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

[i white]

Another possibility might be (e.g.) to treat observations on monozygotic 
twins as repeated measurements on the same individual. This reduces the 
effective number of individuals and the size of the genetic covariance 
matrix. This smaller matrix should be positive definite.

On 04/02/2013 09:52 PM, Gabriel Baud-Bovy wrote:
> 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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