Singularities explained: Response to Klein

Klein [Klein, A. S. (2006). Separating transducer nonlinearities and multiplicative noise in contrast discrimination. Vision Research, 46, 4279-4293] questions the existence of intrinsic singularities in two-alternative force-choice (2AFC) Signal Detection Theory (SDT) models, suggesting that the singularities found in Katkov et al. [Katkov, M., Tsodyks, M., & Sagi, D. (2006a). Singularities in the inverse modeling of 2AFC contrast discrimination data. Vision Research, 46, 259-266; Katkov, M., Tsodyks, M., & Sagi, D. (2006b). Analysis of two-alternative force-choice Signal Detection Theory model. Journal of Mathematical Psychology, 50, 411-420] are due to discarding higher order terms in the Taylor expansion of d′ and/or limited to steep psychometric functions. Here we provide some simple intuitive examples that illustrate the results described in Katkov et al. (2006a, 2006b). We show, for the constant noise model, that singularities exist when exact values of d′ are computed and that the singularities are not limited to steep psychometric functions. In these cases the disambiguation of the different models requires millions of trials.