lmer - BLUP prediction intervals

Hi,

Sorry to interfer with my ideas despite I am not a specialist, I try
it to check if I understood things properly or not, I would be
grateful to be corrected if I say uncorrect things.

May be the reason lies in the somehow ? hierarchical ? structure of
the mixed effects model.

I mean, in such a model, you select a set of patients (let's say) in
a whole population - the B random vector (?) - : first random
space. Then, you measure the value you're interested in - the Y
random vector : second random space.

After that, you use Y values (conditionned on the B values you
sampled) to determine the fixed effects estimations and BLUPs for b
values (if I am still correct) ==> these values are a function of Y
hence are themselves random variables, whose law depends on both Y and
B laws --- hence, is valid on the whole population and not only on the
subset of patients you sampled.

In this population, it is then possible to use a kind of covariance
between fixed values and BLUPs to calculate things. But adding them
is then strange, because you are working for "any patient" and add a
term, the BLUP, constructed for a specific patient: it makes no real
sense, and directly using the estimated variances value for B would
be more logical, I think.

Conversely, in the case you want to use really the BLUP for a given
patient, it means you assume you have already done the first step of
randomisation, and are working only within the second subset. But in
this case, the common law for fixed effects and BLUPs estimates does
not hold any more (since it is based on the two randomisation
stages), hence the impossibilty of formalizing covariances between
them...

Hope this is not too far from the real problem behind all of this!

Best regards,

On Thu, Feb 07, 2013 at 10:02:49AM -0500, Doran, Harold 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.
? 
? If the following exists
? 
? \hat{\theta} = \hat{\beta}} + \hat{b_i}
? 
? Then, does var(\hat{\theta}) also exist?
? 
? 
? From: dmbates at gmail.com [mailto:dmbates at gmail.com] On Behalf Of Douglas Bates
? Sent: Thursday, February 07, 2013 9:57 AM
? To: Doran, Harold
? Cc: Andrew Robinson; dcarov at gmail.com; r-sig-mixed-models at 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 at air.org<mailto: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.  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 at r-project.org<mailto:r-sig-mixed-models-bounces at r-project.org> [mailto:r-sig-mixed-<mailto:r-sig-mixed->
? > models-bounces at r-project.org<mailto:models-bounces at r-project.org>] On Behalf Of Andrew Robinson
? > Sent: Tuesday, February 05, 2013 9:15 PM
? > To: dcarov at gmail.com<mailto:dcarov at gmail.com>
? > Cc: r-sig-mixed-models at r-project.org<mailto:r-sig-mixed-models at 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 at stat.wisc.edu<mailto:bates at stat.wisc.edu>>
? > wrote:
? > > > On Tue, Feb 5, 2013 at 2:14 PM, Andrew Robinson <
? > > > A.Robinson at ms.unimelb.edu.au<mailto:A.Robinson at 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 at stat.wisc.edu<mailto:bates at 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<tel:%2B61-3-8344-6410>
? > Department of Mathematics and Statistics            Fax: +61-3-8344 4599<tel:%2B61-3-8344%204599>
? > University of Melbourne, VIC 3010 Australia
? > Email: a.robinson at ms.unimelb.edu.au<mailto:a.robinson at 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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