[R-sig-ME] Nested Effects and Correlation

[Ben Bolker]

On 13-03-11 01:36 AM, Pantelis Hadjipantelis wrote:
> 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.

  I think it of as the other way around (although this may be just a
difference in the way we're describing the same thing).  The theta
vector represents the primary model parameters; L is derived from theta
when computing likelihoods etc..