-criterion experiments (e.g., yes/no, two-alternative forced choice), however, the corrected law yields identical d' values and linear ZetaROCs whether criterion variance is nonzero or zero. Performance differences observed in such experiments can always be attributed equally well to altered perceptual sensitivity or to modified criterion variance. The Law of Categorical Judgment (Corrected) offers to resolve this ambiguity through rating experiments.