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| 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. | 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). |
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| 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. A partial correlation can be used if x differs between groups and is related to y (as in the example above) and a semi-partial correlation if either x only differs between group and is not releated to y or if x does nto differ between groups and is only related 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 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 x differs between groups and is ''also'' related to y (as in the example above) and a semi-partial correlation if either x only differs between group and is not related to y or if x does nto differ between groups and is only 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 [http://www.psypress.com/multivariate-analysis/medical-examples/chapter06/med_partial_corr_analysis.pdf Chapter 6 here] or alternatively in pdf form [attachment:partial.pdf here.] Howell DC (1997) Statistical methods for psychologists. Fourth edition. Wadsworth:Belmont,CA (see page 529). |
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 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 x differs between groups and is also related to y (as in the example above) and a semi-partial correlation if either x only differs between group and is not related to y or if x does nto differ between groups and is only 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 [http://www.psypress.com/multivariate-analysis/medical-examples/chapter06/med_partial_corr_analysis.pdf Chapter 6 here] or alternatively in pdf form [attachment:partial.pdf here.]
Howell DC (1997) Statistical methods for psychologists. Fourth edition. Wadsworth:Belmont,CA (see page 529).