[R-sig-ME] crossed effects formulation

Douglas Bates bates at stat.wisc.edu
Mon Feb 4 15:05:27 CET 2008

On Feb 4, 2008 6:15 AM, Nikos Fyllas <nfyllas at gmail.com> wrote:
> Dear all,

> I have a model like: y.m1 <- lmer(y~1+factor(Site)+(1|Species))

> so a fixed site effect is crossed with a random effect of  a species.

I'm not sure I understand what you mean.  That model is an additive
model with fixed effects for Site and random effects for Species.

The random effects have an expected value of zero and, by default, the
Site factor will be represented as an "Intercept" and N-1 contrasts
labelled Site2, Site3, ..., SiteN.  The "Intercept" coefficient is the
prediction of the mean response at Site1 for a Species effect of 0.
The coefficient labeled Site2 is the difference between Site2 and
Site1.  That is, Site 1 is the reference level and all the other
coefficients are relative to that reference level.

> Does the actual equation yield to:
> y(site, species) = mean + SIte1_effect + Site2_effect + ...... SiteN_effect
> + Species_effect + error

Not quite.  The parameters are as I described them above.

> ie allowing different Sites to have different variances?

No, the different Sites have different predicted means but not
different variances.

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