S-multiplicativity of a stochastic matrix and applications to visual identification

A stochastic matrix is called S-multiplicative if the conditional probabilities in a specific subset S of its cells can be factorized into the product of two functions depending on the rows and columns, respectively. The concept of S-multiplicativity is related to Goodman's concept of quasi-independence and generalizes Falmagne's concept of a multiplicative confusion matrix, where S consists of the cells lying outside the main diagonal ($\text{D}$-multiplicativity). In the present study, the notion of S-multiplicativity is applied to the analysis of visual identification performance. The confusion matrices obtained from three identification experiments are tested for predictions of S-multiplicativity derived from the multicomponent theory of perception. If the stimuli do not overlap, i.e., different stimuli have no features in common, then the theory predicts a $\text{D}$-multiplicative confusion matrix. Alternatively, in the case of overlapping stimuli the theory predicts S-multiplicativity with respect to a more severely restricted subset S. These predictions are confirmed by the data.