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Principle Foundations Home Page
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Goodness of Fit (R2) and Correlation-Coefficient (r) |
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The closer the observations fall to the regression line (ie the smaller
the residuals), the greater is the variation in Y "explained" by
the estimated regression equation. The total variation in Y is equal to
explained plus the residual variation:
Total variation
Explained variation
Residual variation in Y (or total
in Y (or regression
in Y (or error sum sum of squares) sum
of squares)
of squares) TSS
=
RSS
+ ESS Dividing both
sides by TSS gives
The
coefficient of determination, or R2, is then defined as the
proportion of the total variation in Y "explained" by the
regression of Y on X:
R2
can be calculated by
R2
ranges in value from 0 (when the estimated regression equation explains
none of the variation in Y) to 1 (when all points lie on the regression
line). Thus, R2
is unit -free and
Example When
for instance we have estimated the value of R2, and we have
found that R2= 0.9710 or 97.10% we say that: The
regression equation explains about 97% of the total variation in Y (eg.
corn output). The remaining 3% is attributed to factors included in the
error term.
The correlation-coefficient, r, measures the degree of association between two or more variables. In the two-variable case, the simple linear correlation coefficient for a set of sample observations is given by
Its value
varies form -1 to +1, ie
Copyright
© 2002
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