PARTIAL CORRELATION

What it does:

It measures the linear relationship between two interval/ratio scale variables controlling for (holding constant) a third interval/ratio scale variable. One way of thinking about the partial correlation coefficient is this: suppose you calculate the bivariate correlation coefficient for each value of the control variable and you calculate some sort of a weighted average of these bivariate correlation coefficients.

Words of caution:

1. Unlike in tabular analysis, here you end up with a single number (and not as many as categories the control variable has). As a result, partial correlation is not a good way of detecting specification (statistical interaction).

2. In the world of correlation (bivariate or partial) all relationships are assumed to be linear including the relationship between the original variables and the control variable.

3. The partial correlation is symmetric only with respect to the original two variables. The partial correlation between y and x controlling for t is the same as the partial correlation between x and y controlling for t. But the partial correlation between y and x controlling for t is not the same as the partial correlation between y and t controlling for x, or between x and t controlling for y.

Where r_yx.t = the (first-order) partial correlation of y and x controlling for t

r_yx = the bivariate (zero-order) correlation of y and x

r_yt = the bivariate (zero-order) correlation of y and t

r_xt = the bivariate (zero-order) correlation of x and t

The formula suggests that the partial correlation is the function of the bivariate correlations among the three variables.

The partial correlation can be generalized to situations where you control for more than one variable. As a result, we distinguish between first-order partial correlation (controlling for a single variable), second-order partial correlation (controlling for two) and so on. In this class we only cover first order partial correlations.