Page 191 - DMGT404 RESEARCH_METHODOLOGY

Unit 9: Correlation and Regression

Notes
nS - nS 2 1
2
i . i jk  S - 2
2
= i S . i jk .... (13)
2
nS 2 (nS - nS 2 ) S i
i i . i jk
Square of R is known as the coefficient of multiple determination.
i×jk
2
×
2
R 2  1 S - S 2 × ) 1 - S i jk .... (14)
×
i jk 2 ( i i jk 2
S S
i i
S 2 ×
i jk
It may be noted here that 2 is proportion of unexplained variation. Thus, we can also write
S
i
x 2 ×
R 2 i jk  1 - i jk
2 .
×
x i
Further, we can write R 2 i jk in terms of the simple correlation coefficients.
×
2
2
2
2
S 2 i (1 r- ij 2 - r - r + 2r r r ) r + r - 2r r r
jk
ij ik jk
ik
ik
ij ik jk
ij
2
R i jk  1 - 2 2 )  1 r 2
-
×
S
i (1 r- jk jk
×
×
2
Notes If there are m variables, R 1 23....m  1 - S 2 1 23....m  1- å x 2 1 23....m
×
S 1 2 å x 2 1
Self Assessment
Fill in the blanks:
14. The coefficient of ………………correlation in case of regression of xi on xj and xk, denoted
by Ri×jk
15. The coefficient of multiple correlation in case of regression of xi on xj and xk is defined as
a ………………coefficient of correlation between xi and xic.
9.3 Partial Correlation
In case of three variables x , x and x , the partial correlation between x and x is defined as the
i j k i j
simple correlation between them after eliminating the effect of x . This is denoted as r .
k ij×k
We note that x = x – b x is that part of x which is left after the removal of linear effect of x on
i×k i ik k i k
it. Similarly, x = x – b x is that part of x which is left after the removal of linear effect of x on
j×k j jk k j k
it. Equivalently, r can also be regarded as correlation between x and x . Thus, we can write
ij×k i×k j×k
å x x
r ij k ×  i k × j k × .
å x x 2 j k ×
2
×
i k
Using property III of residual products, we can write
Sx x = Sx x = S(x – b x )x = Sx x – b Sx x
i×k j×k i×k j i ik k j i j ik j k
= nS S r - r S i nS S r  nS S r - r r )
ik
j ij
i
S j k jk i j ( ij ik jk
k
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