Random vs. fixed effects

The answer is effect-size dependent, is it not?

If you fit the random effect and the algorithm 
works without failure, why not use it?

If it doesn't work, you have a faulty tool for 
estimation. Punting to a fixed model is one way 
out of the problem. Another is matched-on-the-random-factor data analysis.

Pragmatism is certainly an issue. But what if you 
have 10 centers as a factor with known 
correlation issues. If you analyze with one set 
of predictors, missing values leaves you with 
only 5 centers, so you treat centers as a fixed 
effect with 5 levels. If you use another set of 
predictors, you have all 10 levels, so you use 
centers as a random effect with a variance. Isn't 
intellectual consistency an issue here too? How 
do you explain this in the executive summary?

One thing you can do if the mixed modeling fails 
is to use the standard deviation among levels of 
the random-treated-as-fixed factor as an estimate 
of the random effect. This would at least maintain consistency of concept.

Note that I'm not a mixed modeling expert, so my 
opinions may not be worth much.

Here's my question for the group:   Given that 
it is a reasonable *philosophical* position to 
say 'treat philosophically random effects as 
random no matter what, and leave them in the 
model even if they don't appear to be 
statistically significant', and given that with 
small numbers of random-effect levels this 
approach is likely to lead to numerical 
difficulties in most (??) mixed model packages 
(warnings, errors, or low estimates of the variance), what should one do?

(Suppose one is in a situation that is too 
complicated to use classical method-of-moments 
approaches -- crossed designs, highly unbalanced data, GLMMs ...)

1. philosophy, schmilosophy.  Fit these factors 
as a fixed effect, anything else is too dangerous/misleading/unworkable.

2. proceed with the 'standard' mixed model 
(lme4, nlme, PROC MIXED, ...) and hope it doesn't break.  Ignore warnings.

3. use Bayesian-computational approaches 
(MCMCglmm, WinBUGS, AD Model Builder with 
post-hoc MCMC calculation? Data 
cloning?)?  Possibly with half-Cauchy priors on 
variance as recommended by Gelman [Bayesian 
Analysis (2006) 1, Number 3, pp. 515?533]?

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

"Vere scire est per causas scire"
================================================================