Page 218 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Unit 10: Correlation

We note that x × k = x – bikxk is that part of xi which is left after the removal of linear effect of x Notes
i i k
on it. Similarly, x × k = x – bjkxk is that part of xj which is left after the removal of linear effect of
j j
xk on it. Equivalently, r × k can also be regarded as correlation between x × k and x × k. Thus, we
ij i j
can write .
 x x
r  i k  j k 
Using property III of residual products, we can write ij k 2 2
 x x j k
i k


Sx x = S j = S(x – b x )x = Sx x – b Sx x
i×k j×k xi×kx i ik k j i j ik j k
S
= nS S r  r i nS S r  nS S r  r r 
i j ij ik j k jk i j  ij ik jk
S k
Further, using property III, we can write
x 2 i×k = Sxixi×k = Sxi(xi – bikxk) = Sxi – bikSxixk
2
S
2
= nS  r ik i nS S r  nS 2 i 1 r ik 2 
i
k ik
i
S
k

Similarly, x 2 i×k = nS 2 j  1 r jk 2  .
nS S r  r r r  r r
Thus, we have ri = i j  ij ik  jk  ij ik jk
j×k 2 2 2 2 2 2
nS 1 r   1nS  r 1 r  1 r 
i ik j  jk ik jk
Did u know? What is Zero order, First order, and Second order Partial Correlation?
Simple correlation between two variables is called the zero order co-efficient since in simple
correlation, no factor is held constant. The partial correlation studied between two variables by
keeping the third variable constant is called a first order co-efficient, as one variable is kept
constant. Similarly, we can define a second order co-efficient and so on. The partial correlation
co-efficient varies between –1 and +1. Its calculation is based on the simple correlation
co-efficient.

10.4 Multiple Correlations

The coefficient of multiple correlations in case of regression of xi on xj and xk, denoted by Ri×jk,
is defined as a simple coefficient of correlation between xi and xic.


Cov  ,x x ic   x x  x x  x . i jk
i
 i
i ic
i
Thus R =  
i × jk 2 2 2
 Var x
Var x i    x i  x ic  x 2 i   i x . i jk 
x 
ic
 x   x x i jk  x   x x i jk
2
2
i
i
i
=   i  (Using property III)
 x
 x 2 i  x  x i jk x i  x 2 i  2 i –  x i x i jk 


i
2
nS  nS 2 . i jk 1
i
2
=  S  S 2 . i jk
i
2
nS 2 i nS  nS 2 . i jk  S i
i
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