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Coefficient of multiple determination
and
Partial Correlation-Coefficient
THE
COEFFICIENT OF MULTIPLE DETERMINATION
The coefficient of multiple determination, R2, is defined as
the proportion of the total variation in Y "explained" by the
multiple regression of Y on X1 and X2, and it can be
calculated by
Explained
variance
where k is number of
estimated parameters
Unexplained
variance
.
Example
The
calculated F ratio or statistic for the case of a simple regression and
for a regression with n=15, k=3 (ie a multiple regression), we get:
where
the subscripts of F denote the number of degrees of freedom in the
numerator and denominator, respectively. In this simple regression case,
F1, n-2 = t2n-2 for the
same level of significance. For a multiple regression with n= 15 and k=13,
.
It is possible for the calculated F statistic to be "large" and yet none of the estimated parameters to be statistically significant. This might occur when the independent variables are highly correlated with each other. The F test is often of limited usefulness because it is likely to reject the null hypothesis, regardless of whether or not the model explains "a great deal" or the variation of Y.
Since
the inclusion of additional independent or explanatory variables is likely
to increase the
for the same
, R2 increases. To take into consideration the reduction in the
degrees of freedom as additional independent or explanatory variables are
added, the adjusted R2, or
is computed
where
n= the number of observations
k= the number of parameters estimated
PARTIAL-CORRELATION
COEFFICIENT
The partial-correlation coefficient measures the net correlation between
the dependent variable and one independent variable after excluding the
common influence of (ie, holding constant) the other independent variables
in the model. For example rYX1X2
is the partial correlation between Y and X1, after removing the
influence of X2 from both Y and X1
where
rYX1= simple-correlation coefficient between Y and X1
and rYX2 and rX1X2 are analogously defined.
Partial-correlation coefficient range in value from -1 to +1 (as do
simple-correlation coefficients), have the sign of the corresponding
estimated parameter and are used to determine the relative importance of
the different explanatory variables in a multiple regression.
For
example, rYX1X2= -1 refers to the case where there is an exact
or perfect negative linear relationship between Y and X1 after
removing the common influence of X2 from both Y and X1.
However, rYX1X2= 1 indicates a perfect positive linear net
relationship between Y and X1. And rYX1X2= 0
indicates no linear relationship between Y and X1 when the
common influence of X2 has been removed from both Y and X1.
As a result, X1 can be omitted from the regression.
The
sign of partial correlation coefficients is the same as that of the
corresponding estimated parameter. For example, for the estimated
regression equation
, rYX1X2 has the same sign as
and rYX2X1 has the same sign as
.
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© 2002
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Evgenia Vogiatzi <<Previous Next>>
