On Thu, Feb 7, 2013 at 9:02 AM, Doran, Harold <HDoran@air.org> wrote:

> Doug:****
>
> ** **
>
> Sorry if this is too simple a question. Call \hat{\beta}} the estimate of
> a fixed parameter and \hat{b_i}  the BLUP for unit i.
>

But the BLUPs are not estimates of parameters so the notation \hat{b_i} is,
at best, misleading.


> If the following exists****
>
> ** **
>
> \hat{\theta} = \hat{\beta}} + \hat{b_i}
>

Technically you need to distinguish between a parameter estimate and the
estimator of that parameter.  The estimate is a number for which
var(\hat{\beta}) is not meaningful because it is always zero.  The
estimator is a random variable and does have a variance but the variance
often depends on the parameter being estimated or on other parameters.
 That's the reason that there is a distinction between the standard
deviation of the estimator and the standard error of the estimator.  To
evaluate the standard deviation you need to know the values of at least
some of the parameters.  The standard error is "the estimated standard
deviation of the parameter estimator".  In the Gaussian cases the standard
deviation of the estimator of \beta involves \sigma but not \beta.  The
standard error is the same expression evaluated with s in place of \sigma.

A possible way out is to assume that the variance components are known (or,
alternatively, the ratios of the variance components to the variance of the
noise term are known). You can then create an equivalent linear model in
which both the random effects and the fixed-effects are parameters, using
pseudo-data to apply the penalty on the least squares estimates of the
random effects coefficients.  That's essentially what the Henderson
mixed-model equations do.  We write them in a slightly different form that
is easier computationally.  Then it would make sense to talk about the
variance of the estimator of the random effects and you would calculate it
using the Henderson matrix or, preferably, the representation of the
Cholesky factor that we use, to evaluate the joint distribution of the
fixed-effects and random-effects parameter estimators.

This would be roughly the same as regarding the distribution of the sample
mean or the coefficients in a linear model as being Gaussian rather than
Student's T.  You are conditioning on an estimated value of some parameters
instead of taking into account the fact that they are being estimated.

The problem with this approach is that those parameters on which you are
conditioning are not very precisely estimated when there is a moderate
number of levels in the random-effects grouping factor.  For your
applications that probably wouldn't be a problem but others may find it
useful to run some simulations to see how well the conditional results
mirror the underlying variability.

 **
>
> Then, does var(\hat{\theta}) also exist?**
>
> * *
>
> ** **
>
> *From:* dmbates@gmail.com [mailto:dmbates@gmail.com] *On Behalf Of *Douglas
> Bates
> *Sent:* Thursday, February 07, 2013 9:57 AM
> *To:* Doran, Harold
> *Cc:* Andrew Robinson; dcarov@gmail.com; r-sig-mixed-models@r-project.org;
> Ben Bolker
>
> *Subject:* Re: [R-sig-ME] [R] lmer - BLUP prediction intervals****
>
> ** **
>
> On Wed, Feb 6, 2013 at 9:36 AM, Doran, Harold <HDoran@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.  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.
>  (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.****
>
> ** **
>
>
> If that term exists under the model, but it is ignored for purposes of
> variance estimation, what would be the reason for doing so?****
>
>
> > -----Original Message-----
> > From: r-sig-mixed-models-bounces@r-project.org [mailto:r-sig-mixed-
> > models-bounces@r-project.org] On Behalf Of Andrew Robinson
> > Sent: Tuesday, February 05, 2013 9:15 PM
> > To: dcarov@gmail.com
> > Cc: r-sig-mixed-models@r-project.org; Ben Bolker
> > Subject: Re: [R-sig-ME] [R] lmer - BLUP prediction intervals
> >
> > I think that it is a reasonable way to proceed just so long as you
> interpret the
> > intervals guardedly and document your assumptions carefully.
> >
> > Cheers
> >
> > Andrew
> >
> > On Wednesday, February 6, 2013, Daniel Caro wrote:
> >
> > > Dear all
> > >
> > > I have not been able to follow the discussion. But I would like to
> > > know if it makes sense to calculate prediction intervals like this:
> > >
> > > var(fixed effect+random effect)= var(fixed effect) + var(random
> > > effect) + 0 (i.e., the cov is zero)
> > >
> > > and based on this create the prediction intervals. Does this make
> sense?
> > >
> > > All the best,
> > > Daniel
> > >
> > > On Tue, Feb 5, 2013 at 8:54 PM, Douglas Bates <bates@stat.wisc.edu>
> > wrote:
> > > > On Tue, Feb 5, 2013 at 2:14 PM, Andrew Robinson <
> > > > A.Robinson@ms.unimelb.edu.au> wrote:
> > > >
> > > >> I'd have thought that the joint correlation matrix would be of the
> > > >> estimates of the fixed effects and the random effects, rather than
> > > >> the things themselves.
> > > >>
> > > >
> > > > Well, it may be because I have turned into a grumpy old man but I
> > > > get
> > > picky
> > > > about terminology and the random effects are not parameters - they
> > > > are unobserved random variables.  They don't have "estimates" in the
> > > > sense of parameter estimates.  The quantities returned by the ranef
> > > > function are
> > > the
> > > > conditional means (in the case of a linear mixed model, conditional
> > > > modes in general) of the random effects given the observed data
> > > > evaluated with the parameters at their estimated values. In the
> > > > Bayesian point of view none of this is problematic because they're
> > > > all random variables but otherwise I struggle with the
> > > > interpretation of how these can be
> > > considered
> > > > jointly.   If you want to consider the distribution of the random
> effects
> > > > you need to have known values of the parameters.
> > > >
> > > >
> > > >> The estimates are statistical quantities, with specified
> > > >> distributions, under the model.  The model posits these different
> > > >> roles (parameter,
> > > random
> > > >> variable) for the quantities that are the targets of the estimates,
> > > >> but
> > > the
> > > >> estimates are just estimates, and as such, they have a correlation
> > > >> structure under the model, and that correlation structure can be
> > > estimated.
> > > >>
> > > >> An imperfect analogy from least-squares regression is the
> > > >> correlation structure of residual estimates, induced by the model.
> > > >> We say that the errors are independent, but the model creates a
> > > >> (modest) correlation structure than can be measured, again,
> conditional
> > on the model.
> > > >>
> > > >
> > > > Well the residuals are random variables and we can show that at the
> > > > least squares estimates of the parameters they will have a known
> > > > Gaussian distribution which, it turns out, doesn't depend on the
> > > > values of the coefficients.  But those are the easy cases.  In the
> > > > linear mixed model
> > > we
> > > > still have a Gaussian distribution and a linear predictor but that
> > > > is for the conditional distribution of the response given the random
> > effects.
> > >  For
> > > > the complete model things get much messier.
> > > >
> > > > I'm not making these points just to be difficult.  I have spent a
> > > > lot of time thinking about these models and trying to come up with a
> > > > coherent
> > > way
> > > > of describing them.  Along the way I have come to the conclusion
> > > > that the way these models are often described is, well, wrong.  And
> > > > those descriptions include some that I have written.  For example,
> > > > you often
> > > see
> > > > the model described as the linear predictor for an observation plus
> > > > a "noise" term, epsilon, and the statement that the distribution of
> > > > the random effects is independent of the distribution of the noise
> > > > term.  I
> > > now
> > > > view the linear predictor as a part of the conditional distribution
> > > > of
> > > the
> > > > response given the random effects so it wouldn't make sense to talk
> > > > about these distributions being independent.  The biggest pitfall in
> > > transferring
> > > > your thinking from a linear model to any other kind (GLM, LMM, GLMM)
> > > > is
> > > the
> > > > fact that we can make sense of a Gaussian distribution minus its
> > > > mean so
> > > we
> > > > write the linear model in the "signal plus noise" form as Y = X\beta
> > > > + \epsilon where Y is an n-dimensional random variable, X is the n
> > > > by p
> > > model
> > > > matrix, \beta is the p-dimensional vector of coefficients and
> > > > \epsilon is an n-dimensional Gaussian with mean zero.  That doesn't
> > > > work in the other cases, despite the heroic attempts of many people
> > > > to write things in that way.
> > > >
> > > > Here endeth the sermon.
> > > >
> > > >
> > > >> On Wed, Feb 6, 2013 at 6:34 AM, Douglas Bates <bates@stat.wisc.edu>
> > > wrote:
> > > >>
> > > >>> It is possible to create the correlation matrix of the fixed
> > > >>> effects
> > > and
> > > >>> the random effects jointly using the results from lmer but I have
> > > >>> difficulty deciding what this would represent statistically.  If
> > > >>> you
> > > adopt
> > > >>> a B
> >
> >
> >
> > --
> > Andrew Robinson
> > Director (A/g), ACERA
> > Senior Lecturer in Applied Statistics                      Tel:
> > +61-3-8344-6410
> > Department of Mathematics and Statistics            Fax: +61-3-8344 4599
> > University of Melbourne, VIC 3010 Australia
> > Email: a.robinson@ms.unimelb.edu.au    Website:
> > http://www.ms.unimelb.edu.au
> >
> > FAwR: http://www.ms.unimelb.edu.au/~andrewpr/FAwR/
> > SPuR: http://www.ms.unimelb.edu.au/spuRs/
> >****
>
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> >
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>
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> ** **
>

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