Florian,

In practice, won't be confounded the preparation x animal interaction with residual error?

Thanks.
Walmes Zeviani.




________________________________
De: Florian Klinglmueller <float@lefant.net>
Para: r-sig-mixed-models@r-project.org
Enviadas: Quarta-feira, 10 de Março de 2010 9:45:56
Assunto: [R-sig-ME] problem with estimation of individual and interaction term with small residual variance

dear list,

i have a problem with estimating the variance components of the following model:

i have 10 animals, from each of which 4 different tissue preparations are measured in replicates of 5. i'm interested in testing if there are any differences between the 4 preparations.
so i have a fixed effect for the preparation, then a random effect for the individual and i'd like to include a random interaction term since there might be slight differences in how the preparations are done from each animal. this leads me to the model:

y(ijk) = 1 + x(i) + z(j) + z(ij) + e(ijk)

where y(ijk) is the measurement from each of the samples, x(i) is the fixed effect for the 4 preparations, z(j) is the individual random effect, z(ij) is the interaction random effect of animal and preparation and finally e(ijk) the residual.

i want to estimate the variances of z(j), z(ij) and e(ijk).

i did a little simulation study (see code below) over several scenarios for the different variance components. if i set the residual standard error to a very small value (~0.01) keeping individual and interaction variance at one, i observe is that the individual effect is underestimated as being close to zero (mean sd for z(ij) estimate over 1000 simulations: .01), although it should be 1. for se=0 i get false convergence, however not for small values. consequently the estimates are negatively biased, with the amount of bias dependent on the amount of residual standard deviation. This also doesn't get better if i increase the sample sizes.

so my question is: is this a numerical problem or do i have misspecified something?

thanks and regards
florian

--- Simulation Code -----

sim = function(rsd){
   sb = 1 # sd of individual random effect
   sg = 1 # sd of interaction random effect
   se = rsd  # residual standard deviation
   a = 1:4 # preparation fixed effecs
   ia = factor(rep(1:4,each=10*5)) # dummy variable for preparation
   ib = factor(rep(1:10,each=5,4))  # dummy variable for individual
   y = a[ia]+rnorm(10,0,sb)[ib] +rnorm(4*10,0,sg)[ia:ib]+rnorm(4*10*5,0,se)

   lmer(y~1+ia+(1|ib)+(1|ia:ib))
}

## 1000 runs of the simulation
foo <- replicate(1000,attr(VarCorr(sim(2))[[2]],'stddev'))

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