On Tue, 19 Dec 2000, Ray Webster wrote:
> I'm sure there's a simple explanation for this...
>
> The manuals outline an example of fitting multivariate loess models using
> the ethanol data (page 352 of Guide to Statistics V1, S-Plus 2000 for
> windows). They fit the following model:
>
> > ethanol.m <- loess(NOx ~ C * E, data = ethanol)
>
> As far as I can tell, this is equivalent to fitting the model without the
> interaction, ie NOx ~ C + E, or to be more precise, the fitted values of the
> latter model indicate an interaction term has been included also.
>
> Can someone tell me why this should be the case, and if I can actually
> test for an interaction between two predictors with these models (using
> anova function)?
The formula notation is confusing you (and `multivariate' is confusing me,
as this is like multiple linear regression, not multivariate linear
regression: seems the person who wrote that S-PLUS manual made this up as
it is not in the White Book, that I can see).
Loess fits a smooth curve or surface in all the input variables. It does
not fit an additive model (a sum of smooth functions of individual
inputs), but gam can.
`Interactions' of continuous variables like this make little sense,
but in a linear model they stand for cross-product terms, so
y ~ (x + z)^2
expands to
y ~ x + z + x:z
which is
y = b_0 + b_1 x + b_2 z + b_3 xz
and b_3 xz is the `interaction' x:z in the model. This is not usually
what you want!
For a gam, you can contrast
gam(y ~ lo(x) + lo(z))
gam(y ~ lo(x, z))
and even do an anova on the pair of models. However, as they are not
nested, that will make little sense.
--
Brian D. Ripley, ripley@stats.ox.ac.uk
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272860 (secr)
Oxford OX1 3TG, UK Fax: +44 1865 272595
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