A Phi Coefficient (sometimes called a mean square contingency coefficient) is a measure of the association between two binary variables.
For a given 2×2 table for two random variables x and y:
The Phi Coefficient can be calculated as:
Φ = (AD-BC) / √(A+B)(C+D)(A+C)(B+D)
Example: Calculating a Phi Coefficient
Suppose we want to know whether or not gender is associated with political party preference. We take a simple random sample of 25 voters and survey them on their political party preference. The following table shows the results of the survey:
We can calculate the Phi Coefficient between the two variables as:
Φ = (4*4-9*8) / √(4+9)(8+4)(4+8)(9+4) = (16-72) / √24336 = -0.3589
Note: We could have also calculated this using the Phi Coefficient Calculator.
How to Interpret a Phi Coefficient
Similar to a Pearson Correlation Coefficient, a Phi Coefficient takes on values between -1 and 1 where:
- -1 indicates a perfectly negative relationship between the two variables.
- 0 indicates no association between the two variables.
- 1 indicates a perfectly positive relationship between the two variables.
In general, the further away a Phi Coefficient is from zero, the stronger the relationship between the two variables.
In other words, the further away a Phi Coefficient is from zero, the more evidence there is for some type of systematic pattern between the two variables.