 Linear-by-linear association test

For a two-way contingency table whose row and column variables are ordinal categorical variables, assign row scores $u_i$ and column scores $v_j$. The linear-by-linear association statistic measures the weighted covarianceAfter normalization, its square is asymptotically ${\chi }_1^2$ under independence. A signed, one-sided version tests a specified direction of ordinal association and can have greater statistical power than an omnibus independence test against a monotone alternative.