Dependency structure

Just to clarify, you mean that you want to specify a correlation structure
between the individual level error term in the model (also called the
residuals) and the random intercept or group-level error.

This doesn't make a lot of sense to me because the random intercept is
literally the product of a decomposition of the general model's error
structure into the within group (R matrix) and between group (G matrix)
components of the error. They are uncorrelated by construction. The only
way that they could possibly be correlated would be if you had an
exchangability problem in the random effects structure. You could have a
fuzzy boundaries issue like US counties are correlated by space. But you
wouldn't solve that by correlating the lower level error term with the
random intercept. You'd build a group boundary spatial weights matrix and
include it in the model.

I must be missing something in the translation.

On Tue, Nov 6, 2018 at 1:11 PM Yashree Mehta <yashree19 at gmail.com> wrote:

Hi,

Regarding the question on dependency structure, is there a way to allow
for
the possibility of the error term and random intercept being correlated? I
need to define the covariance matrix between these two terms and estimate
the values which should go into this matrix.

Thank you

Regards,
Yashree

On Wed, Oct 17, 2018 at 2:37 AM Ben Bolker <bbolker at gmail.com> wrote:

Hi,

Is there literature on how to specify the dependency structure between

the

random intercept and the statistical noise error term in a random

intercept

model?
It would be useful to also know how to implement using R...

  Can you be more specific about what you want?  Suppose you have
observations j within groups i, and you have an epsilon_{0,ij} for each
observation (error term) and an epsilon_"1,i} for each group (random
intercept).  Typically the epsilon_{0,ij} values are iid with
homogeneous variance sigma_0^2, and epsilon_{1,i} are iid with variance
sigma_1^2.  What kind of correlation structure are you looking for?

While we're at it, you previously asked:

===
I am working with a random intercept model. I have the usual "X" vector
of covariates and one id variable which will make up the random
intercept. For example,

Response variable: Production of maize
Covariate: Size of plot
ID variable: Household_ID

I need to acknowledge that there is correlation between the FIXED EFFECT
coefficient of plot size and the estimated random intercept. It is my
model assumption.

Does lme4 assume this correlation or do I have to make changes in the
formula so that it gets considered?
===

  The short answer to this one is "no", I think -- I don't know that
there's a way to allow for correlation between fixed effect coefficients
and random intercepts. (This actually seems like a weird question to me;
in the frequentist world, as far as I know, you can only specify
correlation models for *random variables* within the model.  In the
context of LMM fitting, I don't think parameters are random effects in
this sense.

On 2018-10-16 01:03 PM, Yashree Mehta wrote:

Thank you

Yashree

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Thanks,
John


John Poe, Ph.D.
Postdoctoral Scholar / Research Methodologist
Center for Public Health Services & Systems Research
University of Kentucky
www.johndavidpoe.com