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Re: st: about residuals and coefficients


From   David Hoaglin <dchoaglin@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: about residuals and coefficients
Date   Mon, 2 Sep 2013 08:25:35 -0400

Hi, Kayla.

Your questions seem to be fairly basic ones about multiple regression.

I hope you have looked at the three scatterplots (y vs. x1, y vs. x2,
and x2 vs. x1) to see how the data behave.

R^2 provides information equivalent to
[sum(residual^2)]/[sum((y-ybar)^2)], often abbreviated as SSE/SST.
R^2 = 1 - (SSE/SST) is the percentage of the (squared) variation in y
that is accounted for by the regression model (i.e., by x1 and x2
together).

In general, it is not possible to express R^2 as the sum of a
percentage accounted for by x1 and a percentage accounted for by x2.
The obstacle is correlation (in the data) between x1 and x2.  Thus,
you can say how much variation x2 accounts for after adjustment for
x1, and you can say how much variation x1 accounts for after
adjustment for x2.  To get those percentages, you can fit the simple
regressions involving only x1 and only x2 and subtract the values of
R^2 for those regressions from the value of R^2 for the regression
involving both x1 and x2.  If x1 and x2 are uncorrelated (technically,
orthogonal), usually by design, it is possible to express the R^2 of
the two-variable model as the sum of the contributions of x1 and x2.

I hope this discussion helps.

David Hoaglin

On Mon, Sep 2, 2013 at 5:57 AM, Kayla Bridge <kayla.bridge@outlook.com> wrote:
> Dear all,
> I am currently running a simple regression, and try to explain the coefficients. The model and estimation results are the following.
> y=5.41+1.24*x1+.28*x2, R2=0.7, N=20
>  (0.58) (3.4)   (2.56)
> The t-stats are in parentheses.
> I'd like to know how much (in terms of percentage) of the change in y is accounted for by change in x1, and how much change in y by change in x2.
> Another question is: can I use [sum(residual^2)]/[sum((y-ybar)^2)], where ybar is the mean value of the dependent variable, to say something about percentage of residual, like smaller percentage of residuals implies that x1 and x2 are good explanatory factors for y?
> Any suggestion is greatly appreciated.
> Best,
> Kayla

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