[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"
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
> [[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/
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