I have been following your R discussion list on mixed modeling for a few
weeks, in hopes of understanding mixed modeling better.  And it has
helped.  I was not aware of the controversy surrounding degrees of
freedom and the distribution of test statistics.  I have just been
trusting the ANOVA output from software (Minitab, JMP) that reported F
tests.  JMP uses Kenward-Roger, Minitab's ANOVA reports an F-statistic,
followed by "F-test not exact for this term". 

A recent mention by Douglas Bates of George Box, though, hit upon an
aspect of mixed models that has confused me.  I'm an industrial
statistician, and studied statistics at Iowa State and the University of
Minnesota.  I have had 3 courses in DOE, 2 at the graduate level, and
none of them mentioned blocking factors could (should?) be modeled as
random effects.  **Exception: the whole plots in a split plot design
were taught as random effects.**

The 2005 update to Box Hunter Hunter discusses blocking as does Wu &
Hamada (2000).  Both texts model blocking factors such as Days and
Batches as fixed effects.  Montgomery's DOE text, 2009 rev., pretty
consistently states that blocks can be either random or fixed.  Don't
have a consensus from that small sample. 

I'm trying to understand the implications if I consistently used random
effects for DOE analysis.

I'm quite willing to use R for mixed models, seeing as Minitab, JMP etc.
appear to use degrees of freedom calculations that are questionable.
But as Douglas points out --- Box said, "all models are wrong, some are
useful" => Box's latest text doesn't bother with random effects for DOE
=>  does it follow that for practical purposes it's OK to consider
blocks as fixed?  There are certainly several advantages to keeping it
simple (i.e. fixed only):
* The analyses we (my statistics group) provide to our chemists and
engineers are more easily understood
* The 2-day short courses we teach in DOE to these same coworkers
couldn't realistically get across the idea of mixed model analysis ---
they would become less self-sufficient, where we're trying to make them
more self-sufficient
* We have a handful of softwares (Minitab, JMP, Design Expert) that can
perform DOE and augment the results in a number of ways:
   *** fold-over the design to resolve aliasing in fractional designs
   *** add axial runs to enable Response Surface methods
   *** add distributions to the input factors, enabling
Robustness/Sensitivity analyses
   *** running optimization algorithms to suggest the factor settings
that simultaneously consider multiple objectives
 *****  Not to mention the loss of Sample Size Calculations, far and
away my most frequent request
None of these softwares recognize random factors to perform these
augmentations

Replacing this functionality with R is going to be a high learning
curve, and probably not entirely possible.  My coding skills in R
consist of cutting and pasting what others have done.

I don't really expect that there's a "right" answer to the question of
random effects in DOE.  But I do believe that beyond the loss of
p-values, there are other ramifications for advising experimenters,
'"You can't trust results from your blocking on Days (or Shifts or RM
Lots or Batches, etc) unless they are modeled as random effects."

There's statistical significance, and practical significance.  My hope
is that blocks while random effects are statistically "truer", their
marginal worth over fixed effects in DOE is ignorable. Again, I don't
want this to come across as shooting the messenger, you are only laying
out the current state of art and the work that remains to be done.  But
any insight you can provide into what's practical right now would be
highly interesting.

Thank you for your time and consideration,
Paul Prew

651-795-5942     fax 651-204-7504
Ecolab Research Center
Mail Stop ESC-F4412
Lone Oak Drive
Eagan, MN 55121-1560


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