I'd use mcmcsamp() to examine the posterior distribution, under
a relatively uninformative prior, of of the parameter estimates.
For estimates that are based on four or five or more "degrees of
freedom", I'd surmise that the prior will not matter too much.
With estimates where the number of "degrees of freedom" is
one or two or three, the posterior distribution may vary greatly
from one run of mcmcsamp() to another.  Of course, the definition
of "degrees of freedom" becomes quite fraught if there is severe
imbalance; e.g., some of the items that contribute to the estimate
based on much more data than others.

Subject to such caveats as just noted, I'd expect that credible
intervals derived from the posterior distributions would be close
to the usual frequentist confidence intervals.  The main effect of
the non-normality may be that the estimates are "inefficient", i.e.,
the variance may be larger, or the distributions more dispersed,
than for "true" maximum likelihood estimates, were you able to
obtain them!
John Maindonald

John Maindonald             email: john.maindonald@anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Mathematical Sciences Institute, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.


On 27 Jun 2006, at 8:00 PM, r-help-request@stat.math.ethz.ch wrote:

> From: "Bruno L. Giordano" <bruno.giordano@music.mcgill.ca>
> Date: 27 June 2006 11:21:25 AM
> To: r-help@stat.math.ethz.ch
> Subject: [R]  Robustness of linear mixed models
>
>
> Hello,
>
> with 4 different linear mixed models (continuous dependent) I find  
> that my residuals do not follow the normality assumption  
> (significant Shapiro-Wilk with values equal/higher than 0.976;  
> sample sizes 750 or 1200). I find, instead, that my residuals are  
> really well fitted by a t distribution with dofs' ranging, in the  
> different datasets, from 5 to 12.
>
> Should this be considered such a severe violation of the normality  
> assumption as to make model-based inferences invalid?
>
> Thanks a lot,
>    Bruno


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