significance test of random and fixed effects in (quasi) poisson GLMM

Dear Vincent,

some time ago I posted a question on Poisson GLMM for overdispersed
data, including a simple simulation in order to compare the reliability
of glmmPQL and glmer.
See
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2010q1/003289.html 


While glmmPQL returns the correct estimates, glmer largely
overestimated the sigma (corresponding to the overdispersion), producing
an inflated within-group residual variance.
This odd behaviour seems to be confirmed by your analysis.

As pointed out in the response I had to my question, the quasi-Poisson
is not a distribution and the results are not grounded on an appropriate
statistical theory. Anyway, as in your case, the quasipoisson family is
currently used and I would expect the command to return (approximate)
correct results.

My suggestion is to repeat the analysis with glmmPQL, even if this
doesn't solve your problem to run a test. To my knowledge, the
approximation used by the penalized quasi-likelihood method is
reasonable for Poisson data and a moderate number of counts (McCulloch &
Searle say with mean count of 7 or higher). Interestingly, the command
always estimates the sigma (not fixed to 1 as in Poisson) even with the
simple poisson family.

I hope this helps

Antonio Gasparrini
Public and Environmental Health Research Unit (PEHRU)
London School of Hygiene & Tropical Medicine
Keppel Street, London WC1E 7HT, UK
Office: 0044 (0)20 79272406 - Mobile: 0044 (0)79 64925523

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

Message: 3
Date: Fri, 12 Mar 2010 15:46:23 +0100
From: Vincent Kint <Vincent.Kint at ees.kuleuven.be>
To: "r-sig-mixed-models at r-project.org"
<r-sig-mixed-models at r-project.org>
Subject: [R-sig-ME] significance test of random and fixed effects in
(quasi) poisson GLMM
Message-ID:
<562EA47F252E594B826D3B440E0B34A21288E5803C at ICTS-S-EXC2-CA.luna.kuleuven.be>

Content-Type: text/plain

Dear list members,

I am new to this list, and new to generalised mixed modelling.

My aim is to develop a model for tree branchiness (number of branches
per tree, with trees measured in different plots) with both tree and
plot-level predictors. My choice was for a generalised model using the
poisson family, since I have count data. And for a mixed approach since
I have a nested design.

I built a first model using the lme4 package (see below). My question
is: is there an approximate test for the significance of the random
effect? From previous posts on this list, I understand that such a test
is not always reliable, and good alternatives are not implemented yet.
But from my perspective of an applied modeller, even an approximate test
(or even a rule of thumb) would be helpful in making a decision. Indeed,
if the random effect turns out to be likely not significant, I could do
with a more simple GLM.

In a second step I tried to correct for overdispersion by running the
same model as a quasi GLMM. The output is also given below. Here I have
the same question as before, but now also concerning the fixed effects.
Additionally, I wonder whether I may have made a mistake in implementing
this model, since I get a result where nearly all the variation is
attributed to the error term, and (at a first glance) the random effect
and all the fixed predictors seem to be irrelevant.

I attach the output of both models below.
Thanks for all suggestions on how to proceed.
Vincent


#1. The GLMM model
> form1<-formula(response ~ TreeHeight + DBH + TreeAge + Vplot + mF +
mL + (1 | plots))
> M.glmm<-lmer(form1, data=data, family=poisson)
> summary(M.glmm)
Generalized linear mixed model fit by the Laplace approximation
Formula: form1
   Data: data
AIC  BIC logLik deviance
976 1008   -480      960
Random effects:
Groups Name        Variance Std.Dev.
plots  (Intercept) 0.044913 0.21193
Number of obs: 399, groups: plots, 30

Fixed effects:
              Estimate Std. Error z value Pr(>|z|)
(Intercept)  6.0090427  1.4143768   4.249 2.15e-05 ***
TreeHeight  -0.0398146  0.0125816  -3.165 0.001553 **
DBH          0.0032600  0.0006599   4.940 7.80e-07 ***
TreeAge     -0.1193541  0.0242996  -4.912 9.03e-07 ***
Vplot       -0.0060713  0.0016115  -3.768 0.000165 ***
mF          -0.4838699  0.1672462  -2.893 0.003814 **
mL           0.4878563  0.1731139   2.818 0.004831 **
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.?
0.1 ? ? 1



#2. The same GLMM model with overdispersion
> M.glmm.q<-lmer(form1, data=data, family=quasipoisson)
> summary(M.glmm.q)
Generalized linear mixed model fit by the Laplace approximation
Formula: form1
   Data: data
AIC  BIC logLik deviance
978 1014   -480      960
Random effects:
Groups   Name        Variance Std.Dev.
plots    (Intercept)  1.6478  1.2837
Residual             36.6894  6.0572
Number of obs: 399, groups: plots, 30

Fixed effects:
             Estimate Std. Error t value
(Intercept)  6.009043   8.567129   0.701
TreeHeight  -0.039815   0.076209  -0.522
DBH          0.003260   0.003997   0.816
TreeAge     -0.119354   0.147187  -0.811
Vplot       -0.006071   0.009761  -0.622
mF          -0.483870   1.013039  -0.478
mL           0.487856   1.048581   0.465

_____________________________________
dr. ir. V. KINT
Forest Ecology and Management
Division Forest, Nature and Landscape
K.U.Leuven
Celestijnenlaan 200E - B-3001 Leuven
Tel.: +32 16 32 97 69
Fax:  +32 16 32 97 60
vincent.kint at ees.kuleuven.be 

www.kuleuven.be/forecoman<http://www.kuleuven.be/forecoman>

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