Re: "...the sum of squares removed by fitting..."

kj <socyl@xxxxxxxxxxxxxxxxx> wrote in
news:efchnv$mrl$1@xxxxxxxxxxxxxxxxx:

In <Xns984AC92A71C0Ddwtttttt@xxxxxxxxxxxxxx> David Winsemius
<doe_snot@xxxxxxxxxxx> writes:

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".

Minor correction: The F statistic is compared to the ratio of mean model
sum of squares over mean residual (error) 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.

Thanks. Here's a link:

http://www.amazon.com/gp/reader/0387201912

Well, almost. The above is a link to the amazon.com reader for
the book in question, but one still needs to search for "regression",
and choose the hit on p. 63. NB: the use of the above feature
requires having an account with Amazon.com.

OK, I get the idea of splitting the total variance as a sum of
model squares and a sum of error squares. Thank you. Still, I
don't see how the author derived the particular expression I quoted
in my original post. (BTW, I assume this expression is for the
model sum of squares, not the error sum of squares.)

That is how I read it.

I thought
that there may be a standard expression for the model sum of squares,
and that the author's derivation for the equation transcribed above
consisted basically of plugging in the right values in this general
expression. At any rate, that's what I'm hoping to find.

Idea #1
The material I saw did not have much statistical content ... no F
statistics or considerations of probability distributions. Perhaps the
statistics content comes later. I did not see where expression 2.61 was
obvious, either. From my understanding of regression, sigma_squared (of
anything) would need to be a quadratic expression in X (or X's). You may
want to take the definition of a_u given in eq 2.60 on the prior page and
use it to expand the equation that you question. You may see that it
becomes a quadratic form of the data.

Idea #2:
Send an abbreviated record of your struggles to the email address in his
webpage:
http://www.bio.upenn.edu/faculty/ewens/

--
David Winsemius
.

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