*50*

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

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

**Chi-Square Test for Independence:Â **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 doesnâ€™t tell us the effect size of the test.

There are three ways to measure effect size: Phi (Ï†), Cramerâ€™s V (V), and odds ratio (OR).

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

**Phi (Ï†)**

**How to CalculateÂ **

Phi is calculated asÂ Ï†Â =Â âˆš(*X*^{2} / n)

where:

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

n = total number of observations

**When to Use**

Itâ€™s appropriate to calculateÂ Ï† only when youâ€™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 ofÂ Ï†Â = 0.1 is considered to be a small effect, 0.3 a medium effect, and 0.5 a large effect.

**Cramerâ€™s V (V)**

**How to CalculateÂ **

Cramerâ€™s V is calculated asÂ VÂ =Â âˆš(*X*^{2} / n*df)

where:

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

n = total number of observations

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

**When to Use**

Itâ€™s appropriate to calculate V when youâ€™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 CalculateÂ **

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**

Itâ€™s appropriate to calculate the odds ratio only when youâ€™re working with a 2 x 2 contingency table. Typically the odds ratio is calculated when youâ€™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 theÂ further away the odds ratio is from 1, the higher the likelihood that the treatment has an actual effect.

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