Partial Correlation

Partial correlation, calculates the correlation between two variables, while excluding the effect of a third variable. This makes it possible to find out whether the correlation r_xy between variables x and y is produced by the variable z.

The partial correlation r_xy,z tells how strongly the variable x correlates with the variable y, if the correlation of both variables with the variable z is calculated out.

Calculate Partial Correlation

For the calculation of the partial correlation, the three correlations between the individual variables are required. The partial correlation then results in

• r_xy = Correlation between variable x and y
• r_xz = Correlation of the third variable z with the variable x
• r_yz = Correlation of the third variable z with the variable y

There are several ways to measure the correlation between two variables, the most common being the Pearson correlation and the Spearman correlation.

Partial Correlation Example

Probably the most prominent example of partial correlation is that of storks and babies from Robert Matthew's "Storks Deliver Babies" The correlation between the number of nesting storks and the birth rate is r=0.62. However, the area per inhabitant correlates both with the number of nesting storks r=0.58 and with the birth rate r=0.92.

To calculate the partial correlation, the individual correlations are inserted into the equation

The result is that the number of nesting storks and the birth rate correlate only with 0.329 if the area per inhabitant is excluded.

Interpret partial correlation

Partial correlation is a concept closely related to correlation. It shows that a correlation between two variables does not necessarily mean that there is a causal relationship between the two variables.

That is, if there is a correlation between two variables, it may be that this correlation can be partially explained by a third variable. After removing the third variable, what remains is the partial correlation between the two variables.

2^nd order partial correlation

A 2^nd order partial correlation occurs when two variables are subtracted from the correlation of two variables, rather than just one variable. The equation to calculate the 2^nd order partial correlation is:

where x and y are the two variables of which we want to know the correlation excluding variables z1 and z2.