A Clarification of Some Current Multiplicative Confusion Models

Consider the class of experimental situations where there are n possible stimulus presentations and n possible responses and there is an expelimenter-defined one-one stimulus-response mapping. Then a confusion matrix is a table with the following properties: The stimuli are given row identification such that the cell entries are conditional frequencies that the column response is made to the row stimulus. More formally, {Cj;}represents a confusion matrix with positive entries in its cells such that :L:=l Cj; = 1, for 1 ~ i ~ n, where i identifies the stimulus and the j the response. Confusion matrices are useful in the study of psychological similarity in perception and cognition. Some examples are Miller and Nicely (1955), Luce (1963), Townsend (1971), Geyer and DeWald (1973) and examples with special emphasis on application of multiplicative confusion models are Falmagne (1972) and Wandmacher (1975). We will consider here this special type of confusion matrix (designated multiplicative confusion matrices). In these, Cufor i =1= j can be factored into an element that is a function of i and one that is a function of j. Since certain degenerate conditions arise when n ~ 2 (see. e.g., Falmagne, 1972), it will henceforth be assumed that n ~ 3. Most contemporary mathematical models of confusion matrices, including nonmultiplicative models, can be captured by the following definition of confusion models.