= Explaining Part (also known as semi-partial) correlations =

Try to assess the importance of phonological awareness on predicting
Reading Ability which is independent of dysphraxia. You can do this
using General Linear Model:univariate. Suppose putting both phonological 
awareness and dysphraxia as covariates gives 

Suppose we have

Dependent Variable=Reading Ability

||||||||||||<16% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">  Source || df || Type III SS || MS || F || p ||
||||||||||||<style="TEXT-ALIGN: center"> Dysphraxia || 1 || 13.59 || 13.59 || 9.44 || .013 ||
||||||||||||<style="TEXT-ALIGN: center"> Phonological Awareness || 1 || 8.45 || 8.45 ||5.87 || .038 ||
||||||||||||<style="TEXT-ALIGN: center"> Error || 9 || 12.96 || 1.44 ||  || ||
||||||||||||<style="TEXT-ALIGN: center"> Corrected Total || 11 || 35.00 ||  ||  || ||
R-squared=0.630


This tells us that phonological Awareness has a statistically significant
influence on reading ability after taking dysphraxia into account 
(F(1,9)=8.45, p<0.05).

Fitting just dysphraxia gives

Dependent Variable=Reading Ability

||||||||||||<16% style="VERTICAL-ALIGN: top; TEXT-ALIGN: center">  Source || df || Type III SS || MS || F || p ||
||||||||||||<style="TEXT-ALIGN: center"> Dysphraxia || 1 || 15.50 || 15.50 || 7.95 || .018 ||
||||||||||||<style="TEXT-ALIGN: center"> Error || 10 || 19.50 || 1.95 ||  || ||
||||||||||||<style="TEXT-ALIGN: center"> Corrected Total || 11 || 35.00 ||  ||  || ||
R-squared=0.443 

Comparing the two R-squareds tells us that phonological awareness accounts for 0.630-0.443 = 0.187 or 18.7% of total variance in reading ability over and 
above that predicted by dysphraxia. The signed square root of this Sqrt(0.187)=sgn(0.432) is the '''Part correlation''', also called the '''semi-partial correlation''' of phonological awareness adjusted for dysphraxia with reading ability.

In other words:
The square of the (Part) correlation which relates aspects of phonological 
awareness, unrelated to dysphraxia, to reading ability is the difference in
 R-squareds of a model with dysphraxia and phonological awareness and
the R-squared of a model with dysphraxia only with reading ability as
dependent (outcome) variable.

R-squared (or equivalently its signed square root, the part correlation) is 
often given as a measure of the strength of an association between one or 
more predictor variables of interest, adjusted for other confounding predictors, with an outcome. Since this is a regression term R-squared can also be used to describe analysis of (co)variance.