Further considerations of a general d′ in multidimensional space

Thomas (Psychonom. Bull. Rev. 6 (1999) 224) proposed a generalization of d′,d_g′ for multidimensional distributions and demonstrated that it is not equivalent to Euclidean distance as had been assumed in some previous studies. In this note, it is further shown not to be a metric in the general sense as it fails the triangle inequality. Also, a rigorous proof is offered of the claim (found in Thomas, Psychonom. Bull. Rev. 6 (1999) 224) that in order for the definition of d_g′ to correspond to the quantity that is estimated from data in the traditional way (using hits and false alarms) one must assume a distance classifier (proved in the case of equal covariance matrices). If one assumes optimal responding instead, then the estimated d′ corresponds to (the square root of) Mahalanobis distance. This latter observation clears up an apparent paradox between the fact that d_g′ is not a metric and Ashby and Perrin's (Psychol. Rev. 95 (1988) 124) statement relating the weighted Euclidean scaling model and a signal detection model of similarity that would yield a distance classifier.