# [R-sig-ME] Non normal random effects

John Maindonald john.maindonald at anu.edu.au
Sat Nov 27 23:18:18 CET 2010

```You can have this sort of situation:
'Normal' effect                       .                                       .                                  .
Observations                   .. .  .  .    .                        ..  .    .    .                      ..  .    .     .

The large contribution from the random effect means that,
until it is accounted for, you will not see the non-normality.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 27/11/2010, at 11:43 PM, Eric Edeline wrote:

> Dear John,
>
> thanks for your feed back and for the useful tutorial. Actually the random effect in question is normally distributed (I did not check before, sorry), so the problem comes from somewhere else. I am modeling fish body size from a large dataset as a function of many covariates, and adding a "species" effect (be it fixed or random) skews the residuals but drops the AIC:
>
> m1<-lmer(log(Length) ~log(Slope)+log(Width)+Temp*log(D)+Temp*log(Compint2)+Temp*log(Predln102)+Temp*Year
> +(1|Species/Station),
> data=Data, na.action=na.omit, REML=TRUE) #AIC 73427, skewed residuals
>
> m2<-lmer(log(Length) ~log(Slope)+log(Width)+Temp*log(D)+Temp*log(Compint2)+Temp*log(Predln102)+Temp*Year
> +(1|Station),
> data=Data, na.action=na.omit, REML=TRUE) #AIC 147157, Gaussian residuals
>
> This looks puzzling to me. Would you have an idea for why introducing a normally distributed effect shews the residuals?
>
>
>
>
> On 11/26/2010 10:51 PM, John Maindonald wrote:
>> Contrary to what is often claimed, it is not the normality of the
>> random effects themselves that matters, but the normality of
>> the sampling distribution of the relevant fixed effect.  In mixed
>> models, there is by comparison with iid models the additional
>> complication that normality can affect the trade-offs between
>> the different components in the fitted model.   Opportunities
>> for such trade-offs are large if there are several random effects
>> and there is imbalance or incompleteness (some combinations
>> of factor levels missing) in the fixed effects structure.  Non-normality
>> in the random effects can then be both hard to detect and have
>> implications for inference.
>>
>> There is an examination of a data set with a relatively complicated
>> random effects structure in the overheads at:
>> http://www.maths.anu.edu.au/%7Ejohnm/r-book/2edn/xtras/mlm-ohp.pdf
>>
>> John Maindonald             email: john.maindonald at anu.edu.au
>> phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
>> Centre for Mathematics&  Its Applications, Room 1194,
>> John Dedman Mathematical Sciences Building (Building 27)
>> Australian National University, Canberra ACT 0200.
>> http://www.maths.anu.edu.au/~johnm
>>
>> On 27/11/2010, at 7:04 AM, Eric Edeline wrote:
>>
>>
>>> Dear list,
>>>
>>> is non normality of random effects a serious issue for inference on the fixed effects? I am having a non normal random effect that tremendously improves model AIC.
>>>
>>> Thanks!
>>>
>>> --
>>> Eric Edeline
>>> Assistant Professor
>>> UPMC-Paris6
>>> UMR 7618 BIOEMCO
>>> Ecole Normale Supérieure
>>> 46 rue d'Ulm
>>> 75230 Paris cedex 05
>>> France
>>>
>>> Tel: +33 (0)1 44 32 38 84
>>> Fax: +33 (0)1 44 32 38 85
>>>
>>> http://www.biologie.ens.fr/bioemco/biodiversite/edeline.html
>>>
>>> _______________________________________________
>>> R-sig-mixed-models at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>>
>>
>
> --
> Eric Edeline
> Assistant Professor
> UPMC-Paris6
> UMR 7618 BIOEMCO
> Ecole Normale Supérieure
> 46 rue d'Ulm
> 75230 Paris cedex 05
> France
>
> Tel: +33 (0)1 44 32 38 84
> Fax: +33 (0)1 44 32 38 85
>
> http://www.biologie.ens.fr/bioemco/biodiversite/edeline.html
>

```