Phi Coefficient Calculator

Given a 2×2 contingency table with cells a, b, c, d, the Phi coefficient is computed as:

$$\phi = \frac{ad - bc}{\sqrt{(a+b)(c+d)(a+c)(b+d)}}$$

This formula computes the cross-product difference (ad − bc) and normalizes it by the geometric mean of the four marginal totals. The result ranges from −1 to +1.

The relationship to chi-square is direct:

$$\chi^2 = n \cdot \phi^2$$

where n = a + b + c + d is the total sample size. Conversely, φ = √(χ²/n) when only the magnitude is needed.

The sign of φ indicates direction: a positive φ means that the two variables tend to co-occur (high values of one variable are associated with high values of the other in the binary coding), while a negative φ indicates an inverse relationship. The magnitude follows Cohen's conventions for correlation-like effect sizes: small (0.1), medium (0.3), and large (0.5).

Phi can also be expressed in terms of the 2×2 table's probabilities:

$$\phi = \frac{p_{11}p_{00} - p_{10}p_{01}}{\sqrt{p_{1\cdot}p_{0\cdot}p_{\cdot 1}p_{\cdot 0}}}$$

This probabilistic formulation shows that Phi is indeed the Pearson product-moment correlation between two indicator variables.

The Phi coefficient is interpreted using standard effect size benchmarks:

• |φ| < 0.10: Negligible association
• 0.10 ≤ |φ| < 0.30: Small association
• 0.30 ≤ |φ| < 0.50: Medium association
• |φ| ≥ 0.50: Large association

A positive φ indicates that the diagonal cells (a and d) dominate, meaning the two variables tend to agree. A negative φ indicates that the off-diagonal cells (b and c) dominate, showing an inverse relationship. The corresponding chi-square statistic can be used to test whether the association is statistically significant.