Poisson signal-detection theory: Link between threshold models and the Gaussian assumption

The Gaussian model of signal detection cannot fit asymmetric data as long as the variances of the distributions are kept equal. It is therefore common practice to assume unequal variances in order to fit these data. But this assumption leads to the well-known crossover problem. The present paper provides new arguments for the abandonment of the Gaussian model with unequal variances. In its stead, this paper reevaluates multiple-parallel-threshold models. In particular, the Poisson model turns out to be very useful: it can handle data with any degree of asymmetry, giving a reasonable interpretation of the two parameters of the receiver-operating characteristic. The three-state-threshold model (Krantz, 1969) is given a new interpretation in light of the Poisson model. The slope of Poisson double-probability plots turns out to be much closer to unity than is predicted by the Gaussian approximation.