On Mar 21, 2012, at 09:04 , Rolf Turner wrote:

> On 21/03/12 20:19, Gerrit Eichner wrote:
>> Dear Ben, or anybody else, of course,
>> 
>> I'd be grateful if you could point me to a reference (different from ch. 4 
>> "Linear models" in "Statistical Models in S" (Chambers & Hastie (1992))) 
>> regarding the (asserted F-)distributional properties of the test statistic 
>> (used, e.g., by anova.lm()) to compare model 1 with model 2 using the MSE of 
>> model 3 in a sequence of three nested (linear) models? (A short 
>> RSiteSearch() and a google search didn't lead me far ...)
>> 
>> Thx in advance!
> 
> A good, if somewhat dry, reference on this is "Theory and Application of the
> Linear Model" by Franklin A. Graybill, Duxbury, 1976.
> 
> There are of course many, *many* other such books.

The whole thing is of course a fairly straightforward consequence of 
Fisher-Cochran's theorem. This says that, in the absence of systematic effects, 
the Sums of Squares in the ANOVA tables are proportional to independent 
chi-square variables. Hence, the ratio of any pair of Mean Squares or pooled 
Mean Squares has an F distribution. This holds for the sort of ANOVA tables 
that decompose the total sum of squares (i.e, not the drop1 or TypeII-IV style 
of table)

The convention of dividing by the "overall error" term rather than successively 
pooling terms according to model reductions is pervasive both in statistical 
literature and in software. This stems from the emphasis on balanced designs in 
the days before electronic computers: When you know that the sums of squares 
are independent of the testing order, you rather like to have the same property 
for the F tests, so that all conclusions can be conveniently read from a single 
ANOVA table. 

By and large, it doesn't make a whole lot of difference whether you gain a few 
denominator DF. It does, however, imply that the F tests with a common 
denominator are not independent, as opposed to the "proper" successive F tests.

-- 
Peter Dalgaard, Professor
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd....@cbs.dk  Priv: pda...@gmail.com

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