Hi,
Quoting Margaret Evans mekev...@yahoo.com on Mon, 24 Sep 2012
22:56:48 +0100 (BST):
Hello all,
I have a few questions concerning the specification of flat priors
(on the probability scale) for a phylogenetic logistic regression in
MCMCglmm.
1) First, I'd like to verify my understanding of the default
priors in MCMCglmm. Specifically, are they flat on the probability
scale or not? It
seems like, from reading section 2.6 in the MCMCglmm course notes, that the
default priors for the fixed effects (intercept term, as well as
predictor[s]) are not flat on the probability scale.
You are correct, the default priors for the fixed effects are flatish
(as long as the predictors don't have very small variances) on the
logit scale but not on the probability scale. Generally this only
becomes an issue when you have (near) complete separation. For
categorical predictors this means there are very few 0's or 1's for
some level of the fixed effects. Gelman 2008 discusses priors for the
fixed effects in logistic regression. In
http://permalink.gmane.org/gmane.comp.lang.r.lme4.devel/8608 I provide
a function (prior.scale) for obtaining a Gelman like prior which can
be passed to B$V without having to rescale the predictors. B$V needs
to be scaled by the total variance + pi^2/3 with logit link.
2) I'm trying to use the alternative priors (flatter on the
probability scale) suggested in section 2.6 of the MCMCglmm course
notes.
For a model (phylogenetic logistic regression) with two fixed effects
(no intercept), the prior is as shown in the course notes:
prior.flat - list(B = list(mu = c(0, 0), V = diag(2)*(1 + pi^2/3)), R
= list(V = 1, fix=1), G = list(G1 = list(V = 1, nu = 0.002)))
For a model with a single fixed effect (no intercept), would it be:
prior.flat - list(B = list(mu = 0, V = (1 + pi^2/3)), R
= list(V = 1, fix=1), G = list(G1 = list(V = 1, nu = 0.002)))
DID I GET THIS RIGHT??? I have a bad feeling that the V term in the
B list is not right.
See above. In the special case of an intercept only, prior.scale will
return a 1 which should be scaled by the variance (rcov variance +
random effect variance + pi^2/3). This is what you have done except
you have set the random effect variances to zero.
3) Removing the intercept term is recommended in section 2.6 of the
MCMCglmm course notes, but this is justified in terms of
separation...all (or nearly all) zeros or ones in one treatment
class. For one, my predictor is continuous, and for two I don't
have this separation problem. Can anyone provide further insight
into the rationale for removing the intercept? I should decide to
include it or not based on the usual criteria (posterior
distribution of the intercept term strongly overlaps zero, BIC-type
information criterion??). If I exclude the intercept term, does that
mean I am forcing the regression through the origin?
Removing the intercept is not necessary if you use prior.scale.
Removing the intercept in the example of a two level fixed predictor
(with levels A and B) results in two parameters that refer to the
logit probability of being 1 in group A and the logit probability of
being 1 in group B. The prior is roughly flat for these probabilities.
If the intercept is not removed the default contrasts in R are such
that the intercept is the logit probability of being 1 in group A and
the other parameter is the *difference* in probability on the logit
scale of being 1 in group B versus group A. Using a prior from
prior.scale will result in the same prior regardless of the
parameterisation (i.e. whether the intercept is dropped or not).
prior.scale will also deal with continuous predictors in the same way
that Gelman recommends. Complete separation can be hard to detect when
there are multiple and/or continuous predictors. In the simplest case
no overlap in the predictor values associated with the zeros and the
ones would result in complete separation.
Generally I find that it is the prior on the variance components
rather than the fixed effects which have a greater influence on the
results. Pierre Villemereuil is about to publish a paper in Methods
in Ecology and Evolution looking at prior specifications in binary
models when estimating heritabilities from pedigreed populations which
may be useful.
Cheers,
Jarrod
with thanks,
Margaret
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