Regression - 6 Calculation of Coefficients of Regression
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Case
From the following information find out coefficient of regression, lines of regression and estimate the value of y when x = 10 and value of x when y = 10.
X 6 2 10 4 8
Y 9 11 5 8 7
Coefficients of Regression
According to Karl Pearson, “The coefficient of regression is the product of the coefficient of correlation with the ratio of the standard deviation of the dependent variable to the standard deviation of the independent variable.”
Thus, Coefficient of Regression of the dependent variable on the independent variable is
CoR = CoC X (SD of dependent variable/SD of independent variable)
= r x (SDdep / SDindep)
Since, in correlation, the variable ‘x’ is considered as dependent on the variable ‘y’ and variable ‘y’ is considered as dependent on variable ‘x’, there should be two coefficients of regression,
(i) The coefficient of regression of ‘y’ on ‘x’ = byx (when ‘y’ is considered as dependent on ‘x’ and ‘x’ is considered as independent) and
(ii) The coefficient of regression of ‘x’ on ‘y’ = bxy (when ‘x’ is considered as dependent on ‘y’ and ‘y’ is considered as independent).
Relationship between byx, bxy and r:
The product of coefficient of regression is equal to the square of the coefficient of correlation
byx = r x Sy/Sx , bxy = r x Sx/Sy
byx .bxy = r x Sy/Sx . r x Sx/Sy (We can cancell Sx and Sy)
So, (byx) (bxy) = r^2
√byx . bxy = r
Thus, coefficient of correlation is geometric mean of coefficient of regression.
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