[R-sig-ME] Fixed effects cannot be interpreted if Y is transformed in (g)lme's [was: conditional vs. marginal coefficients in GLMM]

[Jarrod Hadfield]

Dear Lorenz,

I meant to reply to your earlier posts but forgot.  Section 2.5 of the  
MCMCglmm CourseNotes "Prediction with random effects" provides some  
solutions to your problem that I have found in the literature. In the  
logit case Equations 2.12 or 2.13 can be used in order to get the  
difference between two groups while marginalising any random effects  
(including observational-level effects). From my experience the  
approximations are generally very accurate. They do rely on the (true)  
distribution of the random effects being normal though, something  
which I believe true marginal models are more robust to, although I'm  
not entirely sure about this . Getting confidence intervals etc on  
these marginal differences may be difficult outside of an MCMC  
framework, but again, I'm not sure.

Cheers,

Jarrod.





On 14 Mar 2011, at 06:50, <lorenz.gygax at art.admin.ch> <lorenz.gygax at art.admin.ch 
 > wrote:

> Dear Mixed-modelers,
>
> I am aware that the point is not as simple as I have stated in the  
> subject line but I hope to pique your curiosity and thus provoke  
> some answers.
>
> I have come across this issue due to a reviewer request in one of  
> our publications and as you all know it is sometimes worthwhile to  
> argue with the reviewers and for other aspects one just tries to  
> accommodate the reviewer's wishes. Nevertheless, I think there was  
> one fundamental point raised by the reviewer or rather by the  
> comments I received in response to my original posting that is worth  
> discussing (and that unsettles me quite a bit).
>
> In our research group, we are mostly using mixed-modeling techniques  
> because we need to accommodate dependencies in the data of our  
> experimental designs which involve repeated measurements and  
> hierarchical nesting. We are usually only marginally interested in  
> the actual (relative) size of the random effects.
>
> Let's take a somewhat simplified version of our real problem: we  
> have observed the occurrence of lameness in three consecutive  
> seasons for 10 cows each on 36 farms. The farms additionally  
> differed in the type of flooring they provide in the barns. Thus, we  
> are interested in the risk of lameness in dependence of season  
> (within effect) and type of flooring (between effect). To  
> accommodate dependencies we have included a nested random effect of  
> cows (repeated measurement) in farms (hierarchical nesting). We have  
> originally implemented this model in glmmPQL (family= quasibinomial)  
> but may switch to glmer (family= binomial, including an additional  
> observation-level random effect to check for over-dispersion)  
> because we have been asked to conduct LR-tests.
>
> Now, of course, in such a generalized model the response is  
> (logit-)transformed. But the point to be made is as relevant for  
> mixed-models based on the normal distribution if the outcome  
> variable needs to be transformed (e.g. log) to satisfy the  
> statistical assumptions on error and random effects distributions.
>
> The point raised is that the estimated parameters are reflecting the  
> average reaction of the 'population' only on the transformed scale  
> because the average (additive) random effects are zero only on that  
> scale (conditional estimates). If model estimates are back- 
> transformed or e.g. ORs are calculated, as in our example, then  
> these values are biased because the random effects do not longer  
> average out.
>
> If my understanding is correct, the back-transformed model estimates  
> still reflect the relative risk of the observational units, i.e. in  
> our case we can still calculate ORs that reflect the risk in our  
> cows to suffer from lameness if going from one season to the other  
> and we can calculate ORs for our floor types but these are in the  
> sense virtual in that they represent the relative risk as if one  
> would switch from one floor type to another on any of the given  
> farms (as if it were a within effect).
>
> These ORs do not, however, reflect any direct estimate of  
> "population-wide" occurrence of lameness (marginal estimates), i.e.  
> the mean proportion of cows with lameness that you would find on a  
> set of farms with a given floor type. In this case we need some  
> other methods and GEE as been suggested.
>
> How do you weigh these two types of models in respect to the  
> interpretation of back-transformed model-derived measures, i.e.  
> their fixed-effects estimates? Is this an issue that guides your  
> choice of statistical model in practice? How do you deal with  
> interpreting fixed effects in mixed-models if the outcome variable  
> has been transformed? Or don't you interpret these effects at all?
>
> Many thanks for your thoughts, Lorenz
> -
> Lorenz Gygax
> Federal Veterinary Office FVO
> Centre for proper housing of ruminants and pigs
> Tänikon, CH-8356 Ettenhausen / Switzerland
>
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