Consider the artificial example where x (e.g. age) perfectly predicts y ( a test score) 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 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 - (x mean for group 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 predicting y which already contains group as a predictor) 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). The reduction in correlation is caused by the adjusted x - group mean difference not being ordered with respect to y despite x being perfectly monotonic with respect to y.
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 and correlating variation in x which is independent of group with unique variation in y which is independent of group. The result is a correlation between x and y both standardized by group which is simply pooling the two within group zero-order x,y correlations which gives 1.00 in the example above since both intra-group x,y correlations equal 1.00.
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.
Predictor |
B |
Std. Error |
Beta |
t |
Sig. |
Zero-order r |
Partial r |
Part r |
||||||||
group |
0 |
0 |
0 |
NA |
NA |
.870 |
1.000 |
0.000 |
||||||||
x |
1.000 |
0 |
0 |
NA |
NA |
1.000 |
1.000 |
0.492 |
||||||||
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).
In summary, the semi-partial correlation of x with y adjusted for group compares changes in the raw score, y, with changes in x relative to the subject's group mean whereas the analogous partial correlation compares changes in y relative to the subject's group mean with changes in x relative to the subject's group mean. In the above example the group response and age scores are both identical relative to their group response and age means but the raw scores and raw ages are, themselves, different so you only get a perfect (partial) prediction by centering both response (y) and age (x) by their subject group means.
One could better look at the x,y correlation with group by simply comparing the two zero-order group correlations using Fisher's test or a group by x interaction term in a regression on y and pooling across group (ie just using the x,y zero-order correlations for all observations ignoring group) if the correlations are found not to differ, as in the above example.
Dugard, Todman and Staines (2010) recommend that in general 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.
There doesn't seem to be a consensus on this, though, as change in R-squared is often used to remove covariate or group effects and one could argue that the semi-partial correlation removes the x group differences, for example in the case study above essentially standardizing x using the group means, so that any relationship between the covariate and y cannot be due to group differences as the covariate is no longer differing between the groups so the relationship between x and y is no longer confounded or influenced by group membership.
The partial correlation for a covariate may be obtained from an AN(C)OVA table by taking the square root of SS(covariate) / ( SS(covariate) + SS(error) ).
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 Chapter 6 from here or alternatively in pdf format here.
Howell DC (1997) Statistical methods for psychologists. Fourth edition. Wadsworth:Belmont,CA (see page 529).