Maximal Pearson’s correlation coefficient between Likert items from the viewpoint of Hitchcock’s transportation problem

When two variables are Likert-type items and their marginal distributions differ, the coefficient values of \(\pm \,\) $\pm$ 1 in Pearson’s product-moment correlation are untenable. To address this issue, this study proposes a new method for estimating the maximal and minimal Pearson’s correlations between two Likert items whose numbers of categories are \(R\) $R$ and \(C\) $C$ when their marginal distributions are fixed. First, we demonstrate that the maximizing/minimizing correlation problem is identical to Hitchcock’s transportation problem, which is a classic linear programming problem. Second, we prove the necessary and sufficient condition under which the northwest corner rule, which is frequently used to find an initial solution to the transportation problem, provides the optimal solution to the correlation problem. Finally, the results of two simulation studies conducted with \(5\times 5\) $5\times 5$ and \(5\times 7\) $5\times 7$ contingency tables reveal the extent to which the range of Pearson’s correlation shrinks.