Choosing appropriate priors for bglmer mixed models in blme

Just to follow up on Gelman's Cauchy prior, it seems to work quite  
well even in glmms. I don't have any theoretical results as of yet,  
but if you look at the sampling distribution of the fixed effects  
for any model, they cluster rather nicely. You get "sane" estimates  
for when no kind of separation is involved, infinite (or convergence  
failures) for complete/quasi complete separation, and a third group  
exists with large estimates for when a group contains all 0s or 1s.  
In the third case, a random effect can perfectly predict for that  
group, but because they're integrated out the likelihood remains  
well defined. You'll just get really large estimates of random  
effects, which then go with large estimates of fixed effects.

So long as you believe that some effect magnitudes for logistic  
regression pretty much never happen in nature, the Cauchy prior does  
a good job of pulling the extreme cases back down to earth while  
leaving the well-estimated ones roughly in place. That being said,  
using the priors in blme to patch up a data set is really only  
advised for checking the viability of a model (usually one among  
many, rapidly fit). After that, using something like MCMCglmm for a  
fully Bayesian analysis is the way to go.

Vince

On Mar 7, 2015, at 3:09 AM, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote:

Hi Josie,

Regarding the priors on the fixed effects, if complete separation  
is the issue having a diffuse prior is not going to help. Gelman  
(2008) gives some recommendations about priors for logistic  
regression. Although a Cauchy-prior was considered better than a  
t-prior, the latter can be used in blmer and should alleviate  
complete separation issues. I tend to use a normal-prior after  
performing Gelman's rescaling, but this is mainly because MCMCglmm  
only handles normal priors for the fixed effects (this may not be  
true). In a hierarchical model I'm not sure Gelman's advice holds:  
at least with a normal-prior it makes sense to increase the prior  
variance as the random-effect variances increase. If the prior  
variance is approximately v+pi^2/3, where v is the sum of the  
variance components, then the effects on the probability scale are  
quite close to being uniform on the 0,1 interval.

You can use the gelman.prior function to obtain the prior  
covariance matrix for your model. However, note that in the help  
file I say that the scale argument takes the standard deviation. In  
fact it takes the variance, but in the next version of MCMCglmm  
(coming soon) I have fixed this and it will take the standard  
deviation.

Cheers,

Jarrod


Gelman, A. et al. (2008) The Annals of Appled Statistics 2 4 1360-1383


Quoting Josie Galbraith <josie.galbraith at gmail.com> on Sat, 7 Mar  
2015 12:15:41 +1300:

Thanks Ben,
I didn't have problems with singular estimates of variance components with
this data set.  However, I have a few other pathogens/parasites that I'm
looking at (I'm running separate models for each), and after looking at all
of them some do have zero variances for the random effect, either in
addition to large parameter estimates or alongside reasonable parameter
estimates.
Should I be also be imposing a covariance prior in either of these cases?

As a related aside, my data are collected from individual birds - captured
over 4 sampling rounds (6 months apart).  While the majority of
observations are independent, there is a small proportion of birds that
were recaptured in a subsequent sampling round (between 2?15% of
observations, depending on which response variable).  I have modelled my
data both both with and without bird ID as a random effect.  Including it
seems to cause more problems with zero variances.  Is this because too few
of the birds have actually been resampled?

Cheers,
Josie

Josie Galbraith <josie.galbraith at ...> writes:

[snip]

I'm after some advice on how to choose which priors to use.  I gather I
need to impose a weak prior on the fixed effects of my model but no
covariance priors - is this correct?  Can I use a default prior (i.e. t,

or

normal defaults in the blme package) or does it depend on my data?  What

is

considered a suitably weak prior?

  If all you're trying to do is deal with complete separation (and not,
e.g. singular estimates of variance components [typically indicated
by zero variances or +/- 1 correlations, although I'm not sure those
are necessary conditions for singularity]), then it should be OK
to put the prior only on the fixed effects.  Generally speaking a
weak prior is one with a standard deviation that is large relative
to the expected scale of the effect (e.g. we might say sigma=10 is
large, but it won't be if the units of measurement are very small
so that a typical value of the mean is 100,000 ...)

I am running binomial models for epidemiology data (response variable is
presence/absence of lesions), with 2 fixed effects (FOOD: F/NF; SEASON:
Autumn/Spring) and a random effect (SITE: 8 levels).  The main goal of
these models is to test for an effect of the treatment 'FOOD.'  I'm
guessing from what I've read, that my model should be something like the
following:

This seems fairly reasonable at first glance.  Where were you seeing
the complete separation, though?   I would normally expect to
see at least one of the parameters still being reasonably large
if that's the case.

bglmer (LESION ~ FOOD*SEASON +(1|SITE), data = SEYE.df, family =

binomial,

fixef.prior = normal, cov.prior = NULL)

This is the output when I run the model:

Fixef prior: normal(sd = c(10, 2.5, ...), corr = c(0 ...),  

common.scale =

FALSE)
Prior dev  : 18.2419

Generalized linear mixed model fit by maximum likelihood (Laplace
Approximation) [
bglmerMod]
 Family: binomial ( logit )
Formula: LESION ~ FOOD * SEASON + (1 | SITE)
   Data: SEYE.df

[snip]

Random effects:
 Groups Name        Variance Std.Dev.
 SITE   (Intercept) 0.3064   0.5535
Number of obs: 178, groups: SITE, 8

Fixed effects:
                    Estimate Std. Error z value Pr(>|z|)
(Intercept)          -3.7664     1.4551  -2.588  0.00964 **
FOODNF                0.5462     1.6838   0.324  0.74567
SEASONSpring          1.7529     1.4721   1.191  0.23378
FOODNF:SEASONSpring  -0.8151     1.7855  -0.456  0.64803
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

[snip]

------------------------------

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*Josie Galbraith* MSc (hons)

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*University of Auckland *
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