Partial correlation and multiple correlation:
Correlation:
If two quantities vary in such a way that movements in one are
accompanied by movements in the other, these quantities are
correlated. For example, there exists some relationship between
age of husband and age of wife.
The correlation analysis refers to the techniques used in
measuring the closeness of the relationship between the variables
Use:
Most of the variables show some kind of relationship, for example
there is relationship between price and supply, Income and
expenditure etc. with the help of correlation analysis we can
measure in one figure the degree of relationship existing between
the variables.
Types of correlation:
1. Positive or negative
2. Simple, partial and Multiple
3. Linear and non-linear
Simple, partial and Multiple
When only two variables are studied it is a problem of
simple correlation. When three or more variables are
studied it is a problem of either multiple correlation
or partial correlation.
Partial correlation
If there are three variables X1, X2 and X3 there will be three
coefficients of partial correlation, each studying the relationship
between two variables when the third is held constant. If we denote by
r12.3 ie., the coefficient of partial correlation between X1 and X2
keeping X3 constant, it is calculated as
r12.3 
r12  r13r23
2
1  r132 1  r23
r13.2 
r13  r12 r23
2
1  r12 2 1  r23
r23.1 
r23  r12 r13
2
1  r12 2 1  r13
r12  0.7 ,
Problem: In a trivariate distribution , it is found that
r13  0.61 and
r23  0.4 .
Find
the
partial
correlation
coefficients.
r12.3 
r13.2 
r23.1 
r12  r13r23
1  r13
2
1  r232

0.7  (0.61)(0.4)
1  (0.61)
r13  r12 r23
2
1  r12 2 1  r23
=0.504
r23  r12 r13
1  r12 2 1  r132
=-0.048
2
1  (0.4)
2
 0.628
2. Is it possible to get the following from a set of experimental data?
r12  0.6 , r13  0.5 and r23  0.8
r12.3 
r12  r13r23
1  r13
2
1  r232

0.6  (0.5)(0.8)
1  (0.5)
2
1  (0.8)
2
 1.923
Since the value of r12.3 is greater than one, there is some inconsisting
in the given data.
Multiple correlation
The coefficients of multiple correlation with three variables X1 , X2
and X3 are R1.23 , R2.13 and R3.12 .
R1.23 is the coefficient of multiple correlation related to X1 as a
dependent variable X2 and X3 as two independent variables and it can
be expressed in terms of r12 , r23 and r13 as
r122  r232  2r12 r23r13
1  r132
R1.23
r122  r132  2r12 r23r13

; R2.13 
1  r232
R3.12
r132  r232  2r12 r23r13

;It can be noted that R1.23  R1.32 ,
2
1  r12
R2.13  R2.31 , R3.12  R3.21
Problem:The following Zero –order correlation coefficients are given:
r12  0.98 , r13  0.44 and r23  0.54 . Calculate multiple correlation
coefficient treating first variable as dependent and second & third
variables as independent.
R1.23
r122  r132  2r12 r23r13
=0.986

1  r232
Multiple linear Regressions
If the number of independent variables in a regression model is more
than one, then the model is called as multiple regression. In fact,
many of the real-world applications demand the use of multiple
regression models.
A sample application is as stated below:
Y  bo  b1 X 1  b2 X 2  b3 X 3  b4 X 4
Where Y represents the economic growth rate of country, X1
represents the time period, X2 represents the size of the populations
of the country, X3 represents the level of employment in percentage,
X4 represents the percentage of literacy, b is the intercept and b1 , b2 ,
b3 and b4 are the slopes of the variables X1, X2, X3 and X4
respectively. In this regression model X1, X2, X3 and X4 are the
independent variables and Y is the dependent variable.
Regression Model with Two independent variables using Normal
equations:
Suppose the number of independent variables is two, then
Y  bo  b1 X 1  b2 X 2
Normal equations are
Y
 nbo  b1  X 1  b2  X 2
 Y X1
 bo  X 1  b1  X 1  b2  X 1 X 2
Y X 2
 bo  X 2  b1  X 1 X 2  b2  X 2
2
2
Where n is the total number of combinations of observations. The
solution to the above set of simultaneous equations will form the
results for the coefficients b , b1 and b2 of the regression model.
Problem 1: The annual sales revenue(in crores of rupees ) of a product
as a function of sales force(number of salesmen) and annual
advertising expenditure(in lakhs of rupees) for the past 10 year are
summarized in the following table.
Annual sales 20
revenue Y
Sales force 8
X1
23
25
27
21
29
22
24
27
35
13
8
18
23
16
10
12
14
20
Annual
28
advertising
expenditures
X2
23
38
16
20
28
23
30
26
32
Let the regression model be Y  bo  b1 X 1  b2 X 2
Y
20
23
25
27
21
29
22
24
27
35
 y  253
X1
8
13
8
18
23
16
10
12
14
20
 X 1  142
X2
28
23
38
16
20
28
23
30
26
32
 X 2  264
X12
64
169
64
324
529
256
100
144
196
400
X22
784
529
1444
256
400
784
529
900
676
1024
 X12  2246
 X 2 2  7326
X1X2
224
229
304
288
460
448
230
360
364
640
 X1X 2  3617
YX1
160
299
200
486
483
464
220
288
378
700
YX1 142
YX2
560
529
950
432
420
812
506
720
702
1120
YX 2  142
Substituting the required values in the normal equations, we get the
following simultaneous equations
253  10bo  142b1  264b2
3678  142bo  2246 b1  3617 b2
6751  264bo  3617 b1  7326 b2
The solution to the above set of simultaneous equation is
bo  5.1483, b1  0.6190 and b2  0.4304
Therefore, the regression model is Y  5.1483  0.6190 X1  0.4304 X 2