Choosing appropriate priors for bglmer mixed models in blme

Thanks very much Jarrod & Vince for your inputs.
Admittedly this analysis is stretching my level of understanding!

From a practical point of view, given that time is of the essence in
writing up my PhD, if I only want to test the main effects of my model
(rather than make predictions etc), is this something I can achieve in
blme? (ie testing model terms using LRTs).  If so should I be using blme
for this?

Or should I really be working in MCMCglmm (which I haven't used before -
another learning curve!)?  Any further thoughts on using normal priors
rather than Cauchy?

Thanks again,
Josie

Message: 2

Hi Vince,

For a given difference on the logit scale between (lets say) two
treatment groups then the difference on the observed scale depends on
the magnitude of the variance components. For logit effects beta1 and
beta2, the expected difference is approximately:

plogis(beta1/sqrt(1+c2*v))-plogis(beta2/sqrt(1+c2*v))

where v is the variance component and c2 = (16*sqrt(3)/(15*pi))^2.

If a prior (Cauchy or otherwise) was set up that was invariant to v
then it would imply different prior beliefs about the magnitude of the
difference (on the observed scale) depending on v. For the normal
prior it would imply that when v is large we should expect smaller
differences between treatment groups. This maybe OK (I'm not sure) but
if not is there a way to make it invariant for the t/Cauchy prior? For
the normal you can make the scale = sqrt(v+pi^2/3) which seems to work
OKish.

Cheers,

Jarrod




Quoting Vincent Dorie <vjd4 at nyu.edu> on Sat, 7 Mar 2015 09:47:40 -0500:

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]

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

--
*Josie Galbraith* MSc (hons)

PhD candidate
*University of Auckland *
Joint Graduate School in Biodiversity and Biosecurity ? School of
Biological Sciences ? Tamaki Campus ? Private Bag 92019 ? Auckland

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--
*Josie Galbraith* MSc (hons)

PhD candidate
*University of Auckland *
Joint Graduate School in Biodiversity and Biosecurity ? School of
Biological Sciences ? Tamaki Campus ? Private Bag 92019 ? Auckland 1142* ?
P:* 09-373 7599 ext. 83132* ? E:* josie.galbraith at gmail.com* ? W: * UoA Web
Profile <https://unidirectory.auckland.ac.nz/profile/jgal026> and
*www.birdfeedingnz.weebly.com/* <http://birdfeedingnz.weebly.com/>