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# Three Ways to Calculate Effect Size for a Chi-Square Test

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(X2 / n)

where:

X2Â 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(X2 / n*df)

where:

X2Â 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.