Multiple Correlation
Introduction:-
You already learn the concept of the linear regression, correlation in case of two
variable. You learned how to determine the relationship b/w two variable using
correlation and also observe how to predict value of one variable from given
value of other variable. If we are interested to study the more than two variables
which are interrelated in some-way. And we want to observe the relationship b/w
one variable and set of others. In such case we will move towards multiple
correlation.
Why multiple correlation ?
The study of simple correlation and multiple correlation only based on
information. If information on two variables like:
1. Height and weight
2. Income and expenditure
3. Demand and supply
Are available and we want to study the
linear relationship between two
variables, correlation coefficient serves
our purpose which provides the strength
or degree of linear relationship with
direction whether it is positive or
negative.
But in biological, physical and social sciences, often data are available on more than two
variables and value of one variable seems to be influenced by two or more variables.
For example:
1. Crimes in a city may be influenced by illiteracy, increased
population and unemployment in the city, etc.
2. The production of a crop may depend upon amount of rainfall, quality of
seeds, quantity of fertilizers used and method of irrigation, etc.
3. Similarly, performance of students in university exam may depend upon
his/her IQ, mother’s qualification, father’s qualification, parents income,
number of hours of studies, etc.
Whenever we are interested in studying the
joint effect of two or more variables on a
single variable, multiple correlation gives the
solution of our problem.
Definition:-
Multiple correlation is a measure of the linear Suppose if 𝑋1 , 𝑋2
relationship between a single dependent variable and a 𝑎𝑛𝑑 𝑋3 are three
set of explanatory (independent) variables. variables.
we want to measure the combined effect of X2 and X3 on X1, then the Population
correlation coefficient is denoted by ρ 1.23 (read as rho one dot two three) and can
be calculated as from sample information 𝑅1.23 .
2 2
𝑟12 + 𝑟13 − 2(𝑟12)(𝑟23)(𝑟13)
𝑅1.23 =√ 2
1 − 𝑟23
𝑊ℎ𝑒𝑟𝑒 𝑅1.23 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑤ℎ𝑒𝑛 1𝑠𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 as dependent and other two as
independent.
𝑟12 = 𝑟21 (Relationship with 1st dependent and 2nd independent)
𝑟13 = 𝑟31 (Relationship with 1st dependent and 3rd independent)
𝑟23 = 𝑟32 (Relationship with 2nd independent and 3rd independent)
Other two possible Situation we may compute in multiple correlation, which we will
discusses during an example computation.
Range of multiple correlation is 0 to +1
Perfect
No correlation
correlation
0 1
0.5
Week Strong
relation
Moderate
0.4 to 0.60
Example :
Suppose three variable like 𝑋1 , 𝑋2 𝑎𝑛𝑑 𝑋3 given as below. Compute the
multiple correlation using following information.
𝑋1 65 72 54 68 55 59 78 58 57 51
, 𝑋2 56 58 48 61 50 51 55 48 52 42
𝑋3 9 11 8 13 10 8 11 10 11 7
Compute the multiple correlation for three different cases.
2 2
a. 𝑅1.23 = √𝑟12 +𝑟13 −2(𝑟12 )(𝑟23 )(𝑟13 )
1−𝑟 2 23
2 +𝑟 2 −2(𝑟 )(𝑟 )(𝑟 )
𝑟21
b. 𝑅2.13 = √ 23 12
2
1−𝑟13
23 13
2 +𝑟 2 −2(𝑟 )(𝑟 )(𝑟 )
𝑟31
c. 𝑅3.12 = √ 32 12
2
1−𝑟12
23 13
Solution :-
1st we need to compute, correlation 𝑟12 , 𝑟23 , 𝑟13
𝒏 ∑ 𝑿𝟏 𝑿𝟐− (∑ 𝑿𝟏 )(∑ 𝑿𝟐 )
𝒓𝟏𝟐 =
√{𝒏 ∑ 𝑿𝟐𝟏 −(∑ 𝑿𝟏 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟐 −(∑ 𝑿𝟐 )𝟐 }
𝒏 ∑ 𝑿𝟏 𝑿𝟑− (∑ 𝑿𝟏 )(∑ 𝑿𝟑 )
𝒓𝟏𝟑 =
√{𝒏 ∑ 𝑿𝟐𝟏 − (∑ 𝑿𝟏 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟑 − (∑ 𝑿𝟑 )𝟐 }
𝒏 ∑ 𝑿𝟐 𝑿𝟑− (∑ 𝑿𝟐 )(∑ 𝑿𝟑 )
𝒓𝟐𝟑 =
√{𝒏 ∑ 𝑿𝟐𝟐 − (∑ 𝑿𝟐 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟑 − (∑ 𝑿𝟑 )𝟐 }
Required Computation:-
S.
𝑋1 𝑋2 𝑋3 𝑋12 𝑋22 𝑋32 (𝑋1 )(𝑋2 ) (𝑋1 )(𝑋3 ) (𝑋2 )(𝑋3 )
No.
1 65 56 9 4225 3136 81 65*65=3640 65*9=585 56*9=504
2 72 58 11 5184 3364 121 4176 792 638
3 54 48 8 2916 2304 64 2592 432 384
4 68 61 13 4624 3721 169 4148 884 793
5 55 50 10 3025 2500 100 2750 550 500
6 59 51 8 3481 2601 64 3009 472 408
7 78 55 11 6084 3025 121 4290 858 605
8 58 48 10 3364 2304 100 2784 580 480
9 57 52 11 3249 2704 121 2964 627 572
10 51 42 7 2601 1764 49 2142 357 294
Total 617 521 98 38753 27423 990 32495 6137 5178
Relationship b/w 1st and 2nd variable:-
𝒏 ∑ 𝑿𝟏 𝑿𝟐− (∑ 𝑿𝟏 )(∑ 𝑿𝟐 )
𝒓𝟏𝟐 =
√{𝒏 ∑ 𝑿𝟐𝟏 − (∑ 𝑿𝟏 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟐 − (∑ 𝑿𝟐 )𝟐 }
𝟏𝟎∗𝟑𝟐𝟒𝟗𝟓−𝟔𝟏𝟕∗𝟓𝟐𝟏 𝟑𝟒𝟗𝟑
𝒓𝟏𝟐 = = =
√(𝟏𝟎∗𝟑𝟖𝟕𝟓𝟑−𝟔𝟏𝟕𝟐 )(𝟏𝟎∗𝟐𝟕𝟒𝟐𝟑−𝟓𝟐𝟏𝟐 ) √𝟔𝟖𝟒𝟏∗𝟐𝟕𝟖𝟗
𝟑𝟒𝟗𝟑 𝟑𝟒𝟗𝟑
= =0.79960 =
̃ 0.80
√𝟏𝟗𝟎𝟕𝟗𝟓𝟒𝟗 𝟒𝟑𝟔𝟖.𝟎𝟏𝟒
Relationship b/w 1st and 3rd variable:-
𝒏 ∑ 𝑿𝟏 𝑿𝟑− (∑ 𝑿𝟏 )(∑ 𝑿𝟑 )
𝒓𝟏𝟑 =
√{𝒏 ∑ 𝑿𝟐𝟏 − (∑ 𝑿𝟏 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟑 − (∑ 𝑿𝟑 )𝟐 }
𝟏𝟎∗𝟔𝟏𝟑𝟕−𝟔𝟏𝟕∗𝟗𝟖 𝟗𝟎𝟒 𝟗𝟎𝟒
𝒓𝟏𝟑 = = = =
√(𝟏𝟎∗𝟑𝟖𝟕𝟓𝟑−𝟔𝟏𝟕𝟐 )(𝟏𝟎∗𝟗𝟗𝟎−𝟗𝟖𝟐 ) √𝟔𝟖𝟒𝟏∗𝟐𝟗𝟔 √𝟐𝟎𝟐𝟒𝟗𝟑𝟔
𝟗𝟎𝟒
=0.63527 =
̃ 0.64
𝟏𝟒𝟐𝟑.𝟎𝟎𝟐𝟒𝟔
Relationship b/w 2nd and 3rd variable:-
𝒏 ∑ 𝑿𝟐 𝑿𝟑− (∑ 𝑿𝟐 )(∑ 𝑿𝟑 )
𝒓𝟐𝟑 =
√{𝒏 ∑ 𝑿𝟐𝟐 − (∑ 𝑿𝟐 )𝟐 }{𝒏 ∑ 𝑿𝟐𝟑 − (∑ 𝑿𝟑 )𝟐 }
𝟏𝟎∗𝟓𝟏𝟕𝟖−𝟓𝟐𝟏∗𝟗𝟖
𝒓𝟐𝟑 = =𝟕𝟐𝟐⁄ =
𝟐 𝟐
√((𝟏𝟎∗𝟐𝟕𝟒𝟐𝟑−𝟓𝟐𝟏 ))(𝟏𝟎∗𝟗𝟗𝟎−𝟗𝟖 ) √𝟖𝟐𝟓𝟓𝟒𝟒
𝟕𝟐𝟐
=0.79
𝟗𝟎𝟖.𝟔𝟐𝟒𝟓
Square of correlation also known as
Relation-ship b/w two variables
coefficient of determination.
𝒓𝟐𝟏𝟐 = (𝟎. 𝟖𝟎 ∗ 𝟎. 𝟖𝟎)=0.64 𝒓𝟏𝟐 = 𝟎. 𝟖𝟎
𝒓𝟐𝟏𝟑 = (𝟎. 𝟔𝟒 ∗ 𝟎. 𝟔𝟒)=0.4096 𝒓𝟏𝟑 = 𝟎. 𝟔𝟒
𝒓𝟐𝟐𝟑 = (𝟎. 𝟕𝟗 ∗. 𝟕𝟗)=0.6241 𝒓𝟐𝟑 = 𝟎. 𝟕𝟗
Case 1st :-
2 2
𝑟12 + 𝑟13 − 2(𝑟12)(𝑟23 )(𝑟13)
𝑅1.23 =√ 2
1 − 𝑟23
0.64+0.4096−2∗0.80∗0.64∗0.79 0.24064
=√ =√ == √0.64017 = 0.80
1−0.6241 0.3759
Case 2nd :-
2 2
𝑟21 + 𝑟23 − 2(𝑟12 )(𝑟23)(𝑟13 )
𝑅2.13 =√ 2
1 − 𝑟13
0.64 + 0.6241 − 2 ∗ 0.80 ∗ 0.64 ∗ 0.79 0.45514
=√ =√
1 − 0.4096 0.5904
= √0.770901 = 0.8780
Case 3rd :-
2 2
𝑟31 + 𝑟32 − 2(𝑟12 )(𝑟23)(𝑟13)
𝑅3.12 =√ 2
1 − 𝑟12
0.4096 + 0.6241 − 2 ∗ 0.80 ∗ 0.64 ∗ 0.79 0.22474
=√ =√
1 − 0.64 0.36
= √0.6242 = 0.7901
