Page 219 - DMGT209_QUANTITATIVE_TECHNIQUES_II
P. 219
Quantitative Techniques-II
Notes Square of R is known as the coefficient of multiple determination.
i×jk
1 S 2 i jk
2
R 2 = S S 2
1
i jk 2 i i jk 2
S S
i i
S 2
i jk
It may be noted here that 2 is proportion of unexplained variation. Thus, we can also write
S
i
2
x i jk
R 2 1
i jk 2 .
x i
Further, we can write R 2 i jk in terms of the simple correlation coefficients.
S 2 1 r 2 r r 2r r r r r 2r r r
2
2
2
2
R 2 i jk = 1 i ij ik jk ij ik jk ij ik 2 ij ik jk
S 2 i 1 r jk 2 1 r jk
S 2 x 2
Notes If there are m variables, R 2 1 1 23....m 1 1 23....m
1 23....m 2 2
S 1 x 1
Coefficient of Multiple Correlations
The multiple correlation coefficient generalizes the standard coefficient of correlation. It is used
in multiple regression analysis to assess the quality of the prediction of the dependent variable.
It corresponds to the squared correlation between the predicted and the actual values of the
dependent variable. It can also be interpreted as the proportion of the variance of the dependent
variable explained by the independent variables. When the independent variables (used for
predicting the dependent variable) are pair wise orthogonal, the multiple correlation coefficient
is equal to the sum of the squared coefficients of correlation between each independent variable
and the dependent variable. This relation does not hold when the independent variables are not
orthogonal. The significance of a multiple coefficient of correlation can be assessed with an F
ratio. The magnitude of the multiple coefficient of correlation tends to overestimate the
magnitude of the population correlation, but it is possible to correct for this overestimation.
!
Caution Strictly speaking we should refer to this coefficient as the squared multiple
correlation coefficient, but current usage seems to ignore the adjective “squared,” probably
because mostly its squared value is considered.
Task Distinguish between correlation and regression.
Self Assessment
Fill in the blanks:
5. ....................... correlation is used as a measure of the degree of association in situations
where the nature of population, from which data are collected, is not known.
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