Re: "...the sum of squares removed by fitting..."
- From: David Winsemius <doe_snot@xxxxxxxxxxx>
- Date: Tue, 26 Sep 2006 18:46:29 -0500
kj <socyl@xxxxxxxxxxxxxxxxx> wrote in
news:efbt8a$be7$1@xxxxxxxxxxxxxxxxx:
On p. 63 of W. J. Ewens' "Mathematical Population Genetics, v. 1",
the author uses the phrase "the sum of squares removed by fitting",
which I don't understand. FWIW here's the full sentence, rendered
in bright LaTeX:
Standard regression theory shows that the sum of squares removed
by fitting the $\alpha_j$ values in (2.57), that is the additive
genetic variance $\sigma_A^2$, is given by
\begin{equation}
\sigma_A^2 = 2 \sum_u x_u a_u \alpha_u .
\end{equation}
The context is the computation of a set of coefficients $\alpha_1,
..., \alpha_k$, called the "average effects", by a least-squares
procedure.
I can *guess* possible meanings for what the author's phrase, but
in any can't figure out how to derive the expression above. The
author says this stuff is standard, but I can't find it in my stats
book (it could be there under a different guise, though). Where
can I find a more explicit derivation of this "sum of squares
removed by fitting"?
Suggestion: search for a phrase along the lines of "model sum of
squares" or "regression sum of squares" or "partitioned sum of squares".
When you fit a model, you get a model_sum_of_squares and an
error_sum_of_squares which sum to a total_sum _of_squares or the total
variance. The F statistic is just the expected distribution of the ratio
of "model sum of squares" to the error sum of squares in a linear model.
Another phrase for the "error sum of squares" is the "residual sum of
squares". Your citation suggests (but does not imply) an underlying
linear model, but even if it is not using a linear model your searches
may be more fruitful with the altermative search phrases. If you want to
get a more specific response, you will need to post a link to the article
or describe the methods in more detail. (Or wait for another respondent
to offer more explication.)
--
David Winsemius
.
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