Peter Dalgaard wrote: [snip]
As Peter implies, when I did the simulation in SAS with PROC VARCOMP, using the model option 'Type1' which is an extended estimator, estimates for level 1 average to 0:
Or, try looking at a smaller example where things can be worked out explicitly: One-way ANOVA with random btw.group variation. Say 5 groups and 3 obs per group. If I got this right (please do check!), the estimate of the between-group variance is 1/3 times the difference between two chi^2/f distributed variables with 4 and 10 DF respectively. This will become negative about half the time, and lme (and similar code) will set it to zero in that case. Now
sd.sim <- sqrt(pmax(1/3*(rchisq(1000,4)/4 - rchisq(1000,10)/10),0)) summary(sd.sim)
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0000 0.0000 0.0000 0.2017 0.3955 1.2160
does not seem to be too far from what Gary has been experiencing.
Obviously, the fact that the estimator is censored at zero will make
it biased, but an extended estimator (allowing negative values) is
unbiased.
var.sim <- 1/3*(rchisq(1000,4)/4 - rchisq(1000,10)/10) summary(var.sim)
Min. 1st Qu. Median Mean 3rd Qu. Max. -0.796100 -0.196900 -0.042720 -0.007667 0.138500 1.104000
http://david.science.oregonstate.edu/~allisong/R/sas_Type1_5.pdf
and the range is still quite large.
Thanks to the help of Drs. Lumley, Bates and Dalgaard, I'm beginning to understand what's going on. But now it seems that much harder to interpret my experimental results -- high variance in both level 1 and level 2 at relatively low sample sizes -- given all this. If variance at one level can 'bleed' into another, sometimes to a large degree, is this analysis even useful at relatively small sample sizes?
thanks again, Gary
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