Contingency table

The simplest measure of association for a 2 × 2 contingency table is the odds ratio. Given two events, A and B, the odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the odds ratio is 1; if the odds ratio is greater than 1, the events are positively associated; if the odds ratio is less than 1, the events are negatively associated.

The odds ratio has a simple expression in terms of probabilities; given the joint probability distribution:

${\displaystyle {\begin{array}{c|cc}&B=1&B=0\\\hline A=1&p_{11}&p_{10}\\A=0&p_{01}&p_{00}\end{array}}}$

the odds ratio is:

${\displaystyle OR={\frac {p_{11}p_{00}}{p_{10}p_{01}}}.}$

A simple measure, applicable only to the case of 2 × 2 contingency tables, is the phi coefficient (φ) defined by

${\displaystyle \phi =\pm {\sqrt {\frac {\chi ^{2}}{N}}},}$

where χ² is computed as in Pearson's chi-squared test, and N is the grand total of observations. φ varies from 0 (corresponding to no association between the variables) to 1 or −1 (complete association or complete inverse association), provided it is based on frequency data represented in 2 × 2 tables. Then its sign equals the sign of the product of the main diagonal elements of the table minus the product of the off–diagonal elements. φ takes on the minimum value −1.0 or the maximum value of +1.0 if and only if every marginal proportion is equal to 0.5 (and two diagonal cells are empty).[2]

Two alternatives are the contingency coefficient C, and Cramér's V.

The formulae for the C and V coefficients are:

${\displaystyle C={\sqrt {\frac {\chi ^{2}}{N+\chi ^{2}}}}}$ and

${\displaystyle V={\sqrt {\frac {\chi ^{2}}{N(k-1)}}},}$

k being the number of rows or the number of columns, whichever is less.

C suffers from the disadvantage that it does not reach a maximum of 1.0, notably the highest it can reach in a 2 × 2 table is 0.707 . It can reach values closer to 1.0 in contingency tables with more categories; for example, it can reach a maximum of 0.870 in a 4 × 4 table. It should, therefore, not be used to compare associations in different tables if they have different numbers of categories.[3]

C can be adjusted so it reaches a maximum of 1.0 when there is complete association in a table of any number of rows and columns by dividing C by ${\displaystyle {\sqrt {\frac {k-1}{k}}}}$ where k is the number of rows or columns, when the table is square , or by ${\displaystyle {\sqrt[{\scriptstyle 4}]{{r-1 \over r}\times {c-1 \over c}}}}$ where r is the number of rows and c is the number of columns.[4]

Another choice is the tetrachoric correlation coefficient but it is only applicable to 2 × 2 tables. Polychoric correlation is an extension of the tetrachoric correlation to tables involving variables with more than two levels.

Tetrachoric correlation assumes that the variable underlying each dichotomous measure is normally distributed.[5] The coefficient provides "a convenient measure of [the Pearson product-moment] correlation when graduated measurements have been reduced to two categories."[6]

The tetrachoric correlation coefficient should not be confused with the Pearson correlation coefficient computed by assigning, say, values 0.0 and 1.0 to represent the two levels of each variable (which is mathematically equivalent to the φ coefficient).

The lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level. Values range from 0.0 (no association) to 1.0 (the maximum possible association).

Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.

The uncertainty coefficient, or Theil's U, is another measure for variables at the nominal level. Its values range from −1.0 (100% negative association, or perfect inversion) to +1.0 (100% positive association, or perfect agreement). A value of 0.0 indicates the absence of association.

Also, the uncertainty coefficient is conditional and an asymmetrical measure of association, which can be expressed as

${\displaystyle U(X|Y)\neq U(Y|X)}$.

This asymmetrical property can lead to insights not as evident in symmetrical measures of association.[7]

• Gamma test: No adjustment for either table size or ties.

• Kendall's tau: Adjustment for ties.
  • Tau-b: Used for square tables.
  • Tau-c: Used for rectangular tables.