[R-sig-ME] Choosing appropriate priors for bglmer mixed models in blme
Josie Galbraith
josie.galbraith at gmail.com
Mon Mar 9 22:30:12 CET 2015
Hi Jarrod,
I'm pretty sure it is a complete separation issue. This is the xtab of
counts for the main factors:
LESION 0 1
SEASON FOOD
Autumn NF 38 2
F 21 0
Spring NF 27 3
F 76 11
Lesion incidences were low generally, but particularly so in Autumn (and
fewer replicates in Autumn).
Thanks again,
Josie
On Mon, Mar 9, 2015 at 8:50 PM, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote:
> Hi Josie,
>
> Is the problem you are having because of complete separation, either
> because you have some very good predictors of lesions and/or you have low
> replication for some factor levels? If so blmer with Gelman's recommended
> prior (not the diffuse prior) should do a reasonable job of allowing
> sensible inferences to be made. However, as Ben said in an earlier post,its
> not clear that this is the problem.
>
> Similar issues are possible with the random effects, but this tends to be
> rare because they are constrained. I only see it when the variance
> component is very large, not zero as here.
>
> If the perceived problem is zero variance estimates, I'm not sure why this
> is a problem. If the true variances are zero you should expect a MLE of
> zero 50% of the time. With only 8 levels of the random effect, you should
> expect an MLE of zero often, even if the true variance is moderate. The
> same power issues will generate MLE correlations of -1 and 1.
>
> Cheers,
>
> Jarrod
>
>
>
>
>
>
> Quoting Josie Galbraith <josie.galbraith at gmail.com> on Mon, 9 Mar 2015
> 13:08:59 +1300:
>
> 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
>>> 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/>
>>> >>>
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>>> >>>
>>> >>> _______________________________________________
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>>> >>
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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/>
>>
>>
>
>
> --
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
>
>
>
--
*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/>
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