[R-sig-ME] distribution of random effects glmmTMB - covariance structure

[D. Rizopoulos]

Well, AFAIK checking the normality assumption of the prior distribution 
of the random effects using the EB estimates can be problematic. This is 
because they have different distributions that depend on the design 
matrices of the fixed and random effects of each subject. And, also 
because there is the effect of shrinkage that has an impact on their 
distribution.

For more on these points, a nice overview is given in Section 7.8 of 
book of Verbeke and Molenberghs (2000), "Linear Mixed Models for 
Longitudinal Data", Springer-Verlag.

In any case, if we're talking about linear mixed models, it has be shown 
that misspecifying the prior distribution of the random effects has very 
little impact in parameter estimates and standard errors for the fixed 
effects.

Best,
Dimitris


On 9/6/2018 9:05 PM, Ben Bolker wrote:
> 
>     Yes.
> 
>     While the distribution of conditional modes is certainly not assumed
> to be exactly N(0,s^2), informally, if the observed distribution of
> conditional modes is far from zero-centered Gaussian, I might worry
> about misspecification of the model.  I know of the existence of a
> literature on the diagnosis and effects of model misspecification
> (especially of the distribution of conditional modes) in (G)LMMs -- e.g.
> go to http://bbolker.github.io/mixedmodels-misc/glmmbib.html and search
> for "misspec" -- but I don't know its contents well at all.
> 
>   (1) adding group-level covariates (to explain some of the non-Normal
> among-group variability) can help, if you have any such information
>   (2) one more question about your random-effect specification.  Is time
> being treated as categorical or continuous?
>     If categorical:
>       - if there are n time points, us(time+0|Subject) will have
> n*(n+1)/2 parameters, which could get out of hand (you'll be trying to
> estimate the full variance-covariance matrix among all n observations
> for each subject -- you'll need lots of subjects to make this work).
> Could be worth trying an ar1() model instead?
>       - allowing for a *continuous*, fixed effect of time in addition to
> the random effect could help (again, by explaining some of the
> systematic variability)
>     - if continuous: I'm not sure why you would suppress the intercept
> variation?
> 
> On 2018-09-06 02:42 PM, D. Rizopoulos wrote:
>> Logically, the ranef() gives you the empirical Bayes estimates of the
>> random effects. Note that the distribution (and as a result the variance
>> and covariances) of these is not the same as the distribution you
>> specified in the formula of the model. Namely, the distribution you
>> define is the _prior_ distribution of the random effects, whereas the
>> empirical Bayes estimates are coming from the posterior of the random
>> effects.
>>
>> In math terms, the choice of us() of diag() specifies the distribution
>> [b] of the random effects, whereas from ranef() you get the modes or
>> means of the posterior distribution
>>
>> [b | y] which is proportional to [y | b] * [b],
>>
>> where y denotes you Count outcome, and [y | b] denotes the distribution
>> of your outcome.
>>
>> Best,
>> Dimitris
>>
>>
>> On 9/6/2018 7:59 PM, Vidal, Tiffany (FWE ) wrote:
>>> I'm unclear about the distributional assumptions regarding the random effects in glmmTMB, using different covariance structures. It is my understanding that the default is unstructured covariance structure. When estimating a vector of random effects, what is the assumption about the distribution of the factor levels within each grouping? I'm usually assuming normality with a mean of 0 and estimated variance. This doesn't seem to hold looking at the ranef(mod) for the different grouping variables.
>>>
>>> For example:
>>> mod <- glmmTMB(Count ~ us(time + 0|Subject))
>>> or
>>> mod <- glmmTMB(Count ~ diag(time + 0|Subject))
>>>
>>>
>>> Here, I'm modeling (I think) variability among subjects through time (e.g., a different subject variance in each time step), and assuming that the repeated measures within each individual subject at time t, come from some distribution. If the assumed distribution was normal with a mean of 0, I would expect the sum of the Subject BLUPs in each year to approximate 0, but that doesn't appear to be the case. Any clarification on this would be appreciated.
>>>
>>> Thank you,
>>> Tiffany
>>>
>>>
>>> 	[[alternative HTML version deleted]]
>>>
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>>
> 
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-- 
Dimitris Rizopoulos
Professor of Biostatistics
Department of Biostatistics
Erasmus University Medical Center

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