Optimal scaling for ordered categories (Q1965971)

For finding optimal scalings \(x_1,\dots, x_n \) for \(X_k\) and \(y_1,\dots, y_m\) for \(Y\), usually the following maximization problem is solved: \[ x'Py\to max \{x\in R^n, y\in R^m\}\quad \text{under the condition}\quad x'Px=1=y'Cy, \] where \(P=(p_{ij})\) is the \(n\times m\) table of observed relative frequencies \((X_k =i,Y=j),\) \(C\) is the diagonal matrix of the marginal relative frequencies of \(Y\) and \(R\) is the diagonal matrix of the marginal relative frequencies of \(X_k.\) A modification of this procedure is proposed to investigate psychological data. The numerical results are obtained by means of the software MAXCOR.