Differences between partial and semi-partial correlations

Consider the artificial example where x perfectly predicts y in each of two groups (r = 1.00 in both) and all the x and y values are higher in one of the groups than the other (as shown on this powerpoint slide [attachment:xygroups.ppt here]). Suppose we wish to see how x correlates with y adjusting for group differences.

We can simply subtract the appropriate group mean from the x value of each observation depending upon what group it is in and correlate this new value with y (ie correlate y with $$x - \bar{x}_text{G}$$). This is the semi-partial correlation (also equal to the signed square root of the change in R-squared when you add x to a model containing group in predicting y) between x and y adjusting for group differences where only x is adjusted for group differences and will be less than the zero-order x,y correlation (around 0.5).

If we also adjust y for group differences by subtracting the appropriate y group mean from each y value we obtain the partial correlation between x and y adjusted for group with both x and y adjusted for group and obtain a partial correlation equal to 1.00. The partial correlation of 1.00 follows because there is perfect relationship between x and y in each group. The partial correlation is akin to removing group differences from both x and y. The results are summarised in the table below (form the output from running the SPSS linear regression procedure) with the semi-partial and partial correlations for x, as described above, highlighted. Since there is 'perfect' prediction we have zero standard errors.

The statistical test of significance is the same for both partial and semi-partial correlations and is equal to testing the regression coefficient of x in a regression also containing group to predict y (Howell, 1997).

Dugard, Todman and Staines (2010) recommend that a partial correlation should be used if apriori x is expected to differ between groups and be also related to y (as in the example above) and a semi-partial correlation if either apriori x is expected only to differ between groups and not be related to y or if x is not expected to differ between groups and only be related to y.

References

Dugard P, Todman J and Staines H (2010) Approaching multivariate analysis. A practical introduction. 2nd Edition. Psychology Press:London. Chapter 6 in this book advocates the use of partial correlations for adjusting for a covariate when the covariate influences both primary variables. Details are in Chapter 6 of this book and available on-line [http://www.psypress.com/multivariate-analysis/medical-examples/chapter06/med_partial_corr_analysis.pdf Chapter 6 from here] or alternatively in pdf format [attachment:partial.pdf here.]

Howell DC (1997) Statistical methods for psychologists. Fourth edition. Wadsworth:Belmont,CA (see page 529).