[R-sig-ME] [EXT] Re: Choice of distribution for random effects

[Ben Bolker]

   I believe that there are some mathematically natural pairings (i.e. a 
Gamma random effect + a Poisson response, a Beta-distributed random 
effect + a binomial response), but I don't know if there's much 
theoretical justification other than analytical convenience.

   (If you have a categorical response then the natural random effect 
would be Dirichlet, i.e. a Dirichlet-multinomial marginal distribution, 
but do you really want to go down that rabbit hole?)

    I strongly second Andrew's recommendation to check that the 
difference is not a difference between packages/estimation procedures.

On 2022-07-20 5:26 a.m., Andrew Robinson wrote:
> You should really also verify that the hglm is doing what you expect by fitting with Gaussian random effects.
> 
> The random effects in glmer are Gaussian. The effects enter the model via the linear predictor, which is then translated to the mean function of the chosen distribution of the response variable via the link function (NB this is a conceptual explanation, not an algorithmic one).
> 
> Cheers,
> 
> Andrew
> On 20 Jul 2022, 6:40 PM +1000, J.D. Haltigan <jhaltiga using gmail.com>, wrote:
> External email: Please exercise caution
> 
> ________________________________
> Thanks, I will inspect the BLUPS.
> 
> Re: choice of distribution, what I meant was, for example, if my fixed effects family is estimated using a Poisson model, does that inform the choice for random effects as well? (i.e., why would one invoke a gaussian distribution for random effects if the response variable is, say, categorical, or a count?)
> 
> On Wed, Jul 20, 2022 at 3:40 AM Andrew Robinson <apro using unimelb.edu.au<mailto:apro using unimelb.edu.au>> wrote:
> You can use a qq-plot of the BLUPS to guide that decision.
> 
> The difference might also be due to something else, though.  Have you tried the call to hglm2 with a gaussian random effects family?  Does it give the same output as the glmer?
> 
> For: does the choice of distribution for the fixed effects portion of the model inform the choice for the random effects?  I’m not sure what you mean - do you mean the exponential family for the response variable?
> 
> 
> Cheers,
> 
> Andrew
> 
> --
> Andrew Robinson
> Chief Executive Officer, CEBRA and Professor of Biosecurity,
> School/s of BioSciences and Mathematics & Statistics
> University of Melbourne, VIC 3010 Australia
> Tel: (+61) 0403 138 955
> Email: apro using unimelb.edu.au<mailto:apro using unimelb.edu.au>
> Website: https://researchers.ms.unimelb.edu.au/~apro@unimelb/
> 
> I acknowledge the Traditional Owners of the land I inhabit, and pay my respects to their Elders.
> On 20 Jul 2022, 2:27 PM +1000, J.D. Haltigan <jhaltiga using gmail.com<mailto:jhaltiga using gmail.com>>, wrote:
> Hi:
> 
> Is there clear best practice or guidance when it comes to choosing the
> distribution of random effects where multiple choices exist (e.g.,
> gaussian, gamma, etc.)? I ask in the context of extending some analyses
> from an RCT in which the outcome is symptomatic seropositivity (so a
> count). The random effects I am modeling are village cluster [union] (it's
> a cluster randomized trial). I get different results (significance-wise)
> depending on whether I choose a normal or gamma distribution for the random
> effects.
> 
> The basic model (proportional outcome) is:
> 
> lme4_5_B = glmer(posXsymp~ treatment+proper_mask_base+prop_resp_ill_base_2
> + pairID + (1 | union), family = "poisson", nAGQ=0, data = bdata.raw3)#lme4
> package using glmer
> 
> 
> HGLM2_5_A = hglm2(posXsymp~ treatment+proper_mask_base+prop_resp_ill_base_2
> + pairID + (1 | union), family =poisson (link = log), rand.family
> =Gamma(link=log),
> data = bdata.raw3)#HGLM package using hglm2
> 
> Does, for example, the choice of distribution for the fixed effects portion
> of the model inform the choice for the random effects?
> 
> Thank you for any insights.
> 
> Best regards,
> J.D.
> 
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-- 
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
(Acting) Graduate chair, Mathematics & Statistics