Partial Correlation

Partial correlation is the correlation between two variables with the effect of (an)other variable(s) held constant. An often-quoted example is the correlation between the number of storks and the number of babies in a sample of regions, where the correlation disappears when the degree of rurality is held constant. This means that regions showing differences in the numbers of storks will also differ in birth rates, but this covariation is entirely due to differences in rurality. For regions with equal scores on rurality, this relationship between storks and babies should disappear. The correlation is said to be “spurious.”

As the correlation does not have to become zero but can change in whatever direction, we give here the example of the partial correlation between the extracurricular activity of students (variable X1) and their popularity (variable Y), controlling for their scholastic achievement (X2). To clarify the mechanism of partial correlation, we give a simple (fictitious) sample of five students with the following scores on the three variables:

Table 1
Y	X 1	X 2
1	0	4
3	1	4
2	3	1
6	6	2
4	8	0

To calculate the partial correlation, we remove the influence of X2 from X1 as well as from Y, so that the partial correlation becomes the simple zero-order correlation between the residual terms X1 − X^1 and Y − Ŷ. This mechanism involves three steps.

First, a bivariate regression analysis is performed from X1 on X2. The estimated X^1 values are calculated, and the difference of X1 − X^1 yields the residual scores that are freed from control variable X2.

Second, a bivariate regression analysis of Y on X2 is performed. The estimated Y − Ŷ yields the residual scores.

Third, X1 − X^1 and Y − Ŷ are correlated. This zero-order correlation between the residuals is the partial correlation coefficient ry1·2 (i.e., the zero-order product-moment correlation coefficient between Y and X1 after the common variance with X2 is removed from both variables).

This partial correlation coefficient is a first-order coefficient because one controls for only one variable. The model can be extended for higher-order partial correlations with several control variables. In the thirdorder coefficient ry1·234, the partial correlation between Y and X1 is calculated, controlling for X2, X3, and X4. In that case, the first step is not a bivariate but a multiple regression analysis of X1 on X2, X3, and X4, resulting in the residual term X1 − X^1. The second step is a multiple regression analysis of Y on X2, X3, and X4, yielding the residual term Y − Ŷ. The zero-order correlation between X1 − X^1 is, then, the coefficient ry1·234.

For the population, Greek instead of Latin letters are used. The sample coefficients ry1·2 and ry1·234, which are indicated above, are then written as ρy1·2 and ρy1·234.

Calculation of the Partial Correlation

All the aforementioned operations are brought together in Table 2.

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