Contingency table

A contingency table records counts classified by combinations of categorical variables.

The contingency coefficient transforms the Pearson chi-squared statistic as $C=\sqrt{X^2/(X^2+n)}$. It is an unsigned measure of statistical association, not a p-value or a direction of an ordinal relationship. Writing ${\varphi }^2=X^2/n$ gives $C=\varphi /\sqrt{1+{\varphi }^2}$. Unlike Cramér's V, its attainable maximum depends on table dimensions, so comparisons between differently sized tables need care.

For a contingency table with Pearson chi-squared statistic $X^2$ and sample size $n>0$, the nonnegative magnitude $\varphi =\sqrt{X^2/n}$ is sometimes reported as generalized phi. It does not encode an ordinal direction. In a two-row or two-column table it coincides with Cramér's V; in larger tables it can exceed one, so the normalization in Cramér's V matters. In a two-by-two table it is the absolute value of the Pearson correlation coefficient of binary category indicators; a signed binary phi requires a stated coding convention.

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.

For an $r\times c$ contingency table with $n$ observations and positive row and column totals, Cramér's V rescales the Pearson chi-squared statistic to a dimensionless measure of statistical association. It lies between zero and one and is unchanged by reordering category labels. It therefore describes the magnitude of dependence, without an ordinal direction. A small p-value can accompany a small $V$ in a large sample: evidence against independence and the magnitude of the departure answer different questions. The displayed quantity is the usual uncorrected sample measure.

A two-by-two contingency table cross-classifies observations by two binary variables, producing four cell counts and two pairs of marginal totals.