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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).

None: FAQ/p+sp (last edited 2015-08-03 13:54:57 by PeterWatson)