Bob Wheeler wrote...
> It is a matter of emphasis. Both regression and
> ANOVA are techniques for dealing with linear
> models.
> ANOVA focuses on experiments where the
> variables may be random and where there may be
> several error terms. Regression on models which
> tend to have fixed, continuous, independent fixed
> variables and a single error term.
No, regression models can have stochastic and/or discrete
regressors. I can only agree that regression models have
single error term as compared to ANOVA with random effects.
But a more clear way to look at ANOVA with random effects is
as follows.
ANOVA with random effects is a modification of ANOVA with
fixed effects (which is just a plain linear regression) in
which coefficients are random. Hence, coefficients in the
former case are not the parameters to be estimated. Only
parameters of the distribution of the coefficients are
estimated. This gives efficiency gain relative to ANOVA with
fixed effects. But, of course, additional assumptions about
the distribution of coefficients are needed. This is
sometimes called "random coefficients model". Note, that
random coefficients model has a matrix of regressors (matrix
of plan) which is deterministic, fixed.
> To say that
> ANOVA is a special case of regression, in effect
> redefines regression as "linear model analysis,"
> which can be done, but is a stretch.
OK, let it be "linear model analysis".
> As definitive
> a statement as is likely to be find is given by
> Scheff� in The analysis of variance (1959),
Regretfully, I've lost my Scheff�. But, as far as I
remember, Scheff� also uses regression analysis approach
to ANOVA. Am I wrong?
> and a
> good discussion of using multiple regression to
> perform some ANOVA calculations for fixed effect
> models is given in Draper and Smith's Applied
> regression analysis (1966).
Yes, this book greately reduced the confusion which I
previousely had with ANOVA.
> I speculate that this now seems confusing because
> texts in applied areas have perhaps dwelt too
> heavily on the mechanics of ANOVA calculations and
> mentioned the linear model part only briefly.
Exactly.
> The
> upshot is that students in those areas are
> surprised when the linear model part is called to
> their attention, and apparently Jacob Cohen (1968)
> felt strongly enough about it to write a paper
> explaining the connection. Standard statistical
> texts have always insisted on the mathematics, and
> Kempthorne for example in Design and analysis of
> experiments (1952) takes great pains to structure
> ANOVA in terms of linear models -- he even derives
> the normal equations.
I opened this discussion because Elliot Cramer wrote
"Unfortunately many people misuse ANOVA because they
think of it as regression analysis". Do you agree with him?
I think, to the contrary, that ANOVA _must_ be analysed
in the context of regression analysis to avoid confusion.
It might be better not to use term "ANOVA" at all.
Traditional ANOVA approach is vague and even misleading.
ANOVA tables in most cases are not informative. I was
once greatly pusseled by Statgraphics multiple regression
output with ANOVA table. Does anybody use those sums
of squares?
> Alexander Tsyplakov wrote:
> >
> > An interesting problem have arised during discussion of
the
> > origins of "eigenvalue".
> >
> > My own point of view is that ANOVA is just a particular
case
> > of regression analysis with dummy (1/0) regressors and
> > either fixed or random effects. Block orthogonality of
> > regression matrix in the special case of ANOVA makes it
> > possible to decompose the sum of squared residuals (and
> > variance) into several components.
> >
> > If people misuse the term ANOVA then what is it's
correct
> > meaning? Is it a statistical model which is different
from
> > regression model y=Xb+e? Then there must be some clear
> > formal discription.
> >
> > -----------------------------
> > Alexander Tsyplakov
> > Novosibirsk State University
> > http://www.nsu.ru/ef/tsy/
> >
> > Elliot Cramer wrote...
> > > Werner Wittmann
<[EMAIL PROTECTED]>
> > wrote:
> > > : inverting the
> > > : correlation matrix to get the effects was too
> > complicated to compute by
> > > : hand, so Sir Ronald developed the ANOVA shortcut.
> > >
> > > hardly. They do have some mathematics in common
(through
> > use of dummy
> > > variables which some of us think is for dummies).
they
> > are comceptually
> > > completely different/ Unfortunately many people
misuse
> > ANOVA because they
> > > think of it as regression analysis.
> >
> > > : I'm always teasing my colleagues and students, if
you
> > spent one year
> > > : learning ANOVA and one year multiple regression
you've
> > wasted almost one
> > > : year of your life.
> >
> > > you can learn the mathematics of regression analysis
in 10
> > minutes but
> > > you're still a long way from understanding either it
or
> > ANOVA
>
> --
> Bob Wheeler --- (Reply to: [EMAIL PROTECTED])
> ECHIP, Inc.
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