Phi Coefficient (φ) Calculator
This Phi coefficient calculator measures the association between two dichotomous (binary) variables. It is equivalent to Pearson's correlation coefficient when both variables are dichotomous, and is particularly useful for 2×2 contingency tables.
Further Information
The Phi coefficient (φ) is a measure of association for two dichotomous variables. It ranges from -1 to +1, similar to Pearson's correlation coefficient. The Phi coefficient is mathematically equivalent to the square root of the chi-square statistic divided by the sample size.
Requirements
- Two dichotomous (binary) variables
- Data organized in a 2×2 contingency table
- Independent observations
Null Hypothesis
H₀: φ = 0 (no association between the two variables)
The null hypothesis states that there is no association between the two dichotomous variables. A significant result suggests that the observed association is unlikely to be due to chance.
Equation
φ = (ad - bc) / √[(a+b)(c+d)(a+c)(b+d)]
Where a, b, c, and d are the cell frequencies in a 2×2 table:
| Variable 2: Yes | Variable 2: No | |
|---|---|---|
| Variable 1: Yes | a | b |
| Variable 1: No | c | d |
Relationship to Chi-Square
The Phi coefficient is related to the chi-square statistic by the formula:
φ² = χ² / n
This means that φ can be interpreted as a measure of effect size for the chi-square test of independence on a 2×2 table.
Interpretation
- φ = +1: Perfect positive association
- φ = 0: No association (independence)
- φ = -1: Perfect negative association
- |φ| = 0.1: Small effect
- |φ| = 0.3: Medium effect
- |φ| = 0.5: Large effect
