[R-sig-ME] [R] lmer - BLUP prediction intervals

[Andrew Robinson]

On Thu, Feb 07, 2013 at 08:57:28AM -0600, Douglas Bates wrote:
>    On Wed, Feb 6, 2013 at 9:36 AM, Doran, Harold <[1]HDoran at air.org> wrote:
> 
>      Andrew
> 
>      Ignoring the important theoretical question for just a moment on whether
>      it is sensible to do this, there is a covariance term between the BLUPs
>      and the fixed effects.
> 
> But the picky mathematician in me can't understand in what
> distribution this covariance occurs. It makes sense in the Bayesian
> formulation but not in a classical (sampling theory)
> formulation. 

Not to sound glib or flip, but might it make sense in the
likelihood-based formulation of estimation?

> The distribution of the estimator of the fixed effects for known
> values of the parameters is a multivariate normal that depends on
> \beta, \sigma^2 and \Sigma, the model matrices X and Z being
> known. The random variable B doesn't enter into it.  

I'm sorry to draw this out but I'm not seeing the point right here. If
I've interpreted your model appropriately, I'm now wondering about how
to add the random variable B (a BLUP?) to X \beta.

> (One way of writing this variance-covariance of this multivariate
> normal is X'V^{-1}X where V is that matrix that involves Z and
> Sigma^{-1} - I have forgotten the exact form).  I know these
> considerations sound like needless theoretical niceties but to me
> they're not. I have to be able to formulate the theoretical basis
> before I can make sense of the computational results and, after 20
> years or so, I'm still having trouble making sense of this.

-- 
Andrew Robinson  
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