Partial Correlation

A partial correlation is a measure of the relationship that exists between two variables after the variability in each that is predictable on the basis of a third variable has been removed. A partial correlation, like a conventional Pearson product-moment correlation, can range from −1 to +1, butitcan be larger or smaller than the regular correlation between the two variables. In fact, a partial correlation is simply a conventional correlation between two sets of scores. However, the scores involved are not the scores on the original variables but instead are residuals-that is, scores reflecting the portion of the variability in the original variables that could not be predicted by a third variable.

As a concrete example, a researcher might be interested in whether depression predisposes individuals to interpret neutral faces as manifesting a negative emotion such as anger or sadness, but may want to show that the relationship is not due to the effects of anxiety. That is, anxiety might be confounded with depression and be a plausible rival explanatory variable, in that more depressed individuals might tend to have higher levels of anxiety than less depressed individuals, and it may be known that more anxious individuals tend to interpret emotions in neutral faces negatively. A partial correlation would be directly relevant to the researcher's question and could be used to measure the relationship between depression and a variable assessing the tendency to interpret neutral faces negatively, after one has removed the variability from each of these variables that is predictable on the basis of an individual's level of anxiety.

Partial correlations are convenient and have wide applicability, but they are often misunderstood. There are conceptual and statistical issues that must be kept in mind when working with partial correlations. The fomula used in the computation of a partial correlation is first introduced, and then several issues bearing on the interpretation of partial correlations are briefly considered.

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Fomula for Partial Correlation

The partial correlation, denoted r12.3, between a pair of variables (Variables 1 and 2) controlling for a “third” variable (Variable 3) may be readily computed from the pairwise or bivariate correlations among the three variables by using the fomula

Understanding partial correlations is greatly facilitated by realizing how this fomula flows from the definition of a correlation. A correlation may be defined either as the ratio of the covariance between two variables to the product of their standard deviations, or as the average product of the individuals' z scores or standard scores on the two variables:

(The final value of r is unaffected by whether N or N −1 is used in the denominators of fomulas for the covariance and variances because the sample size terms in the numerator and denominator cancel out. This may be less obvious but is still true in computing r as the average product of z scores, given the standard deviations used to compute the z scores could have been calculated using either N or N −1. N is used on the right in the equation to conform to the typical connotation of computing the “average” product of z scores as implying a division by N, although it should be noted that this implies that the “average” or “standard” deviations also are computed using N rather than N −1.)

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