[R] variance-covariance structure of random effects in lme

[Spencer Graves]

      I actually see two violations of the "compound symmetry" 
assumptions in the Oats example numbers you provide below.  You 
mentioned the fact that the 3 different numbers in 
cor(random.effects(f4OatsB)) are all different, when compound symmetry 
would require them to all be the same.  In addition, note that in 
VarCorr(fm4OatsB), Corr does not equal sigma1^2/(sigma1^2+sigma.e^2), as 
suggested  by the theory. 

      One might naively expect that the algorithm might constrain the 
parameter estimates to meet this compound symmetry assumption.  I don't 
know why the algorithm does not produce that, but it doesn't bother me 
much that it doesn't, because the numbers are close, especially since 
this data set includes only 3 varieties and 6 blocks, producing 6 
estimated random effects for each variety. 

      Someone more knowledgeable may provide more detailed comments. 

      Hope this helps. 
      Spencer Graves

Mi, Deming wrote:
> Dear R users,
> I have a question about the patterned variance-covariance structure for the random effects in linear mixed effect model.
> I am reading section 4.2.2 of "Mixed-Effects Models in S and S-Plus" by Jose Pinheiro and Douglas Bates.
> There is an example of defining a compound symmetry variance-covariance structure for the random effects in a
> split-plot experiment on varieties of oats. I ran the codes from the book and extracted the variance and correlation
> components:
>   
>> library(nlme)
>> data(Oats)
>> fm4OatsB <- lme(yield~nitro, data=Oats, random=list(Block=pdCompSymm(~Variety-1)))
>> VarCorr(fm4OatsB)
>>     
> Block = pdCompSymm(Variety - 1) 
>                    Variance StdDev   Corr       
> VarietyGolden Rain 331.5271 18.20788            
> VarietyMarvellous  331.5271 18.20788 0.635      
> VarietyVictory     331.5271 18.20788 0.635 0.635
> Residual           165.5585 12.86695      
>  
> This is a compound symmetry variance-covariance structure.  I then tried to find out the standard deviation and
> correlation matrix of the BLUPs predictors of the random effects and wish all three standard deviations would be close
> to 18.20788 and the correlation structure would be consistent with a compound symmetry structure.
>  
>  
>   
>> sd(random.effects(fm4OatsB))
>>     
> VarietyGolden Rain  VarietyMarvellous     VarietyVictory 
>           16.01352           15.17026           19.83877 
>   
>> cor(random.effects(fm4OatsB))
>>     
>                    VarietyGolden Rain VarietyMarvellous VarietyVictory
> VarietyGolden Rain          1.0000000         0.6489084      0.8848020
> VarietyMarvellous           0.6489084         1.0000000      0.6343395
> VarietyVictory              0.8848020         0.6343395      1.0000000
>
> The correlation structure is far from a compound symmetry structure, and the standard deviation of three random effects are
> all different from 18.20788.  On the contrary, the result is more like the one from a general positive-definite
> variance-covariance structure.  
> Can anyone tell me why I did not see a compound symmetry structure from the BLUPs predictors of the random effects or if
> I am doing something wrong?
> Thank you!
> Deming Mi
> deming.mi at vanderbilt.edu
>
>
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