Page 198 - DMGT404 RESEARCH_METHODOLOGY

Research Methodology

Notes Consider equation (19)
Cov ( ,X Y ) rs s s
d = = X Y = r × X .... (22)
s 2 s 2 s
Y Y Y
Substituting the value of c from equation (15) into line of regression of X on Y we have

X = X - dY + dY or (X - X = d Y - Y ) .... (23)
Ci i Ci ) ( i
s
Y -
or (X - X ) = r × s X Y ( i Y ) .... (24)
Ci
Remarks: It should be noted here that the two lines of regression are different because these
have been obtained in entirely two different ways. In case of regression of Y on X, it is assumed
2
that the values of X are given and the values of Y are estimated by minimising S(Y – Y ) while
i Ci
in case of regression of X on Y, the values of Y are assumed to be given and the values of X are
estimated by minimising S(X – X ) . Since these two lines have been estimated on the basis of
2
i Ci
different assumptions, they are not reversible, i.e., it is not possible to obtain one line from the
other by mere transfer of terms. There is, however, one situation when these two lines will
coincide. From the study of correlation we may recall that when r = ± 1, there is perfect correlation
between the variables and all the points lie on a straight line. Therefore, both the lines of
regression coincide and hence they are also reversible in this case. By substituting r = ± 1 in
equation (12) or (24) it can be shown that the lines of regression in both the cases become

æ Y - Y ö æ X - Xö
ç è i s Y ø ÷ = ± ç è i s X ÷ ø

Further when r = 0, equation (12) becomes Y = Y and equation (24) becomes X = X . These are
Ci Ci
the equations of lines parallel to X-axis and Y-axis respectively. These lines also intersect at the
point ( , )X Y and are mutually perpendicular at this point, as shown in Figure.

9.4.2 Correlation Coefficient and the two Regression Coefficients

s s
Since b = × Y and d = × X , we have
r
r
s s
X Y
s s
×
b d = r Y r × X = r or r = . d × This shows that correlation coefficient is the geometric mean
2
s s b
X Y
of the two regression coefficients.
Remarks: The following points should be kept in mind about the coefficient of correlation and
the regression coefficients:
Y
X
1. Since r = Cov ( , ) , b = Cov ( ,X Y ) and d = Cov ( ,X Y ) , therefore the sign of r, b and d will
s s s 2 s 2
X Y X Y
always be same and this will depend upon the sign of Cov(X, Y).
2
2
2. Since bd = r and 0  r  1, therefore either both b and d are less than unity or if one of them
is greater than unity, the other must be less than unity such that 0  b × d  1 is always true.
Example: Obtain the two regression equations and find correlation coefficient between X
and Y from the following data:
X 10 9 7 8 11
Y 6 3 2 4 5

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