 Goodman and Kruskal's lambda

Predict categorical response $Y$ by a most frequent category. Without knowing $X$, the number of mistakes in a contingency table is $E_0=n-{\max }_j{n}_{+j}$. Allowing a separate most frequent response for each predictor row gives $E_1=n-{\sum }_i{\max }_j{n}_{ij}$. Goodman and Kruskal's lambda is the fractional reduction in those mistakes when $E_0>0$. Its direction matters, and it can vanish despite statistical association: if every row has the same most frequent response category, knowledge of $X$ changes no modal predictions even when the row distributions differ.