R help - Sep 2006 - variance-covariance structure of random effects in lme

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@vanderbilt.edu


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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
>
>
> 	[[alternative HTML version deleted]]
>
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