Correlation Coefficients > Phi Coefficient

## What is the Phi Coefficient?

The Phi Coefficient is a measure of association between two binary variables (i.e. living/dead, black/white, success/failure). It is also called the *Yule phi* or* Mean Square Contingency Coefficient *and is used for contingency tables when:

- At least one variable is a nominal variable.
- Both variables are dichotomous variables.

For a 2×2 contingency table where a, b, c, and d represent the observation frequencies (the cell count). The formula for phi is:

**Example**: Find phi for the following contingency table:

**Solution**: Insert the counts into the formula and solve.

Φ = ad – bc / √((a + b)(c + d)(a + c)(b + d))

Φ = 14*13 – 10*6 / √((14 + 10)(6 + 13)(14 + 6)(10 + 13))

Φ = 182 – 60 / √((24)(19)(20)(23))

Φ = 122/ √((24)(19)(20)(23))

Φ = 122/ 458

Φ = 0.266.

## Interpreting the Phi Coefficient

The phi coefficient is a **symmetrical statistic**, which means the independent variable and dependent variables are interchangeable.

The interpretation for the phi coefficient is similar to the Pearson Correlation Coefficient. The range is from -1 to 1, where:

- 0 is no relationship.
- 1 is a perfect positive relationship: most of your data falls along the diagonal cells.
- -1 is a perfect negative relationship: most of your data is
*not*on the diagonal.

The Political Science Department at Quinnipiac University posted this useful list of the meaning of Pearson’s Correlation coefficients. The same rule of thumb can be used for the Phi coefficient. Note that these are “**crude estimates**” for interpreting strengths of relationships.

r value = | |

+.70 or higher | Very strong positive relationship |

+.40 to +.69 | Strong positive relationship |

+.30 to +.39 | Moderate positive relationship |

+.20 to +.29 | weak positive relationship |

+.01 to +.19 | No or negligible relationship |

0 | No relationship |

-.01 to -.19 | No or negligible relationship |

-.20 to -.29 | weak negative relationship |

-.30 to -.39 | Moderate negative relationship |

-.40 to -.69 | Strong negative relationship |

-.70 or higher | Very strong negative relationship |

**References**:

Quinnipiac University. Maaning of Pearson’s. Retrieved Jun 20, 2016 from: http://faculty.quinnipiac.edu/libarts/polsci/Statistics.html

Yule, G. U. (1912). J. R. Statist. Soc., 75,

576–642. (On the methods of measuring the

association between two variables. The first

identification of the phi-coefficient.)