Page 287 - DMGT404 RESEARCH_METHODOLOGY

Unit 13: Multivariate Analysis

13.1 Multivariate Analysis Notes

In multivariate analysis, the number of variables to be tackled are many.

Example: The demand for television sets may depend not only on price, but also on the
income of households, advertising expenditure incurred by TV manufacturer and other similar
factors. To solve this type of problem, multivariate analysis is required.

Classification

Multiple-variate analysis: This can be classified under the following heads:

1. Multiple regression
2. Discriminant analysis
3. Conjoint analysis
4. Factor analysis
5. Cluster analysis

6. Multidimensional scaling.

13.1.1 Multiple Regression

In the case of simple linear regression, one variable, say, X is affected by a linear combination
1
of another variable X (we shall use X and X instead of Y and X used earlier). However, if X is
2 1 2 1
affected by a linear combination of more than one variable, the regression is termed as a
multiple linear regression.
Let there be k variables X , X ...... X , where one of these, say X , is affected by the remaining k –
1 2 k j
1 variables. We write the typical regression equation as
X = a + b X + b X +......(j = 1, 2,.... k).
jc j×1, 2, .... j–1, j + 1, .... k j 1.2,3, .... j –1, j + 1, ....k 1 j 2.1, 3, .... j – 1, j + 1, ....k 2
Here a , b ...... etc. are constants. The constant a is interpreted as the value of X
j.1,2, .... j1.2, 3, .... j.1,2, .... j
when X , X , ..... X , X ..... X are all equal to zero. Further, b , b etc.,
2 3 j-1 j + 1 k j1.2,3, .... j–1, j + 1, ....k j2.1,3, .... j –1, j +1, ....k
are (k – 1) partial regression coefficients of regression of X on X , X ...... X , X ...... X .
j 1 2 j – 1 j + 1 k
For simplicity, we shall consider three variables X , X and X . The three possible regression
1 2 3
equations can be written as
X = a + b X + b X .... (1)
1c 1.23 12.3 2 13.2 3
X = a + b X + b X .... (2)
2c 2.13 21.3 1 23.1 3
X = a + b X + b X .... (3)
3c 3.12 31.2 1 32.1 2
Given n observations on X , X and X , we want to find such values of the constants of the
1 2 3
n
2
å
regression equation so that ( X - X ijc ) , j = 1, 2, 3, is minimised.
ij
i= 1
For convenience, we shall use regression equations expressed in terms of deviations of variables
from their respective means. Equation (1), on taking sum and dividing by n, can be written as
å X 1c = + å X 2 + å X 3
n a 1.23 b 12.3 n b 13.2 n

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