3420
Comment:
|
4275
|
Deletions are marked like this. | Additions are marked like this. |
Line 16: | Line 16: |
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. | 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 and one could argue that the semi-partial correlation removes the x group differences, 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. |
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 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. 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).
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 and one could argue that the semi-partial correlation removes the x group differences, 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.
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).