Lecture 5

Multiple Correlation
➢ If information on two variables like
✓ height and weight,
✓ income and expenditure,
✓ demand and supply, etc.
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,
✓ crimes in a city may be influenced by illiteracy, increased population
and unemployment in the city, etc.
✓ The production of a crop may depend upon amount of rainfall, quality
of seeds, quantity of fertilizers used and method of irrigation, etc.
✓ 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.

➢ In fact, multiple correlation is the study of combined influence of two or

more variables on a single variable.

➢ Suppose, X1 , X2 and X3 are three variables having observations on N

individuals or units. Then multiple correlation coefficient of X1 on X2 and X3
is the simple correlation coefficient between X1 and the joint effect of X2
and X3.
➢ It can also be defined as the correlation between X1 and its estimate based
on X2 and X3.

1
 ➢ Multiple correlation coefficient is the simple correlation coefficient
between a variable and its estimate 𝒆𝟏.𝟐𝟑 . It is usually denoted by 𝑅1.23 and
is defined as
2 +𝑟 2 −2𝑟 𝑟 𝑟
𝑟12
2 13 12 13 23
𝑅1.23 = 2
1−𝑟23

2 2
𝑟12 + 𝑟13 − 2𝑟12 𝑟13 𝑟23
𝑅1.23 =√ 2
1 − 𝑟23

which is required formula for multiple correlation coefficient.

where, 𝑟12 is the total/zero order correlation coefficient between variable X1 and X2,
𝑟23 is the total/zero correlation coefficient between variable X2 and X3,
𝑟13 is the total/zero correlation coefficient between variable X1 and X3.
Example: From the following data, obtain 𝑅1.23 and 𝑅2.13 .
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
Solution: To obtain multiple correlation coefficients 𝑅1.23 and 𝑅2.13 , we use
following formulae
2 +𝑟 2 −2𝑟 𝑟 𝑟
𝑟12
2 13 12 13 23
𝑅1.23 = 2 and
1−𝑟23
2 +𝑟 2 −2𝑟 𝑟 𝑟
𝑟12
2 23 12 13 23
𝑅2.13 = 2
1−𝑟13

We need 𝑟12 , 𝑟13 and 𝑟23 which are obtain from the following table:
Sl. # 𝑋1 𝑋2 𝑋3 (𝑋1 )2 (𝑋2 )2 (𝑋3 )2 𝑋1 𝑋2 𝑋1 𝑋3 𝑋2 𝑋3
1 65 56 9 4225 3136 81 3640 585 504
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 S472 408
7 78 55 11 6084 3025 121 4290 858 608
8 58 48 10 3364 2304 100 2784 580 480

2
 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

Now we get total correlation coefficient 𝑟12 , 𝑟13 and 𝑟23

𝑁(∑ 𝑋1 𝑋2 ) − (∑ 𝑋1 ∑ 𝑋2 )
𝑟12 =
√{𝑁(∑ 𝑋12 ) − (∑ 𝑋1 )2 }{𝑁(∑ 𝑋22 ) − (∑ 𝑋2 )2 }
(10 × 32495) − (617)(521)
𝑟12 =
√{(10 × 38753) − (617)2 }{(10 × 27423) − (521)2 }
3493 3493
𝑟12 = = = 0.80
√{684} × {2789} 4368.01

𝑁(∑ 𝑋1 𝑋3 ) − (∑ 𝑋1 ∑ 𝑋3 )
𝑟13 =
√{𝑁(∑ 𝑋12 ) − (∑ 𝑋1 )2 }{𝑁(∑ 𝑋32 ) − (∑ 𝑋3 )2 }
(10 × 6137) − (617)(98)
𝑟13 =
√{(10 × 38753) − (617)2 }{(10 × 990) − (98)2 }
904 904
𝑟13 = = = 0.64
√{684} × {296} 1423.00

and
𝑁(∑ 𝑋2 𝑋3 ) − (∑ 𝑋2 ∑ 𝑋3 )
𝑟23 =
√{𝑁(∑ 𝑋22 ) − (∑ 𝑋2 )2 }{𝑁(∑ 𝑋32 ) − (∑ 𝑋3 )2 }
(10 × 5178) − (521)(98)
𝑟23 =
√{(10 × 27423) − (521)2 }{(10 × 990) − (98)2 }
722 722
𝑟23 = = = 0.79
√{2789} × {296} 908.59

3
Now, we calculate 𝑅1.23
We have, 𝑟12 = 0.80, 𝑟13 = 0.64, 𝑎𝑛𝑑 𝑟23 = 0.79, 𝑡ℎ𝑒𝑛
2 2
2
𝑟12 + 𝑟13 − 2𝑟12 𝑟13 𝑟23
𝑅1.23 = 2
1 − 𝑟23
0.802 + 0.642 − 2 × 0.80 × 0.64 × 0.79
=
1 − 0.792
0.64 + 0.41 − 0.81
=
1 − 0.62
2
0.24
𝑅1.23 = = 0.63
0.38
Then
𝑅1.23 = 0.79
2 2
2
𝑟12 + 𝑟23 − 2𝑟12 𝑟13 𝑟23
𝑅2.13 = 2
1 − 𝑟13
0.802 + 0.792 − 2 × 0.80 × 0.64 × 0.79
=
1 − 0.642
0.64 + 0.62 − 0.81
=
1 − 0.49
0.45
= = 0.88
0.51
Thus,
𝑅2.13 = 0.94

4
Theorem: The coefficient of correlation is independent of change of scale and
origin of the variable X and Y. Change of origin means some value has been added
or subtracted in the observation.
Example: The following data is given:
𝑋1 60 68 50 66 60 55 72 60 62 51
𝑋2 42 56 45 64 50 55 57 48 56 42
𝑋3 74 71 78 80 72 62 70 70 76 65
Obtain 𝑅1.23 , 𝑅2.13 , 𝑎𝑛𝑑 𝑅3.12
Solution: To obtain multiple correlation coefficients 𝑅1.23 , 𝑅2.13 , 𝑎𝑛𝑑 𝑅3.12 , we use
following formulae
𝒓𝟐𝟏𝟐 +𝒓𝟐𝟏𝟑 −𝟐𝒓𝟏𝟐 𝒓𝟏𝟑 𝒓𝟐𝟑
𝑹𝟐𝟏.𝟐𝟑 = 𝟏−𝒓𝟐𝟐𝟑

𝒓𝟐𝟏𝟐 +𝒓𝟐𝟐𝟑 −𝟐𝒓𝟏𝟐 𝒓𝟏𝟑 𝒓𝟐𝟑

𝑹𝟐𝟐.𝟏𝟑 = and
𝟏−𝒓𝟐𝟏𝟑

𝒓𝟐𝟏𝟑 +𝒓𝟐𝟐𝟑 −𝟐𝒓𝟏𝟐 𝒓𝟏𝟑 𝒓𝟐𝟑

𝑹𝟐𝟑.𝟏𝟐 =
𝟏−𝒓𝟐𝟏𝟐

We need 𝑟12 , 𝑟13 and 𝑟23 which are obtain from the following table:
Sl. # 𝑋1 𝑋2 𝑋3 𝑑1 𝑑2 𝑑3 𝑑12 𝑑22 𝑑32 𝑑1 𝑑2 𝑑1 𝑑3 𝑑2 𝑑3
= 𝑋1 − 60 = 𝑋2 − 50 = 𝑋3 − 70
1 60 42 74 0 -8 4 0 64 16 0 0 -32
2 68 56 71 8 6 1 64 36 1 48 8 6
3 50 45 78 -10 -5 8 100 25 64 50 -80 -40
4 66 64 80 6 14 10 36 196 100 84 60 140
5 60 50 72 0 0 2 0 0 4 0 0 0
6 55 55 62 -5 5 -8 25 25 64 -25 40 -40
7 72 57 70 12 7 0 144 49 0 84 0 0
8 60 48 70 0 -2 0 0 4 0 0 0 0
9 62 56 76 2 6 6 4 36 36 12 12 36
10 51 42 65 -9 -8 -5 81 64 25 72 45 40
Total 4 15 18 454 499 310 325 85 110

Here, we can also use shortcut method to calculate 𝑟12 , 𝑟13 and 𝑟23
Let 𝑑1 = 𝑋1 − 𝑋̅1 = 𝑋1 − 60
𝑑2 = 𝑋2 − 𝑋̅2 = 𝑋2 − 50
𝑑3 = 𝑋3 − 𝑋̅3 = 𝑋3 − 70
5
We now get the total/zero order correlation coefficient 𝑟12 , 𝑟13 and 𝑟23
𝑁(∑ 𝑑1 𝑑2 ) − (∑ 𝑑1 ∑ 𝑑2 )
𝑟12 =
√{𝑁(∑ 𝑑12 ) − (∑ 𝑑1 )2 }{𝑁(∑ 𝑑22 ) − (∑ 𝑑2 )2 }
(10 × 325) − (4)(15)
𝑟12 =
√{(10 × 454) − (4)2 }{(10 × 499) − (15)2 }
3190 3190
𝑟12 = = = 0.69
√{4524}{4765} 4642.94

𝑁(∑ 𝑑1 𝑑3 ) − (∑ 𝑑1 ∑ 𝑑3 )
𝑟13 =
√{𝑁(∑ 𝑑12 ) − (∑ 𝑑1 )2 }{𝑁(∑ 𝑑32 ) − (∑ 𝑑3 )2 }
(10 × 85) − (4)(18)
𝑟13 =
√{(10 × 454) − (4)2 }{(10 × 310) − (18)2 }
778 778
𝑟13 = = = 0.22
√{4524}{2776} 3543.81

and
𝑁(∑ 𝑑2 𝑑3 ) − (∑ 𝑑2 ∑ 𝑑3 )
𝑟23 =
√{𝑁(∑ 𝑑22 ) − (∑ 𝑑2 )2 }{𝑁(∑ 𝑑32 ) − (∑ 𝑑3 )2 }
(10 × 110) − (15)(18)
𝑟23 =
√{(10 × 499) − (15)2 }{(10 × 310) − (18)2 }
830 830
𝑟13 = = = 0.23
√{4765}{2776} 3636.98

6
We now calculate 𝑹𝟏.𝟐𝟑
We have, 𝑟12 = 0.69, 𝑟13 = 0.22, 𝑎𝑛𝑑 𝑟23 = 0.23, 𝑡ℎ𝑒𝑛
2 2
2
𝑟12 + 𝑟13 − 2𝑟12 𝑟13 𝑟23
𝑅1.23 = 2
1 − 𝑟23
0.692 + 0.222 − 2 × 0.69 × 0.22 × 0.23
=
1 − 0.232
0.48 + 0.05 − 0.53
=
1 − 0.05
2
0.45
𝑅1.23 = = 0.48
0.95
Then
𝑅1.23 = 0.69
2 2
2
𝑟12 + 𝑟23 − 2𝑟12 𝑟13 𝑟23
𝑅2.13 = 2
1 − 𝑟13
0.692 + 0.232 − 2 × 0.69 × 0.22 × 0.23
=
1 − 0.222
0.46
= = 0.48
0.95
Thus,
𝑅2.13 = 0.69
We now calculate 𝑹𝟐.𝟏𝟑
2 2
2
𝑟12 + 𝑟13 − 2𝑟12 𝑟13 𝑟23
𝑅2.13 = 2
1 − 𝑟13
0.692 + 0.222 − 2 × 0.69 × 0.22 × 0.23
=
1 − 0.222
2
0.454
𝑅3.12 = = 0.478
0.95
𝑅2.13 = 0.69

7
We now calculate 𝑹𝟑.𝟏𝟐
2 2
2
𝑟13 + 𝑟23 − 2𝑟12 𝑟13 𝑟23
𝑅3.12 = 2
1 − 𝑟12
0.222 + 0.232 − 2 × 0.69 × 0.22 × 0.23
=
1 − 0.692
2
0.03
𝑅3.12 = = 0.06
0.52
Then
𝑅3.12 = 0.25

Properties of Multiple Correlation Coefficient

The following are some of the properties of multiple correlation coefficient:
1. Multiple correlation coefficient is the degree of association between observed
value of the dependent variable and its estimate obtained by multiple
regression.
2. Multiple correlation coefficient lies between 0 and 1.
3. If multiple correlation coefficient is 1, then association is perfect and multiple
regression equation may said to be perfect prediction formula.
4. If multiple correlation coefficient is 0, dependent variable is uncorrelated with
other independent variables. From this, it can be concluded that multiple
regression equation fails to predict the value of dependent variable when
values of independent variables are known.
5. Multiple correlation coefficient is always greater than or equal to any
total/zero order correlation coefficient. If 𝑅1.23 is the multiple correlation
coefficient than 𝑅1.23 ≥ 𝑟12 𝑜𝑟 𝑟13 𝑜𝑟 𝑟23 , and
6. Multiple correlation coefficient obtained by method of least squares would
always be greater than the multiple correlation coefficient obtained by any
other method.