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

[Ben Bolker]

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On 11-01-07 12:48 PM, Martin Maechler wrote:
>>>>>> Ben Bolker <bbolker at gmail.com>
>>>>>>     on Fri, 07 Jan 2011 11:49:31 -0500 writes:
> 
> 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 ...
>> }

  Hmm.  I agree that if you just specify the correlation structure of
the residuals (e.g. by using a corStruct in glmmPQL), it's not clear
what the underlying statistical model would be, or whether/when it would
make sense.
> 
>> 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) )

  Yes, that's more or less what I meant, although I forgot the inverse
link function entirely and you put the link function (logit) where I
think you wanted the inverse link function (logistic) ...
> 
> 
>> ??
> 
> 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.
> 

  [snip]
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