-----Original Message-----
From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-
models-bounces at r-project.org] On Behalf Of Andrew Robinson
Sent: Tuesday, February 05, 2013 9:15 PM
To: dcarov at gmail.com
Cc: 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>
On Tue, Feb 5, 2013 at 2:14 PM, Andrew Robinson <
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
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
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
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,
variable) for the quantities that are the targets of the estimates,
but
estimates are just estimates, and as such, they have a correlation
structure under the model, and that correlation structure can be
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
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
still have a Gaussian distribution and a linear predictor but that
is for the conditional distribution of the response given the random
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
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
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
view the linear predictor as a part of the conditional distribution
of
response given the random effects so it wouldn't make sense to talk
about these distributions being independent. The biggest pitfall in
your thinking from a linear model to any other kind (GLM, LMM, GLMM)
is
fact that we can make sense of a Gaussian distribution minus its
mean so
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
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>
It is possible to create the correlation matrix of the fixed
effects
the random effects jointly using the results from lmer but I have
difficulty deciding what this would represent statistically. If
you