A partial correlation is a measure of linear association between two variables after variability in at least one other variable is removed from both variables. The traditional formula for the partial correlation between, say, variables X and Y after controlling for Z, denoted rXY·Z, is
where r is the Pearson correlation between two variables.
rX(Y·Z) is sometimes referred to as a first-order partial correlation to note that the correlation only controls for one other variable. If controlling for two variables, it would be a second-order, and so on. Zero-order correlations do not control for any other variables.
While not obvious from Equation 1, the partial correlation is really the correlation between the residuals of X and Y after regressing both variables on Z. A path model representing this ...
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