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Non normal random effects

12 messages · Eric Edeline, Valerio Bartolino, Douglas Bates +3 more

Original
Eric Edeline
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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!
John Maindonald
# 
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:

            

      
      
      
    

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Eric Edeline
    

    

      
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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:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        
      

    

    

      
      

        


  

        

      Douglas Bates
    

    

      
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On Sat, Nov 27, 2010 at 6:43 AM, Eric Edeline <edeline at biologie.ens.fr> wrote:

            

      
      
      
    

        
Just in terms of the model specification, do you really mean
(1|Species/Station)?  That expands to

 (1|Species) + (1|Species:Station)

and wouldn't reduce to (1|Station) in a model specification.  I think
you meant "species within station", which would be written as
(1|Station/Species) although I prefer the more explicit form
(1|Station) + (1|Station:Species)

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      valerio.bartolino
    

    

      
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Dear Eric,
I'm impressed by the drop in the AIC due to the 'Species' variable. I
think it could be useful comparing the coefficients estimated for the
fixed effects between the two models. It may be the case that they are
rather different between m1 and m2. I would pay great attention to the
interpretation of the effect of each variable.

Hope this could help

Valerio

        
On Fri, 2010-11-26 at 21:04 +0100, Eric Edeline wrote:

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Eric Edeline
    

    

      
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Here are model coeffs, which are indeed very different between the two 
models for some effects:

1> m1 at fixef
         (Intercept)          log(Slope)          
log(Width)                Temp
       64.3355546922        0.0176278497       -0.0301275431       
-3.8692860013
              log(D)       log(Compint2)      
log(Predln102)                Year
        0.0021372925        0.0023000751        0.0327498020       
-0.0294303421
         Temp:log(D)  Temp:log(Compint2) Temp:log(Predln102)           
Temp:Year
       -0.0032965368       -0.0013422775       -0.0008383696        
0.0019082777
1> m2 at fixef
         (Intercept)          log(Slope)          
log(Width)                Temp
        49.601725055        -0.113952871        -0.013659799        
-2.285783668
              log(D)       log(Compint2)      
log(Predln102)                Year
         0.023140502         0.055044993        -0.252180423        
-0.023767542
         Temp:log(D)  Temp:log(Compint2) Temp:log(Predln102)           
Temp:Year
        -0.006871451        -0.011123770         0.010086872         
0.001163320

In m1 I indeed meant "Station" within "Species". Actually, the full 
model I have selected so far is much more complex and includes several 
nested random effects + variance functions in lme:

m3<-lme(log(Length) 
~log(Slope)+log(Width)+log(Fcl)+Nb.species+Temp*log(D)+Temp*log(Compint2)+Temp*log(Predln102)+Temp*Year,
data=Data, na.action=na.omit, random=list(Species=pdDiag(form=~1), 
Strategy=pdDiag(form=~1), Method=pdDiag(form=~1),
Region=pdDiag(form=~1), Station=pdDiag(form=~1)), control=list(maxIter=100),
weights=varComb(varPower(form=~D), varPower(form=~Predln102), 
varPower(form=~Compint2)))#AIC 50945.01

Maybe this is too complex, but this is the best structure in terms of 
AIC. Still, the problem of skewed residuals from having a species effect 
remains (even as a fixed effect in simple lm models), and I really can't 
figure out why...

        
On 11/27/2010 07:14 PM, valerio.bartolino wrote:

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      John Maindonald
    

    

      
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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:

        

            

      
      
      
    

        

      
      
    

            

      
      
      
    

  
  


  


      

    

    

      
      

        


  

        

      John Maindonald
    

    

      
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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:

        

            

      
      
      
    

        

      
      
    

            

      
      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Eric Edeline
    

    

      
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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:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

  
    

      
      
    

  


  


      

    

    
1 day later

    

      
      

        


  

        

      Gustavo Betini
    

    

      
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Lets say that I want to look at the interaction between Species and some other variable...Temperature. The right specification would be 

(Temperature|Species) + (Temperature|Station:Species) 

or simply 

(1|Species) + (Temperature|Station:Species) 

Thanks, 

Gustavo S. Betini 
Dept. of Integrative Biology 
University of Guelph, Canada

        
On 10-11-27 12:04 PM, Douglas Bates wrote:

        
Just in terms of the model specification, do you really mean
(1|Species/Station)?  That expands to

 (1|Species) + (1|Species:Station)

and wouldn't reduce to (1|Station) in a model specification.  I think
you meant "species within station", which would be written as
(1|Station/Species) although I prefer the more explicit form
(1|Station) + (1|Station:Species)

  
  


  


      

    

    

      
      

        


  

        

      Douglas Bates
    

    

      
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On Mon, Nov 29, 2010 at 9:02 AM, Gustavo Betini <betinig at uoguelph.ca> wrote:

            

      
      
      
    

        
That depends on whether you feel that the changes in temperature
coefficient would be associated with the Species or with the
Station:Species combination or both.  I would try to start simply,
perhaps with

(1 + Temperature|Species) + (1|Station:Species)

It is difficult to estimate a large number of variance components
without a lot of data.  Starting with the "sumo" model that has every
possible term and interaction then seeing if you can pare it back
makes more sense in fixed-effects models than in mixed-effects models.
 Deletion strategies are easier to describe and evaluate when working
with fixed-effects only.  Also, in the fixed-effects case having a
model with too many terms in it doesn't usually prevent estimation of
coefficients, it just make the estimators less precise.

In a mixed-effects model, adding interaction terms increases the
number of variance-component parameters quadratically, not linearly.
And it is a lot more difficult to determine that the
variance-covariance matrix of the random effects is singular than to
detect that a model matrix is rank deficient.

So I would recommend building mixed-effects models using forward
selection, not backward selection, and proceeding cautiously.