[R-sig-ME] lmer and semi-definite covariance matrices for random effects

[Gabriel Baud-Bovy]

Dear all,

I am trying to use pedigreemm approach to analyze twin data. The model
includes semidefinite covariance matrices for random effects of the
form  sigma*corA

where corA is semi-definite. In this example,

      [,1] [,2] [,3] [,4]
[1,]    1    1  0.0  0.0
[2,]    1    1  0.0  0.0
[3,]    0    0  1.0  0.5
[4,]    0    0  0.5  1.0

the two blocks represent the covariance structure for the additive 
genetic component
of a monozygote and dizygote pairs of twins respectively. A real example 
might involve
1000 pairs. In this case, this matrix would be 2000 x 2000 and coded as 
a sparse
symmetric matrix ("dsCMatrix").

I have seen that covariance matrices can given to lmer using the 
pedigreemm:::ZStar
function. I found an  example with positive definite matrices here :

http://dysci.wisc.edu/sglpge/posters/Using%20the%20R%20package%20pedigreemm%20for%20traditional%20and%20marker-based%20genetic%20evaluations%20-%20An%20application%20to%20a%20wheat%20population%20-%20Vazquez.pdf

The problem is that ZStar requires the Cholesky factor of corA and, if I 
am nost mistaken, the chol
function in the Matrix package deals only with positive definite matrices.

My questions are

1) can lmer deal with semi-definite covariance matrix for the random 
effects ?

2) how can compute the required Cholesky decomposition for a 
semi-definite symmetric
and sparse matrix ?

One reason I am asking the first question is that D. Bates wrote in the 
vignette (PLS versus GLS)
that it is important to allow for a positive semidefinite covariance 
matrix of the random effects and
the implementation vignette says also that this covariance matrix is 
positive semidefinite (p. 3).
However,  in another older document (MixedEffects.pdf) from 2004, I see 
a mention that these matrices
are restricted to being positive definite and, I also do find obvious 
way of compute the Cholesky
factor of a positive semidefinite matrix.

http://cran.r-project.org/web/packages/lme4/vignettes/PLSvGLS.pdf
http://cran.r-project.org/web/packages/lme4/vignettes/Implementation.pdf
http://pages.cs.wisc.edu/~bates/reports/MixedEffects.pdf

Thank you,

Gabriel

P.S. I found that I could specify my model with the regress function 
(regress package)
but it does not work well with large dataset because it uses dense 
matrices. I
also tried with the lmekin (package coxme) but it gives an error message 
because
the matrix is not positive definite.


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