Francis_Bruey@cytec.com writes:
> Consider the following experiment:
> Factor A with 8 levels applied to 24 experimental units (3 per
> level) in completely randomized fashion.
> Each unit is measured at 4 stages (initially and then after each of
> 3 consecutive applications of a standardized stress).
> Dataframe has 96 rows, with columns Y (numeric response), uniq.unit
> (factor, unique unit identifier), A (factor), and stage (factor).
> Fit with aov() using model:
> aov(Y ~ (A/uniq.unit)*stage)
> has zero residual error.
> Fit with lme() using model:
> lme(fixed = Y ~ A*stage, ran = ~ stage | uniq.unit)
> has nonzero residual error. Is this model not saturated (am I
> misunderstanding lme syntax)? If it is saturated, what is the
> residual error estimating?
> Using S-Plus 2000 on PC with Win95.
Although there are four random effects being estimated for each
uniq.unit, the number of parameters associated with these random
effects is only 10 (four variances and six covariances).
The random effects themselves are not completely free to reduce the
sum of squares to zero. The presence of the variance-covariance
matrix for the random effects imposes a "shrinkage" effect that brings
them closer to zero than corresponding fixed effects would be.
Another way of thinking of this is that the random effects are the
solutions of a penalized least squares problem, not an ordinary least
squares problem, as described in
http://nlme.stat.wisc.edu/CompMulti.pdf
The residual sum of squares is the amount given up for this shrinkage
or penalty.
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