Page 226 - DMGT209_QUANTITATIVE_TECHNIQUES_II

Unit 11: Multiple Regression and Correlation Analysis

Notes
 x y i
i
or b = 2 .... (7)
 x i
where x and y are deviations of values from their arithmetic mean.
i i
Dividing numerator and denominator of equation (6) by n we have

1
  X  X  Y  Y 
i
i
n Cov X,Y 
b =  2 .... (8)
1  X  2  X
i
n  X 
The expression for b, which is convenient for use in computational work, can be written from
equation (5) is given below:
 X i  Y i  X i  Y i 
 X Y  n   X Y 
i
i
i
i
b = n n  n 2
2
  X   X 
2
2
 X  n  i   X  i
i
i
 n  n
Multiplying numerator and denominator by n, we have
n  X Y   X i  Y i 
i
i
b = 2 .... (9)
2
n  X   X i 
i
To write the shortcut formula for b, we shall show that it is independent of change of origin but
not of change of scale.
As in case of coefficient of correlation we define
X  A Y  B
i
u = i and v =
i h i k
or X = A + hu and Y = B + kv
i i i i
 X = A  hu and Y = B kv

also  X  X  = h u   u and Y  Y = k v   v
i
i
i
i
Substituting these values in equation (6), we have
hk  u  u v   v k  u  u v   v
i
i
i
i
b = 2  2
h 2  u   u h  u   u
i
i

k  u v   u i  v i  
n
i
i
=  2  .... (10)
h  n  u   u  
2
 i i 
(Note: if h = k they will cancel each other)
LOVELY PROFESSIONAL UNIVERSITY 221