Nested Effects and Correlation

On Mon, Mar 11, 2013 at 3:57 AM, Ben Bolker <bbolker at gmail.com
<mailto:bbolker at gmail.com>> wrote:

    Pantelis H <kalakouentin at ...> writes:

    >
    > Hello,
    >
    > I have a question:
    > In the case of multiple nested effects one can be provided with
    > correlations for a grouping's subgroups; how are these correlations
    > computed?

      I'm not quite sure what you mean here.


    library(lme4)
    example(Pastes)
    fm2

    gives an example of a model with nested grouping factors.
    I see a variance and a (redundant) standard deviation for
    each random component (cask within batch, batch, residual error),
    but no correlations.
      Looking at the Oats example in the Implementation vignette
    http://cran.r-project.org/web/packages/lme4/vignettes/Implementation.pdf
    shows the same thing.

    > What is the correct reference for the computational calculation
    > of the nested random effects and their correlation in lmer()?
    > I believe an LDL^t decomposition of a scaled precision matrix might be
    > involved but I looked in the three lme4 vignettes, Bates & DebRoy

    (2004) in

    > J. of Multivariate Analysis as well as the DebRoy & Bates'

    Technical Report

    > No. 1076 but none mentions something explicitly (eg. the

    Implementation.pdf

    > vignette specifically sets "corr = FALSE" in the case of nested

    factors

    > actually). Could someone please suggest to me references in regard

    with

    > that?

       Usually corr=FALSE refers to suppressing the printing of
    the correlations among _fixed_ effect parameters ... where else are
    you looking?

    > Only indirect references I could find was: 1. subsection 2.2.7 on

    Pinheiro

    > & Bates "Mixed-Effects Models in S and S-Plus" but even there

    nothing is

    > stated specifically for correlation calculations and 2. the Bell Labs
    > Technical Memorandum "Computational Methods for Multilevel

    Modelling" by

    > Bates & Pinheiro that mentions correlations in regard with the

    Nonlinear

    > multilevel model. Additionally both references are mostly referring on
    > lme().
    > Finally, the "lme4: Mixed-effects modeling with R" book in

    subsection 3.2.1

    > while looking at nested effect correlations it does not provide
    > computational aspects of it.

      I can see that you've tried to be very specific (thank you), but
    I'm still not sure what you're referring to.  Here are some thoughts:

     * the correlation between _variance estimates_ is not explicitly
    modeled
    in lme4: one might be able to extract them from the Hessian matrix, but
    it would be a bit challenging
     * correlations are modeled, by default, among random effects within
    the same grouping factor.  For example, on p 17 of the implementation
    vignette a correlation is given between the intercept and nitrogen
    random effects at the Block level -- it happens to be 1.0 (suggesting
    the model is overfitted).  These correlations are directly modeled as
    part of the 'theta' vector which contains the elements of the lower
    triangle of the Cholesky factor of the relevant L matrix (which is
    described on pp. 5-6 of the same vignette).


That was exactly the clarification I needed. I could "see" (using
verbose=T) that the "theta" vector contained elements regarding the
correlations of random effects within the same grouping but I was not
entirely sure that their final values were directly modeled and
estimated from matrix L. Thank you.