Mathematical studies of the information in the stimulus-response matrix (Q1598975)

The paper considers the information transmitted in judgements encoded in stimulus-response- (confusion-)matrices [see e.g. \textit{W. R. Garner} and \textit{H. W. Hake}, Psychol. Rev. 58, 446-459 (1951)]. When transmitted information is plotted against the number of stimulus categories in the matrix, one obtains a curve that increases towards a flat maximum information. The following is offered as main result. If in an \(m\times n\) confusion matrix two columns (or rows) are merged , the information calculated from the resulting matrix will not be greater than that from the original matrix. The mathematical tools used are convexity [\(\Phi(\sum_{j=1}^n q_j x_j, \sum_{j=1}^n q_j y_j)\leq\sum_{j=1}^n q_j\Phi(x_j,y_j)\) \((\sum_{j=1}^n q_j=1\); \(q_j>0\), \(j=1,\dots ,n)\)] and subadditivity [\(\varphi(\sum_{j=1}^m x_j,\sum_{j=1}^m y_j)\leq\sum_{j=1}^m\varphi(x_j,y_j)\)] of functions in two variables.