Random or Fixed effects appropriate?

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.

 >
 > 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.