[R-sig-ME] Why BLUP may not be a good thing

[Douglas Bates]

On Wed, Apr 6, 2011 at 4:59 PM, Murray Jorgensen <maj at waikato.ac.nz> wrote:
> I could be wrong-headed but it seems to me that in a GLMM context BLUP falls
> into a class of procedures that has been found to have bad properties in a
> missing-data (EM) context. See
>
> @ARTICLE{lr83,
>  author  = {Little, R. J. A. and Rubin, D. B.},
>  title   = {On jointly estimating parameters and
> missing data by maximizing the complete data likelihood},
>  journal = {Amer. Statist.},
>  volume  = {37},
>  number  = {},
>  pages   = {218-220},
>  year    = {1983}
> }
>
> whose abstract follows:
>
> One approach to handling incomplete data occasionally encountered in the
> literature is to treat the missing data as parameters and to maximize the
> complete-data likelihood over the missing data and parameters. This article
> points out that although this approach can be useful in particular problems,
> it is not a generally reliable approach to the analysis of incomplete data.
> In particular, it does not share the optimal properties of maximum
> likelihood estimation, except under the trivial asymptotics in which the
> proportion of missing data goes to zero as the sample size increases.
>
> In the GLMM context we have the article
>
> Maximum Likelihood Algorithms for Generalized Linear Mixed Models
> Charles E. McCulloch Journal of the American Statistical Association, Vol.
> 92, No. 437 (Mar., 1997), pp. 162-170
>
> McCulloch calls BLUP-like algorithms "joint maximization" methods and finds
> that they have poor properties, as we might expect from the Little-Rubin
> article.
>
> It may be that BLUP is one of those things that looses good properties when
> shifted from a linear to non-linear context.

> On the other hand it's also possible that I have completely misunderstood
> what people mean by BLUP in a GLMM context, in which case I'd like to be
> helped out of my confusion!

The acronym BLUP (Best Linear Unbiased Predictor) is not appropriate
in the case of generalized linear or nonlinear mixed models.  Alan
James once told me, in reference to the Lindstrom and Bates (1990)
article about nonlinear mixed-effects models that he "liked the idea
of finding the random effects values that would be the BLUP's - except
that they are not linear and not unbiased and there is no clear sense
in which they are `best' ".

I prefer to think of these values as the conditional modes.  They are
the values of the random effects that maximize the conditional density
of the random effects, given the observed data (and for a fixed,
"known" values of the parameters).

I think McCulloch may have been referring to algorithms that alternate
between estimating parameters in a linear mixed model and parameters
in a generalized linear or nonlinear least squares model.

The definition of the maximum likelihood estimator isn't up for
debate.  Once you have defined the probability model you have defined
the mle's for the model's parameters given the observed data.  The
question is how you evaluate the likelihood and here I would claim
that the Laplace approximation or more generally adaptive
Gauss-Hermite quadrature are pretty well accepted as the best
approximations.  These do involve determining the conditional modes of
the random effects so, in that sense, they are based on what is
mis-termed the BLUP's.  But I don't think that alone should disqualify
them from consideration.
> Murray
>
>
> --
> Dr Murray Jorgensen      http://www.stats.waikato.ac.nz/Staff/maj.html
> Department of Statistics, University of Waikato, Hamilton, New Zealand
> Email: maj at waikato.ac.nz                                Fax 7 838 4155
> Phone  +64 7 838 4773 wk    Home +64 7 825 0441   Mobile 021 0200 8350
>
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