[R] Block factor as random or fixed effect?

Ben Bolker bolker at ufl.edu
Thu May 14 04:08:36 CEST 2009

Robert A LaBudde wrote:
> At 05:49 PM 5/13/2009, Rob Knell wrote:
>>I apologise for asking a general stats question, but I'm at a bit of a
>>loss as to what to do following some hostile referees' comments. If I
>>have a fully randomised blocked design, with only three blocks, should
>>I treat block as a random or fixed effect? I have read comments about
>>not treating block as a random effect if the number of blocks is less
>>than 6 or 7: is this right?
>>Any advice much appreciated
>>Rob Knell
> If you treat the variable as fixed effects, then inference will only 
> apply to those particular choices of blocks. If you treat the 
> variable as a random effect, you are probably going to estimate a 
> variance for a population distribution plus a mean effect, so 
> inference can be made to the population of all possible blocks.
> The rule you've probably seen quoted could be paraphrased to say: "If 
> you're trying to estimate a random effect (i.e., variance), you will 
> need at least 6 subjects, or you won't get any precision on the 
> estimate. For fewer than 6 subjects, you might as well give up on 
> modeling a random effect, and just settle for doing the fixed effects
> model."
> That being said, if you really need inferences on the population of 
> blocks, model the random effect and bite the bullet on the imprecision.
> Also, remember the assumption that the blocks are chosen randomly 
> (from a normal distribution). If they're not, stick with the fixed 
> effects model.

It depends what you're doing.

If everything is normally distributed, (nearly) balanced, orthogonal, etc.,
and you can successfully use classical method-of-moments  approaches to
ANOVA, then you have the choice whether to treat the 3 blocks as random or
fixed (although you will have a really bad estimate of the block variance). 
If all of the above are not true, then you are almost guaranteed not to be
able to estimate the variance properly -- symptoms will range from an
estimated block variance of 0, to various warnings and errors. (The rule of
thumb quoted above applies.)  See also Andrew Gelman, "Analysis of variance:
why it is more important than ever" for ammunition, if you need it ...
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