The Need for Multiple Regression and Partial Correlation

In deriving the regression coefficients byw. and bya.w, we have kept the variable in the subscript after the full stop constant. For this reason these are known as partial regression coefficients. The simple regression coefficients would have been given by 

It will be noted that these simple regression coefficients are preciably larger than the corresponding partial regression coefficients. The explanatioa of this is easy to see. If we calculate the correlation coefficient of Xw and Xa we get 

There is thus a highly significant and very important correlation between Xw and Xa. This is a consequence of the nature of the process, of course. Weather conditions that lead to low air temperatures (Xa) will also tend to give low cooling water temperature (Xw)- 

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