[R-sig-ME] Mixed-model-binary logistic model with dependence between individual repeated measures

[Martin Maechler]

>>>>> Ben Bolker <bbolker at gmail.com>
>>>>>     on Fri, 07 Jan 2011 11:49:31 -0500 writes:

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    > On 11-01-07 11:35 AM, Anna Ekman wrote:
    >> Ben Bolker, thank you for your suggestions.
    >> 
    >> Yes, it is suprising that I in SAS and STATA have to assume
    >> independence between the measurements within an individual.

    > It's fundamentally a bit hard to specify correlation among individuals
    > in a non-normal model. One option is to go completely to the marginal
    > specification (which you said you don't want to do); probably the most
    > sensible statistical formulation is

    > (fixed effects)  eta0 = X*beta
    > (random effects) eta1 ~ MVN(mu=X*beta,Sigma=(something sensible such
    > as AR(1) within individuals))
    > y ~ Bernoulli(eta1)

Interesting... {I've been "taught" in the past that  correlation
                specification for non-normal, i.e. GLME models,
		would not make sense /  be possible,
		something you do not seem to support ...
}

Does the above mean {slight changes}

 (fixed effects)  eta0 = X*beta
 (random effects) eta1 ~ MVN(0, Sigma=(something sensible such
				       as AR(1) within individuals))

  (Y | X,eta1)  ~ Bernoulli( logit(eta0 + eta1) )


??

    > i.e., a hierarchical model with a multivariate normal correlated
    > distribution at the 'lower level', with a level of Bernoulli variation
    > on top of that.

    > The correlation parameters of eta1 will not correspond to the actual
    > correlations among the measurements (which will be smaller due to the
    > extra variation coming from the Bernoulli sampling)

    > I do not
    >> want to assume that. In addition I would like to be able to chose
    >> other distributions than the normal for my random effect, which is
    >> not possible in SAS (proc NLMIXED). 

    > It's not possible in R either as far as I know.


    > The generalized estimating
    >> equation packages are probably not an option as I do not whant
    >> marginal models. I will look at the references you suggested. Thank
    >> you. /Anna
    >> 

    > If you want a non-marginal model with non-normal random effects and
    > within-individual correlation structures other than compound symmetry
    > (i.e. simple block structures), you are probably going to have to
    > construct your own solution with WinBUGS or AD Model Builder or ... ? If
    > you're lucky, MCMCglmm may be able to do what you want -- I would check
    > it out. (Molenbergh and Verbeke's book on longitudinal models describes
    > approaches for non-normal random effects, but in the context of LMMs
    > (i.e. normally distributed errors) -- they may have done something to
    > extend this stuff to GLMMs more recently.  It's possible that someone
    > out there has done what you want and encapsulated it into a canned
    > package, but I doubt it.

    > cheers
    > Ben Bolker
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