Confusion Matrices and the Constant-Ratio Rule

Abstract

A confusion matrix may be defined as a table of conditional probabilities, p(ak|Ai), showing the proportion of instances in which the listener responded ak when the stimulus item Ai was transmitted. The confusion matrix is termed closed when the total set of messages is known to the listener and all his responses are drawn from this set of messages. If a closed matrix is obtained for some particular set of messages, and, under essentially the same physical conditions, a second closed matrix is obtained for some subset of these items, then the relation between the entries in these two matrices is given by the constant-ratio rule. This rule states that the ratio between any two entries in a row of the submatrix is equal to the ratio between the corresponding entries in the master matrix. This statement of the rule assumes that the only variables which differ in obtaining the two matrices are the different sets of messages and the allowable responses. A series of experiments are reported which support this rule and illustrate its use in predicting the entries in a submatrix when the master matrix is known. Other applications of the rule will be discussed. [This research was supported in part by a National Science Foundation Predoctoral Fellowship and in part by the U. S. Air Force under Contract No. AF 19(604)-1962.]