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Phi coefficient

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It has been suggested that this article or section be merged with Matthews correlation coefficient. (Discuss)

In statistics, the phi coefficient φ or rφ is a measure of association for two binary variables. This measure is similar to the Pearson correlation coefficient in its interpretation. In fact, a Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.[1] The Phi coefficient is also related to the chi-square statistic for a 2×2 contingency table (see Pearson's_chi-square_test)[2]

\phi = \sqrt{\frac{\chi^2}{n}}


y = 1 y = 0 total
x = 1 n_{11} n_{10} n_{1\bullet}
x = 0 n_{01} n_{00} n_{0\bullet}
total n_{\bullet1} n_{\bullet0} n

where n11, n10, n01, n00, are non-negative "cell cell counts" that sum to n, the total number of observations. The phi coefficient that describes the association of x and y is

\phi = \frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet0}n_{\bullet1}}}

Maximum values Edit

Although computationally the Pearson correlation coefficient reduces to the phi coefficient in the 2×2 case, the interpretation of a Pearson correlation coefficient and phi coefficient must be taken cautiously. The Pearson correlation coefficient ranges from −1 to +1, where ±1 indicates perfect agreement or disagreement, and 0 indicates no relationship. The phi coefficient has a maximum value that is determined by the distribution of the two variables. If both have a 50/50 split, the range of phi will range from −1 to +1. See Davenport El-Sanhury (1991) [3] for a thorough discussion.

See alsoEdit



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