Should blocking factors be modeled as random effects?

"In a complete randomized block design (CRBD), treating blocks as 
fixed or random should yield identical results."

It depends what you mean by "results".  SEs of effects, for treatments 
that are estimated "within blocks", will be the same.  The between 
block variance does not contribute to this SE.

Estimates of SEs of treatment means may be very different.  The 
between block variance does contribute to this SE.  This is where it 
does matter if there are very few blocks.  The SE will be estimated 
with very poor accuracy (low df).

Of course, the SEs of effects assume that there is no systematic 
change in treatment effect from one block to another.  Unless there 
are super-blocks (sites?), there is no way to estimate the SE of any 
block-treatment interaction.  Look at the kiwishade data in the DAAG 
package for an example where there might well be differences between 
blocks that are affected by the direction in which the blocks face.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 27/01/2009, at 8:51 AM, Juan Pedro Steibel wrote:

Hello,
Treating a block effect as random allows recovering inter-block 
information. In a complete randomized block design (CRBD), treating 
blocks as fixed or random should yield identical results. In an 
incomplete block design (incomplete by design or by missing at random 
some observations), the results will differ.

If the Gaussian assumption regarding the block effects are sound, I 
would expect that treating the block as random will be more efficient 
that fitting block as fixed. Moreover, one could compute the relative 
efficiency of both analyses by comparing the variances of a 
particular treatment difference when block is treated as fixed versus 
when it is treated as a random effect.

The catch is that the relative efficiency depends on the actual 
variance ratios (unknown) and on the assumptions regarding the random 
effects (commonly, Gaussian distribution).

In practice, when analyzing field or lab experiments, I tend to 
specify the block as a random effect. Always.
In some cases there are very few levels, though. In those cases, if 
someone asks "how can you reliably estimate a variance component for 
a (blocking) factor with only (say) 4 or 5 levels?", I just shrug. 8^D

JP



Prew, Paul wrote:

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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-- 
=============================
Juan Pedro Steibel

Assistant Professor
Statistical Genetics and Genomics

Department of Animal Science & Department of Fisheries and Wildlife

Michigan State University
1205-I Anthony Hall
East Lansing, MI
48824 USA
Phone: 1-517-353-5102
E-mail: steibelj at msu.edu