[R-sig-ME] Random or Fixed effects appropriate?

[Douglas Bates]

On 4/8/08, Reinhold Kliegl <reinhold.kliegl at gmail.com> wrote:
> Hi Andrew,
>
>
>  >  > >  lmer( y ~ x + (1|A) + (1|B) + (1|C) + (1|D) + C + x:C) #error:
>  >  > >  Downdated X'X is not positive definite, 82
>  >  > You cannot include C both as a random and a fixed effect
>  >
>  >
>  >  I do not believe that this is generally true.  See, for example,
>  >
>  >  > require(lme4)
>  >  > (fm1 <- lmer(Reaction ~ Days + Subject + (Days|Subject),  sleepstudy))
>  >
>  >  Therefore I am uncertain as to how you can draw this conclusion
>  >  without more information about the design (which the poster really
>  >  should have provided).
>
> I stand corrected. I thought this would force the between-subject
>  variance to zero. So what does the (substantially reduced)
>  between-subjects variance estimated in this model refer to? I noticed
>  that the residual variance stayed the same.

I would regard that result as a numerical accident rather than a
matter of design.  I don't think the same factor A should be present
in both the fixed effects and as a random effect of the form (1|A).
The same factor can be involved in the fixed effects and the random
effects as, for example, the factor Machine in the Machines  data (in
the MEMSS package) for models of the form

lmer(score ~ Machine + (1|Worker/Machine), Machines)

In those cases the role of Machine is different in the fixed effects
and in the random effects.

(Sorry Andrew but I think you are wrong on this one.  As always I am
willing to be convinced otherwise.)
>  >  > The following may not apply to your case, but it might: Sometimes
>  >  > people think that a nested/taxonomic design implies a random effect
>  >  > structure (e.g., schools, classes, students). This is not true. If you
>  >  > have only a few units for each factor, you are better off to specify
>  >  > it as a fixed-effects rather than a random-effects taxonomy. (Of
>  >  > course, you lose generalizability, but if you want this you should
>  >  > make sure you have sample that provides a basis for it.)
>  >
>  >  I can see the sense behind this position but sometimes a few units are
>  >  all that is available, and including them in a model as fixed effects
>  >  muddies the statistical waters, especially if they are the kinds of
>  >  effects that a model user will be unlikely to naturally condition upon.
>
>
> If you have only a few units, how can this muddy the statistical waters?
>
>
>  >
>  >  I do agree that if there are problems with model fitting and/or
>  >  interpretation when the design is rigorously followed, then a more
>  >  flexible approach can and should be adopted, and appropriate
>  >  allowances must be made.
>  >
>  >
>  >  > The interpretation of conditional modes (formerly knowns as BLUPs,
>  >  > that is "predictions") is a tricky business, especially with few
>  >  > units per levels.
>  >
>  >  Sorry, I think I've missed something.  In what sense are the
>  >  conditional modes formerly known as BLUPs?
>
>
> From: "Douglas Bates" <bates at stat.wisc.edu>
>  Date: September 27, 2007 5:00:41 PM GMT+02:00
>  The BLUPs of the random effects (actually as Alan James described
>  the situation, "For a nonlinear model these are just like the BLUPs
>  (Best Linear Unbiased Predictors) except that they are not linear, and
>  they're not unbiased, and there is no clear sense in which they are
>  "best" but, other than that, ...") are not guaranteed to have an
>  observed variance-covariance matrix that corresponds to the estimate
>  of the variance-covariance matrix of the random effects.
>
>         From:     bates at stat.wisc.edu
>         Subject:        Re: [R-sig-ME] [R] coef se in lme
>         Date:   October 17, 2007 10:04:47 PM GMT+02:00
>  Lately I have taken to referring to the "estimates" of the random
>  effects, what are sometimes called the BLUPs or Best Linear Unbiased
>  Predictors, as the "conditional modes" of the random effects.  That
>  is, they are the values that maximize the density of the random
>  effects given the observed data and the values of the model
>  parameters.  For a linear mixed model the conditional distribution of
>  the random effects is multivariate normal so the conditional modes are
>  also the conditional means.
>
>  Thanks,
>  Best
>
> Reinhold
>
>
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