glmer Z-test with individual random effects

Hi John,

I'm not sure which part of the MCMCglmm routine is unclear but in a nut shell:

The data are distributed according to the specified distribution (e.g. Poisson ) with the dsitribution parameter equal to the linear predictor (nu) on the inverse link scale:

y ~ Pois(exp(nu))

A linear (mixed) model is proposed for the linear predictor:

nu ~ N(Xb+Zu, I*Ve)

In all models a "units" term is included which is the same as an observational-level random effect.  This is the residual term in the linear part of the model.

Priors are passed as  a list of the form:

prior=list(B=list(),
              R=list(),
              G=list(G1=list(),
                          G2=list()))


where B is the prior for the fixed effects, R is the prior for Ve (or possibly a residual covariance matrix depending on the model) and the elements of G are the priors for each of the random effect (co)variances.


The prior for b is multivariate normal Pr(b) ~ N(B$mu, B$V)

The prior for Ve is inverse-Wishart Pr(Ve) ~ IW(R$nu, R$V*R$nu) where the first term is the degree of belief and R$V*R$nu is the scale matrix

The prior for the random effects variances (Vg) are set in the same way as Ve, although its possible to also fit parameter expanded priors.

When the variance term is scalar (rather than a covariance matrix) then the inverse-Wishart and inverse-gamma are the same and IW(R$nu, R$V*R$nu) = IG(R$nu/2, R$V*R$nu/2) where the two parameters are the shape and scale. Your specification R=list(V=1, nu=0.002)  is therefore  IG(0.001, 0.001) which used to be used a lot (and probably still is) as a "weak" prior by users of WinBUGS.  The REML estimator for Ve should be equal to the posterior mode in MCMCglmm when the improper prior R=list(V=0, nu=-2)  is used. 0 will not be accepted by MCMCglmm so you would have to use something small (e.g. 1e-6). The prior is improper so you have to be careful when using it, and although the point estimate may have some useful properties (the same as the REML estimate) the full posterior distribution may not be ideal with small sample sizes.

The sampling scheme is:

update nu using MH updates or slice sampling
Gibbs sample b and u together
Gibbs sample (co)variance components sequentially.

For other models  such as ordinal regression or simultaneous/recursive models there are other schemes also, but these are quite specialised.

Cheers,

Jarrod



















On 11 Nov 2010, at 23:45, John Maindonald wrote:

The Wald tests (as represented in glm and glmer output of z-statistics
and p-values) are approximate for both glm models and glmm models
with non-identity links. The approximation can fail badly if the link is
highly non-linear over a region of the response that is relevant for a
parameter of interest.  The Hauck-Donner phenomenon, where the
z-statistic decreases as the effect estimate increases, is an extreme
example.
(This happens, e.g., as one of the levels being compared gives a
fitted value that moves close to a binomial proportion of 0 or 1, or
close to a Poisson estimate of 0.)

The additional complication for a glmm is that the SE may have
two components -- e.g., a poisson or binomial error, and a random
normal error that is added on the scale of the linear predictor.  This
random normal error somewhat alleviates variance change effects
(including Hauck-Donner) that result from non-linearity in the link,
while adding uncertainty to the SE estimate.  If the contribution from
the glm family error term is largish relative to contribution from the
random normal error (> 3 or 4  times as large?), treating the z-statistic
as normal may not in many circumstances be too unreasonable,
even if the relevant degrees of freedom are as small as maybe 4
(e.g., where the test is for consistency across 5 locations).

On data where I seem to have this situation, fairly consistently
across a number of responses, I initially used MCMCglmm().
I was concerned about the contribution from the random normal
error (including a contribution from an observation level random
effect term).  I found I hat I had to choose a somewhat informative
prior (inverse Wishart with V=1, nu=0.002) to consistently get
convergence.  With this prior, the MCMCglmm credible intervals
were remarkably close to glmer confidence intervals, treating the
z-statistics as normal.

No doubt the effect of the chosen prior is to insist that the true
random normal error variance is fairly close to the variance as
estimated by glmer().  A frustration (for me, at least) with using
MCMCglmm is that I do not know just what MCMCglmm() is doing
in this respect, short of doing some careful investigative exploration
of the MCMC simulation results (which is an after-the-event check,
where I'd like to know before the event).  Comments, Jarrod?

All this is to emphasise that we are not in the arena, unless the
relevant degrees of freedom are large, of precise science.

John Maindonald             email: john.maindonald at anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 12/11/2010, at 2:18 AM, Ben Bolker wrote:

On 11/11/2010 09:58 AM, Jens ?str?m wrote:

Dear list,

As I have read (Bolker et al. 2009 TREE), the Wald Z test is only
appropriate for GLMMs in cases without overdispersion.

Assuming we use family=poisson with lmer and tackle overdispersion by
incorporating an individual random effect AND this adequately "reduces"
the overdispersion, is it then OK to use the Wald z test as reported by
lmer?

In other words, are the p-values reported by lmer in those cases
useful/"correct"? Or do they suffer from the usual problems with
figuring out the number of parameters used by the random effects?

They are equivalent to assuming an infinite/large 'denominator degrees
of freedom'.  If you have a large sample size (both a large number of
total samples relative to the number of parameters, and a large number
of random-effects levels/blocks) then this should be reasonable -- if
not, then yes, the 'usual problems with figuring out the number of
parameters' is relevant.  On the other hand, if you're willing to assume
that the sample size is large, then likelihood ratio rests
(anova(model1,model2)) are probably better than the Wald tests anyway.

Secondly, is it good practice to judge lmer's capability of "reducing"
the overdispersion by summing the squared residuals (pearson) and
compare this to a chi square distribution (with N-1 degrees of freedom)?

I would say this is reasonable, although again it's a rough guide
because the true degrees of freedom are a bit fuzzy -- it should
probably be at most N-(fixed effect degrees of freedom)?

Would be happy to hear any conflicting opinions.

Ben Bolker

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