Technical clarification to Silbert and Thomas (2013): "decisional separability, model identification, and statistical inference in the general recognition theory framework".

We offer a minor technical correction to the published proof of part (ii) of the main theorem in Silbert and Thomas (Psychonomic Bulletin & Review, 20, 1-20, 2013) that somewhat limits the scope of the equivalence observed in that work. Specifically, in order for a mean shift integrality with decisional separability to be mimicked by a perceptually separable but nondecisionally separable configuration, one needs to assume stimulus invariance. This holds when all of the covariance matrices in the stimulus configuration are equal to each other. We note that part (i) of the theorem is unaffected by this modification; an empirical finding of perceptual separability and the failure of decisional separability can be mimicked by a perceptually nonseparable, decisionally separable configuration without restricting the covariance matrices to be equal. We also note that stimulus invariance is often assumed in simple designs (e.g., Macmillan & Ornstein in Journal of the Acoustical Society of America, 97, 1261-1285, 1998), due to the implausibility of different perceptual correlations being present within stimuli perched very closely in perceptual space.