[R-sig-ME] glmer vs. MCMCglmm

[Jarrod Hadfield]

Hi,

As David suggested, the problem with binary data is that any residual  
variance in the underlying probability cannot be observed, unlike  
other types of data. For example, if data are generated according to  
the process

y ~ binom(n=1, p=inv.logit(mu + e_i))

then the data look the same irrespective of the variance of the e's:

y1<-rbinom(1000, 1, inv.logit(rnorm(1000, 0, 1)))
y2<-rbinom(1000, 1, inv.logit(rnorm(1000, 0, 10)))

table(y1)
table(y2)

Because there is no information on VE (residual variance) we have to  
make an assumption about its value.  In terms of the fit of the model  
to the data the choice of VE should make no difference because the  
parameter is redundant. I think (if some one could clarify this that  
would be great) that lmer fixes VE to zero, where with MCMCglmm the  
user is free to fix VE at any value.  Other than that I believe the  
models should be identical. However, as VE approaches zero in MCMCglmm  
mixing becomes  problem, and actually when VE=0 the chain no longer  
mixes at all.

With genetic models there are extra restrictions on the parameter  
values which are not met when fitting a simple dam-sire model. Often  
the parameter estimates lie within these restrictions, but with binary  
data this is often not the case.  The issue is that if genetic  
variance (VA) exists, then you know apriori that VE in a dam-sire  
model cannot be zero, because VE under this parameterisation actually  
contains some VA (due to mendelian sampling variation).  The animal  
model, which you should be able to fit using MCMCglmm, not only  
accounts for more complicated patterns of relatedness , but also  
ensures that these restrictions are met.  Even so, you still have no  
information regarding the non-genetic VE and you have to fix it at  
something. My guess is that there are certain things that are  
invariant to the choice of VE (such a as the genetic correlation in  
the Plodia example)  but off-hand I don't know what they are.   It may  
also be possible to obtain post-analysis what the posterior  
distribution of VA/fixed effects would be under a different assumption  
about VE - but again I don't know the literature well enough to say.

One other worrying issue is that comparing DIC for different models  
only seems valid if the same assumption regarding VE is used. Again,  
any ideas/insights into this problem would be great.

Cheers,

Jarrod








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