Computational Approach - Unique Prediction and Partial Correlation

Note: in this equation, the regression coefficients (or B coefficients) represent the independent contributions of each independent variable to the prediction of the dependent variable. Another way to express this fact is to say that, for example, variable X1 is correlated with the Y variable, after controlling for all other independent variables. This type of correlation is also referred to as a partial correlation (this term was first used by Yule, 1907).

Perhaps this example will clarify this issue. One would probably find a significant negative correlation between hair length and height in the population (i.e., short people have longer hair). At first this may seem odd; however, if we were to add the variable Gender into the multiple regression equation, this correlation would probably disappear. This is because women, on the average, have longer hair than men; they also are shorter on the average than men. Thus, after we remove this gender difference by entering Gender into the equation, the relationship between hair length and height disappears because hair length does not make any unique contribution to the prediction of height, above and beyond what it shares in the prediction with variable Gender. Put another way, after controlling for the variable Gender, the partial correlation between hair length and height is zero.