[R-sig-ME] Random vs. fixed effects

Robert A LaBudde ral at lcfltd.com
Fri Apr 23 18:13:28 CEST 2010

The question isn't whether an effect is "random" or "fixed". Anything 
can be considered either under different conditions. The real 
question is one of inference: Are you willing to make statements only 
about "these" blocks, or do you want to make inferences about "those" 
blocks too.

Consider the simplest case: a randomized block design. Suppose you 
have 3 blocks upon which you apply the treatments you want to 
measure. The treatments are applied in replicate within blocks.

If you treat the blocks as a fixed effect, you are saying you only 
care about inferences of treatment effect on "these" blocks, but no 
others. The blocks are then just an artifact of the design used to 
reduce variance of the treatment effect.

If the treat the blocks as a random effect, you are saying you want 
your inferences on treatment effect to apply not only to "these" 
blocks, but also hypothetically all "those" other blocks of similar 
type. You would also then be interested in the standard deviation of 
the block effect of that population of blocks.

Now we get to the practical side. You can treat the blocks as random 
with only 3 blocks, but you will get a very imprecise estimate of the 
block variance. This is not doing something wrong, it's only a fact 
of estimation. You will still control for block in measuring 
treatment effect. The bad news is that the imprecision in block 
variance will lead to imprecision in treatment effect. What then do 
you want to do? Treat blocks correctly as random, but get imprecise 
estimates? Or treat blocks as fixed, but get a  precise treatment 
estimate, but with a confidence interval that's a little too liberal?

Personally, I'm a purist. If blocks are random, I want to model them 
as random, even if I've only got 3 or 4 of them. I rather have a 
correct model (low bias) than precise, biased estimates. Others have 
different opinions. Hence the 6 blocks minimum rule of thumb.

Note that any effect can be theoretically fixed or random. Consider a 
1-way ANOVA with 3 drugs tested in replicate (mice, say). Normally 
this is treated as a fixed effect model, because we only care about 
the effects of "these" three drugs. But suppose I hypothecate the 3 
drugs were chosen at random from some sampling frame of drugs, and I 
want to make inferences about variability of treatment effect within 
"that" population of drugs. Then  I might use a random effect model 
to get a standard deviation of treatment effect.

The "elephant in the room" on this question is that rarely do people 
actually have random sampling behind the choice of the random effect 
subjects/blocks/centers. That introduces considerable selection bias 
and confusion about what population is really being represented. If 
your blocks aren't being randomly selected or at least representative 
of the population of inference, that is a more important issue than 
whether you model the effect as fixed or random. So this is more 
justification for the ad hoc 6 blocks minimum rule, and creates more 
acceptability for the fixed effect model.

At 09:38 AM 4/23/2010, Schultz, Mark R. wrote:
>I just read a post by Andrew Dolman suggesting that a factor with only 3
>levels should be treated as a fixed effect. This seems to be a perennial
>question with mixed models. I'd really like to hear opinions from
>several experts as to whether there is a consensus on the topic. It
>really makes me uncomfortable that such an important modeling decision
>is made with an "ad hoc" heuristic.
>Mark Schultz, Ph.D.
>Bedford VA Hospital
>Bedford, Ma.
>         [[alternative HTML version deleted]]
>R-sig-mixed-models at r-project.org mailing list

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

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