Sometimes the correlation between two variables X1 and X2 may be partly due to the correlation of a third variables, X3 with both X1 and X2. In such a situation, one may want to know what the correlation between X1 and X2 would be if the effect of X3 on each of X1 and X2 were eliminated. This correlation is called the partial correlation and the correlation coefficient between X1 and X2 after the linear effect of X3 on each of them has been eliminated is called the partial correlation coefficient.
The residual X1.3 =X1 - b13 X3 may be regarded as that part of the variable X1 which remains after the linear effect of X3 has been eliminated. Similarly the residual X2.3 may be interpreted as the part of the variable X2 obtained after eliminating the linear effect of X3. Thus the partial correlation coefficient between X1 and X2 usually denoted by r12.3 is given by
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