[R-sig-ME] Non normal random effects

[Eric Edeline]

Dear John,

thanks a lot for your reply. For some reason part of your message 
(illustrations of distributions seemingly) comes up corrupted. Although 
I got the essence of your explanation, I would very much like having it 
in full. Could you send illustrations in another format?

eric

On 11/28/2010 12:01 AM, John Maindonald wrote:
> 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.
>                                                                                ~~~~~~~~~~~
>
> [For the extreme case that is illustrated, "skewness" perhaps
> rather than "non-normality".    But if the contribution from the
> random effect is somewhat weaker, overlap between points
> that correspond to the successive sets of non-normally
> distributed residuals will indeed lead to a distribution that, in
> practice, will be quite hard to distinguish from normal.
> Non-normality at the level of the residuals may or may not
> matter, depending on what it does to the sampling distributions
> that are relevant to the intended inferences.]
>
> 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
>>
>>      
>    

-- 
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