 Correspondence analysis

Correspondence analysis represents a nonnegative contingency table geometrically, displaying how row and column profiles depart from independence. Let $P$ be the table divided by its grand total, and let $r=P1$, $c=P^{T}1$ be positive marginal proportions. Its standardized residual matrix isA singular value decomposition of $S$ provides principal axes; the sum of squared singular values equals the Pearson Pearson chi-squared statistic for contingency tables divided by the grand total. Thus weighted departures from independence become geometric variation.^[ref]

For counts $\left(\begin{array}{cc}30& 10\\ 10& 30\end{array}\right)$, the total is 80 and both margins are $(1/2,1/2)$. Then $S=\left(\begin{array}{cc}1/4& -1/4\\ -1/4& 1/4\end{array}\right)$ has one nonzero singular value $1/2$. The total inertia is therefore $1/4$ and the Pearson chi-squared statistic for contingency tables is 20. The association has one geometric dimension. Zero margins require removal or separate treatment before using the inverse square roots.