Assessing sensitivity in a multidimensional space: Some problems and a definition of a general d′

Abstract

This article provides a formal definition for a sensivity measure, d′g, between two multivariate stimuli. In recent attempts to assess perceptual representations using qualitative tests on response probabilities, the concept of a d′ between two multidimensional stimuli has played a central role. For example, Kadlec and Townsend (1992a, 1992b) proposed several tests based on multidimensional signal detection theory that allow conclusions concerning the perceptual and/or decisional interactions of stimulus dimensions. One proposition, referred to as the diagonal d′ test, relies on specific stimulus subsets of a feature-complete factorial identification task to infer perceptual separability. Also, Ashby and Townsend (1986), in a similar manner, attempted to relate perceptual independence to dimensional orthogonality in Tanner's (1956) model, which also involves d′ between two multivariate signals. An analysis of the proposed d′g reveals shortcomings in the diagonal d′ test and also demonstrates that the assumptions behind equating perceptual independence to dimensional orthogonality are too weak. This d′g can be related to a common measure of statistical distance, Mahalanobis distance, in the special case of equal covariance matrices.