[R-sig-ME] How is the covariance factor computed?

[Vincent Dorie]

I'm not sure if this answers your question, but the parameters of the variance/covariance matrix are stored in the theta slot of a merMod in a form such that they correspond to a Cholesky factorization of the individual components of the var/cov matrix of the random effects, with the diagonal in the first 'd' parts of the vector and the off-diagonal stored in column-major format in the next d * (d - 1) / 2 elements. Given a theta vector, to get to a representation such as that which Tlist gives requires knowing how to map the parameters to matrices. This is currently done by hand, using the knowledge that the cnms slot of a merMod contains the dimension of each grouping "factor"/"level" and the aforementioned Cholesky decomposition storage concept. In the future, however, if lme4 supports different forms of variance/covariances matrices for factors other than "full" (e.g. independent, or correlation only), then that knowledge will need to be referenced instead. I believe there is effort on that front in the "flexLambda" branch on github.

On the other hand, if you were asking where those numbers come from, it turns out that (at least for linear models) those parameters are sufficient to define a likelihood wherein the fixed effects and conditional error term (sigma) are analytically optimized. Since the goal is a maximum likelihood, or REML, the sigma parameters are then simply numerically optimized. You can then easily evaluate the mixed model likelihood at any value of the var/cov matrix of the random effects that you like, provided you are willing to accept maximal values for the fixed effects and sigma. If you wanted to plug those values in as well, it's a bit of a pain but it can be done.

Vince

On Jul 17, 2014, at 11:41 AM, Christian Brauner <christianvanbrauner at gmail.com> wrote:

> Hello,
> 
> I am performing a priori power simulations for mixed-effect models based
> on previous experiments. This works out quite nicely. I extract parts of
> my parameters from a previous model I fitted:
> 
> prev_mod <- lmer(Y ~ A
>                   + (B | Context)
>                   + (B | Subjects),
>                   data =3D data)
> 
> "A" :=3D 2 level factor
> "Context" :=3D 40 level factor
> "Subjects" :=3D 70 level factor
> 
> create design matrices for the fixed- and each random effect use functions from
> the apply-family and bind them to a previously set-up data frame and so on.
> 
> In order to simulate data I draw from two multivariate distributions. One
> for Subjects and one for Context. I previously constructed the
> variance-covariance matrix by using the estimations I get by issuing
> "as.data.frame(prev_mod)". After having read that Doug implemented a new
> method for the "getME()" extractor "Tlist" that gives the covariance
> factors from which the block matrices in "Lambda" are created I figured I
> could get the variance-covariance matrix way easier by doing (code here
> only for the "Context" variance-covariance matrix):
> 
> cov_fac <- getME(prev_mod, "Tlist")
> cov_fac_context <- cov_fac$Context
> 
> sigma(prev_mod)^2*cov_fac_context%*%t(cov_fac_context)
> 
> But the question that has been haunting me for weeks now is how the individual
> covariance factors (better: "matrices" in this case) that I can extract via
> "getME(prev_mod, "Tlist") are computed. Is there some literature on that?  I
> couldn't find any apart from the paper "Fitting linear mixed-effects models
> using lme4" published 23.06.2014. I would be interested in reading up/get an
> explanation how the covariance factor can be computed (mathematically and in
> lme4) and if I need the variance-covariance matrix beforehand or vica versa.
> 
> Thank for any help!
> 
> Best,
> Christian Brauner
> Eberhard Karls Universit=C3=A4t T=C3=BCbingen
> Mathematisch-Naturwissenschaftliche Fakult=C3=A4t - Faculty of Science
> Evolutionary Cognition - Cognitive Science
> Schleichstra=C3=9Fe 4 =C2=B7 72076 T=C3=BCbingen =C2=B7 Germany
> Telefon +49 7071 29-75643 =C2=B7 Telefax +49 7071 29-4721
> christianvanbrauner at gmail.com
> 
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