Semi-partial correlation

In multiple linear regression (MLR), there are two or more independent or predictor variable (IV) and one dependent or response variable (DV).

A correlation (r) indicates the linear relationship between an IV and a DV.

A semi-partial correlation (sr) indicates the unique relation between an IV and the DV. An sr is the variance in a DV explained by an IV and only that IV (i.e., it does not include variance in a DV explained by other IVs).

This Venn diagram represents the variance, rs, and srs involved in an MLR analysis.

Question 1: Which areas represent the squared semi-partial correlations between the IVs and DV? Answer: a and c

Question 2: What do the other labelled areas represent? Answer:

a = semi-partial correlation between IV1 and DV (with the effect of IV2 removed)
b = variance in the DV which is explained by both IV1 and IV2
c = semi-partial correlation between IV2 and DV (with the effect of IV1 removed)
d = variance in DV which is not explained by IV1 and IV2

What are semi-partial correlations useful for?

In MLR, square the sr for each IV, to get the semi-partial correlations squared (sr²).

The sr²s provide the percentage of variance in the DV which is uniquely associated with each IV.

Add the sr² values to get the percentage of variance in the DV which is uniquely explained by the IVs.

R²s (total percentage of variance explained) minus the total of the sr²s equals the percentage of variance explained by the relationship between predictors.

What are semi-partial correlations useful for?

See also