# 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, *r*s, and *sr*s 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?[edit | edit source]

In MLR, square the *sr* for each IV, to get the semi-partial correlations squared (*sr*^{2}).

The *sr*^{2}s provide the percentage of variance in the DV which is *uniquely* associated with each IV.

Add the *sr*^{2} values to get the percentage of variance in the DV which is uniquely explained by the IVs.

*R*^{2}s (total percentage of variance explained) minus the total of the *sr*^{2}s equals the percentage of variance explained by the relationship between predictors.

## See also[edit | edit source]

- Correlation
- Multiple linear regression
- Partial correlation (Wikipedia)