What is F2 Test, X, Y Arithmetic Means The above method of finding out regression equation is tedious. The calculations can very much be simplified if instead of dealing with the actual values of X and Y we take the deviations of X and Y series form their respective means. In such a case the two regression equations are written as follows:
Regression equation of x on Y ; X - X = r σx / σy (Y - Y)
X is the mean of X series. Y is then mean of y series
R σx/ σy is known as the regression coefficient of X on Y.
The regression coefficient of X on Y is denoted by the symbol b xy or b1. It measures the change in X corresponding to a unit change in Y. when deviations are taken from the means of X and Y, the regression coefficient of X on Y is obtained as follows:
b xy or r σ/ σy = Σ x y / Σ y2
Instead of finding out the value of correlation coefficient σx, σ y etc, we can find the value of regression coefficient by calculating Σ x y and Σ y2 and dividing the former by the latter.
Regression equation of y on X; Y - Y =r σy/ σx (X - X)
R σy/ σx is the regression coefficient of Y on X it is denoted by b xy or b2. It measures the change in Y corresponding to a unit change in X. when deviators are taken from actual means the regression coefficient of Y on X can be obtained as follows.
R σy/ σx = Σ xy / Σ x2
It should be noted that ht ender- root of the product of two regression coefficients given us the value of correlation coefficient. Symbolically,
R = √bxy X b xy
Proof, B xy = r σx / σy and b xy = r σy/σx
B xy x b xy = r σx/σy x r σy/ σx = r2 , ∴ r = √bxy X b yx
The following points should be noted about the regression coefficients
Both the regression coefficients will have the same sign. Either they will be positive or negative. It is never possible that one of the regression coefficients is negative and the other positive.
Since the value of the coefficients of correlation cannot exceed one, one of the regression coefficients must be less than one or, in other words both the regression coefficients cannot be greater than one. For example, if b xy = 1.2 and b xy = 1.4 the value of correlating to coefficient would be √(1.2 x 1.4) = 1.296 which is not possible.
The coefficients of correlation will have the same sign as that of regression coefficients if regress coefficients have a negative sign r will also be negative and if regression coefficients have a positive sign, r would also be positive. For example if b xy = - 0.8 and b xy = - 1.2 r would be √(- 0.8 x-1.2) = - 0.98and not + 0.98.
Since b xy = r σx / σy we can find out any of the four values given the other three. For example if we know that r = 0.6 σx = 4 and b xy = 0.8, we can find σy.
B xy = r σx/ σy
Substituting the given values: 0.8 = 0.6 x 4 / σy or σy= 2.4 / 0.8 = 3.
Regression coefficients are independent of change of origin but two of scale.
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