The phi coefficient (φ) is a measure of association between two binary variables in a 2×2 contingency table. It is equivalent to the Pearson correlation coefficient computed on two dichotomous variables. Phi ranges from −1 to +1, where 0 indicates no association, +1 a perfect positive association, and −1 a perfect negative association.
For a 2×2 table with cells a, b, c, d:
Phi is directly related to the chi-square statistic: φ = √(χ²/N). This means you can always compute phi from a chi-square result. For tables larger than 2×2, use Cramér's V instead, which generalizes phi by normalizing for table dimensions.
Cohen's (1988) guidelines for interpreting phi:
For 2×2 tables, |φ| = V. They are identical. For larger tables, only Cramér's V is meaningful because phi can exceed 1. Use phi when you have exactly two binary variables; use V for any table size.
Yes. Unlike Cramér's V (which is always non-negative), phi preserves direction. A negative phi means the diagonal cells (a,d) have fewer observations than expected, indicating a negative association.
Report as: φ = .35, p = .002. CrossTabs.com generates APA-formatted text automatically in the Export tab.