We review three different measures of effect size: Phi φ, Cramer’s V and the Odds Ratio.
For the goodness of fit in 2 × 2 contingency tables, phi, which is equivalent to the correlation coefficient r (see Correlation), is a measure of effect size. Phi is defined by
where n = the number of observations. A value of .1 is considered a small effect, .3 a medium effect and .5 a large effect.
This is the effect size measure (labelled as w) that is used in power calculations even for contingency tables that are not 2 × 2 (see Power of Chi-square Tests).
Cramer’s V is an extension of the above approach, and is calculated as
where df* = min(r – 1, c – 1) and r = number of rows and c = number of columns in the contingency table. Per Cohen, you use the guidelines for phi divided by the square root of df*. Thus, the guidelines are:
Figure 1 – Effect sizes for Cramer’s V
For a 2 × 2 contingency table, we can also define the odds ratio measure of effect size as in the following example.
Example 1: Calculate the odds ratio for the data in Example 2 of Independence Testing.
Figure 2 – Odds ratio effect size
As we saw in Example 2 of Independence Testing, there is a significant difference between those taking therapy 1 and those taking therapy 2. In fact, 26.19% of the people who took therapy 1 were not cured, while 47.22% of those who took therapy 2 were not cured. This shows that those taking therapy 2 were 1.80 times as likely as those taking therapy 1 to remain uncured. This is a meaningful measure of effect size, called the risk ratio or relative risk.
A related measure of effect size is the odds ratio. The odds of a person who took therapy 1 remaining uncured is 11 to 31 or .3548. The odds of a person who took therapy 2 is 51 to 57 or .8947. This means that the odds of remaining uncured is 2.52 times greater for therapy 2 than therapy 1. The ratio 2.52 is the odds ratio.