In statistics, there are two commonly used Chi-Square tests:

**Chi-Square Test for Goodness of Fit**: Used to determine whether or not a categorical variable follows a hypothesized distribution.

**Chi-Square Test for Independence: **Used to determine whether or not there is a significant association between two categorical variables from a single population.

For both of these tests, we end up with a p-value that tells us whether or not we should reject the null hypothesis of the test. The p-value tells us whether or not the results of the test are significant, but it doesn’t tell us the effect size of the test.

There are three ways to measure effect size: Phi (φ), Cramer’s V (V), and odds ratio (OR).

In this post we explain how to calculate each of these effect sizes along with when it’s appropriate to use each one.

**Phi (φ)**

**How to Calculate **

Phi is calculated as φ = √(*X*^{2} / n)

where:

*X*^{2} is the Chi-Square test statistic

n = total number of observations

**When to Use**

It’s appropriate to calculate φ only when you’re working with a 2 x 2 contingency table (i.e. a table with exactly two rows and two columns).

**How to Interpret**

A value of φ = 0.1 is considered to be a small effect, 0.3 a medium effect, and 0.5 a large effect.

**Cramer’s V (V)**

**How to Calculate **

Cramer’s V is calculated as V = √(*X*^{2} / n*df)

where:

*X*^{2} is the Chi-Square test statistic

n = total number of observations

df = (#rows-1) * (#columns-1)

**When to Use**

It’s appropriate to calculate V when you’re working with any table larger than a 2 x 2 contingency table.

**How to Interpret**

The following table shows how to interpret V based on the degrees of freedom:

Degrees of freedom |
Small |
Medium |
Large |
---|---|---|---|

1 |
0.10 | 0.30 | 0.50 |

2 |
0.07 | 0.21 | 0.35 |

3 |
0.06 | 0.17 | 0.29 |

4 |
0.05 | 0.15 | 0.25 |

5 |
0.04 | 0.13 | 0.22 |

**Odds Ratio (OR)**

**How to Calculate **

Given the following 2 x2 table:

Effect Size | # Successes | # Failures |
---|---|---|

Treatment Group |
A | B |

Control Group |
C | D |

The odds ratio would be calculated as:

Odds ratio = (AD) / (BC)

**When to Use**

It’s appropriate to calculate the odds ratio only when you’re working with a 2 x 2 contingency table. Typically the odds ratio is calculated when you’re interested in studying the odds of success in a treatment group relative to the odds of success in a control group.

**How to Interpret**

There is no specific value at which we deem an odds ratio be a small, medium, or large effect, but the further away the odds ratio is from 1, the higher the likelihood that the treatment has an actual effect.

It’s best to use domain specific expertise to determine if a given odds ratio should be considered small, medium, or large.

Thanks for sharing this information! I have been looking for the effect size for a chi squared test of homogeneity, but I have not found one. These two (phi and Cramer’s V) that you mentioned correspond to the independence test. Do you know of one effect size applied to the homogeneity test and what the interpretation is?

Thanks!

It was fine

Could you please elaborate how to explain cramer’s V with example?

The formula for Cramer’s V that you provide is incorrect: it only works for tables with either 2 rows or columns. The real formula is

sqrt(chi-squared / (n * (min(rows, columns)-1))).

(See Sheskin 2011: equation 16.22)