Partial Correlation

What is a partial correlation?

Partial correlation measures the correlation between two variables after statistically controlling for one or more other variables. For example, it can show how the linear association between x and y changes after controlling for variable z.

The partial correlation r_xy,z is the correlation between the parts of x and y that are not linearly explained by z. In other words, the influence of z is removed from both x and y.

The ordinary correlation r_xy, without control variables, is also called the zero-order correlation. A semipartial correlation differs from a partial correlation because it removes the control variable from only one of the two variables, not from both.

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 Matthews' "Storks Deliver Babies." The correlation between the number of nesting storks and the birth rate is r = 0.63. However, the area per inhabitant correlates both with the number of nesting storks (r = 0.57) and with the birth rate (r = 0.88).

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

The partial correlation between the number of nesting storks and the birth rate is 0.329 after statistically controlling for area per inhabitant.

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 statistically explained in part by a third variable. However, partial correlation alone cannot identify causal relationships. The result depends on which variables are controlled for, and controlling for a variable influenced by both variables (a collider) can even create a misleading association.

2^nd order partial correlation

A 2^nd order partial correlation controls for two variables rather than one. 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 after controlling for variables z1 and z2.

Assumptions and significance test

For a Pearson partial correlation, the relationships between the variables should be approximately linear, the observations should be independent, and influential outliers should not dominate the result. For the usual significance test and confidence interval, approximate multivariate normality is also assumed.

To test a partial correlation that controls for k variables, the t-test has n − k − 2 degrees of freedom. For example, controlling for one variable gives n − 3 degrees of freedom. The test statistic is t = r √((n − k − 2) / (1 − r²)). A small p-value provides evidence that the population partial correlation differs from zero; a confidence interval additionally shows the range of plausible values and the precision of the estimate.