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Comparison of crossed ranom effects: lmer vs. MCMCglmm

4 messages · Linus Holtermann, Jarrod Hadfield

Original
Linus Holtermann
# 
Hello,

I read that lmer can handle independent (often labelled as crossed) random effets in mixed models. It seems to be possible with MCMCglmm as long as groups for the random effects are uniquely labelled. I use the "Penicllin" data in the lme4-package to compare both approaches:

library(lme4)
library(MCMCglmm)

str(Penicillin)
attach(Penicillin)

ml <- lmer(diameter~ 1 + (1|plate)+ (1|sample))
summary(ml)

mcmc <- MCMCglmm(diameter~ 1, random=~ plate + sample,verbose=F, nitt=110000,burn=10000,thin=10,data=Penicillin)
summary(mcmc)

Why are the result for the plate-variance differ by a large amount? Is it because MCMCglmm applies Gibbs sampling? Or is MCMCglmm doing something else here, instead of fitting independent random effects?


Best regards,


Linus Holtermann
Hamburgisches WeltWirtschaftsInstitut gemeinn?tzige GmbH (HWWI)
Heimhuder Stra?e 71
20148 Hamburg
Tel +49-(0)40-340576-336
Fax+49-(0)40-340576-776
Internet: www.hwwi.org
Email: holtermann at hwwi.org
 
Amtsgericht Hamburg HRB 94303
Gesch?ftsf?hrer: PD Dr. Christian Growitsch | Prof. Dr. Henning V?pel
Prokura: Dipl. Kauffrau Alexis Malchin
Umsatzsteuer-ID: DE 241849425
Jarrod Hadfield
# 
Hi Linus,

The point estimates are almost identical if the posterior mode is used:

hist(mcmc$VCV[,"plate"], breaks=30)

abline(v=VarCorr(ml)[["plate"]][1], col="red")

The posterior mean (which is reported in the summary) is often not a  
good measure of central tendency for variance components because of  
the skew. Posterior modes have high Monte Carlo error though.

Cheers,

Jarrod




Quoting Linus Holtermann <holtermann at hwwi.org> on Mon, 19 Jan 2015  
18:39:52 +0100:

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Linus Holtermann
    

    

      
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Thanks Jarrod. 
Just to be on the safe side, MCMCglmm indeed fits two independent random effects in the "mcmc"-specification? The different results emerge because the MCMC-Approach treat the variance components as random variables that capture more of the skewness?  It is often claimed that mixed models fitted via Maximum Likelihood underestimate the random effect variance.


Best regards,


Linus Holtermann
Hamburgisches WeltWirtschaftsInstitut gemeinn?tzige GmbH (HWWI)
Heimhuder Stra?e 71
20148 Hamburg
Tel +49-(0)40-340576-336
Fax+49-(0)40-340576-776
Internet: www.hwwi.org
Email: holtermann at hwwi.org

Amtsgericht Hamburg HRB 94303
Gesch?ftsf?hrer: PD Dr. Christian Growitsch | Prof. Dr. Henning V?pel
Prokura: Dipl. Kauffrau Alexis Malchin
Umsatzsteuer-ID: DE 241849425

        

      
      
    

  
  


  


      

    

    

      
      

        


  

        

      Jarrod Hadfield
    

    

      
      #
    

  


  

      
Hi,

Yes, MCMCglmm fits two independent random effects.

Bayesian approaches treat the variance components as random variables,  
and MCMC allows you to estimate their distribution. In general that  
distribution is not known, but if the response is Gaussian, the prior  
conjugate, and all fixed effects known, then the distribution is  
scaled inverse-Chi-squared. This distribution is skewed, particularly  
with low degrees of freedom.

(RE)ML does not posit a distribution for the variance components, it  
simply finds the variance components that maximise the (restricted)  
likelihood. Sometimes an approximate distribution for the *estimates*  
of the variance components is posited: usually normal with mean equal  
to the (RE)ML estimates. This approximation is based on high-n, but in  
reality the sampling distribution will rarely be normal and will also  
have skew.

The underestimation of the variance components via Maximum Likelihood  
is a separate issue. This arises because the deviation of observations  
from the estimated mean will always be smaller than the deviation of  
observations from the true mean. REML corrects for this by accounting  
for the uncertainty in estimated mean.

Cheers,

Jarrod





Quoting Linus Holtermann <holtermann at hwwi.org> on Tue, 20 Jan 2015  
10:50:41 +0100: