Question about zero-inflated Poisson glmer
NBinom was not really successfull unitl now, but will try to tune. Thanks for your help! One point I forgot to mention was that apart from my excess of zeros, the lowest data outcome is 10, so there is a gap between zeri and 10. Could that be somehow a problem?
On 27.06.2016 21:59, Thierry Onkelinx wrote:
If there is overdispersion, then try a negative binomial model or a
zero-inflated negative binomial model. If not try a zero-inflated
Poisson. Adding relevant covariates can reduce overdispersion.
ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature
and Forest
team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
Kliniekstraat 25
1070 Anderlecht
Belgium
To call in the statistician after the experiment is done may be no
more than asking him to perform a post-mortem examination: he may be
able to say what the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does
not ensure that a reasonable answer can be extracted from a given body
of data. ~ John Tukey
2016-06-27 17:46 GMT+02:00 Philipp Singer <killver at gmail.com
<mailto:killver at gmail.com>>:
Well, as posted beforehand the std dev is 9.5 ... so does not seem
too good then :/
Any other idea?
On 27.06.2016 17:31, Thierry Onkelinx wrote:
Dear Philipp,
You've been bitten by observation level random effects. I've put
together a document about it on http://rpubs.com/INBOstats/OLRE.
Bottomline you're OKish when the standard devation of the OLRE
smaller than 1. You're in trouble when it's above 3. In between
you need to check the model carefully.
Best regards,
ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for
Nature and Forest
team Biometrie & Kwaliteitszorg / team Biometrics & Quality
Assurance
Kliniekstraat 25
1070 Anderlecht
Belgium
To call in the statistician after the experiment is done may be
no more than asking him to perform a post-mortem examination: he
may be able to say what the experiment died of. ~ Sir Ronald
Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer
does not ensure that a reasonable answer can be extracted from a
given body of data. ~ John Tukey
2016-06-27 16:17 GMT+02:00 Philipp Singer <killver at gmail.com
<mailto:killver at gmail.com>>:
Here is the fitted vs. residual plot for the
observation-level poisson model where the observation level
has been removed as taken from:
https://stat.ethz.ch/pipermail/r-sig-mixed-models/2013q3/020817.html
So basically the prediction is always close to zero.
Note that this is just on a very small sample (1000 data points).
If I fit a nbinom2 to this smalle sample, I get predictions
that are always around ~20 (but never zero). Both plots are
attached.
What I am wondering is whether I can do inference on a fixed
parameter in my model which is my main task of this study.
The effect is similar in the different models and in general
I am only itnerested in whether it is positive/negative and
"significant" which it is. However, as can be seen, the
prediction looks not too good here.
2016-06-27 15:18 GMT+02:00 Philipp Singer <killver at gmail.com
<mailto:killver at gmail.com>>:
The variance is:
Conditional model:
Groups Name Variance Std.Dev.
obs (Intercept) 8.991e+01 9.4823139
2016-06-27 15:06 GMT+02:00 Thierry Onkelinx
<thierry.onkelinx at inbo.be <mailto:thierry.onkelinx at inbo.be>>:
Dear Philipp,
How strong is the variance of the observation level
random effect? I would trust a model with large OLRE
variance.
Best regards,
Thierry
ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research
Institute for Nature and Forest
team Biometrie & Kwaliteitszorg / team Biometrics &
Quality Assurance
Kliniekstraat 25
1070 Anderlecht
Belgium
To call in the statistician after the experiment is
done may be no more than asking him to perform a
post-mortem examination: he may be able to say what
the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for
an answer does not ensure that a reasonable answer
can be extracted from a given body of data. ~ John Tukey
2016-06-27 14:59 GMT+02:00 Philipp Singer
<killver at gmail.com <mailto:killver at gmail.com>>:
I have now played around more with the data an
the models both using lme4
and glmmTMB.
I can report the following:
Modeling the data with a zero-inflated Poisson
improves the model
significantly. However, when calling predict and
simulating rpoissons, I
end up with nearly no values that are zero (in
the original data there are
96% zero).
When I model the data with overdisperion by
including an observation-level
random effect, I can also improve the model (not
surprisingly due to the
random effect). When I predict outcomes by
ignoring the observation-level
random effect (in lme4), I receive bad prediction
if I compare it to the
original data. While many zeros can be captured
(of course), the positive
outcomes can not be captured well.
Combining zero inflation and overdispersion
further improves the model, but
I can only do that with glmmTMB and then have
troubles doing predictions
ignoring the observation-level random effect.
Another side question:
In lme4, when I do:
m = glm(x~1,family="poisson")
rpois(n=len(data),lambda=predict(m,
type='response',re.form=NA)
vs.
simulate(1,m,re.form=NA)
I receive different outcomes? Do I understand
these function wrongly?
Would appreciate some more help/pointers!
Thanks,
Philipp
2016-06-24 15:52 GMT+02:00 Philipp Singer
<killver at gmail.com <mailto:killver at gmail.com>>:
> Thanks - I started an issue there to answer
some of my questions.
>
> Regarding the installation: I was trying to
somehow do it in anaconda with
> a specific R kernel and had some issues. I am
trying to resort that with
> the anaconda guys though, if I have a tutorial
on how to properly setup
> glmmTMB in anaconda, I will let you know. The
install worked fine in my
> standard R environment.
>
>
> On 24.06.2016 15 <tel:24.06.2016%2015>:40, Ben
Bolker wrote:
>
>> Probably for now the glmmTMB issues page is best.
>>
>> When you go there:
>>
>> - details on installation problems/hiccups
would be useful
>> - a reproducible example for the problem
listed below would be useful
>> - dispformula is for allowing
dispersion/residual variance to vary
>> with covariates (i.e., modeling
heteroscedasticity)
>>
>> cheers
>> Ben Bolker
>>
>>
>> On 16-06-24 09:13 AM, Philipp Singer wrote:
>>
>>> Update, I tried it like that, but receive an
error message.
>>>
>>> Warning message:
>>> In nlminb(start = par, objective = fn,
gradient = gr): NA/NaN function
>>> evaluation
>>>
>>> Error in solve.default(hessian.fixed): Lapack
routine dgesv: system is
>>> exactly singular: U[3,3] = 0
>>> Traceback:
>>>
>>> 1. glmmTMB(y ~ 1 + x + (1 | b),
>>> . data = data, family = "poisson",
dispformula = ~1 + x)
>>> 2. sdreport(obj)
>>> 3. solve(hessian.fixed)
>>> 4. solve(hessian.fixed)
>>> 5. solve.default(hessian.fixed)
>>>
>>> Any ideas on that?
>>>
>>> BTW: Is it fine to post glmmTMB questions
here, or should I rather use
>>> the github issue page, or is there maybe a
dedicated mailing list?
>>>
>>> Thanks,
>>> Philipp
>>>
>>> On 24.06.2016 14:35, Philipp Singer wrote:
>>>
>>>> It indeed seems to run quite fast; had some
trouble installing, but
>>>> works now on my 3.3 R setup.
>>>>
>>>> One question I have is regarding the
specification of dispersion as I
>>>> need to specify the dispformula. What is the
difference here between
>>>> just specifying fixed effects vs. also the
random effects?
>>>>
>>>> On 23.06.2016 23:07, Mollie Brooks wrote:
>>>>
>>>>> glmmTMB does crossed RE. Ben did some
timings in vignette("glmmTMB")
>>>>> and it was 2.3 times faster than glmer for
one simple GLMM.
>>>>>
>>>>>
>>>>> On 23Jun 2016, at 10:44, Philipp Singer
<killver at gmail.com <mailto:killver at gmail.com>> wrote:
>>>>>>
>>>>>> Did try glmmADMB but unfortunately it is
way too slow for my data.
>>>>>>
>>>>>> Did not know about glmmTMB, will try it
out. Does it work with
>>>>>> crossed random effects and how does it
scale with more data? I will
>>>>>> check the docu and try it though. Thanks
for the info.
>>>>>>
>>>>>> On 23.06.2016 19:14, Ben Bolker wrote:
>>>>>>
>>>>>>> I would also comment that glmmTMB is
likely to be much faster
>>>>>>> than the
>>>>>>> lme4-based EM approach ...
>>>>>>>
>>>>>>> cheers
>>>>>>> Ben B.
>>>>>>>
>>>>>>> On 16-06-23 12:47 PM, Mollie Brooks wrote:
>>>>>>>
>>>>>>>> Hi Philipp,
>>>>>>>>
>>>>>>>> You could also try fitting the model
with and without ZI using
>>>>>>>> either
>>>>>>>> glmmADMB or glmmTMB. Then compare the
AICs. I believe model
>>>>>>>> selection
>>>>>>>> is useful for this, but I could be
missing something since the
>>>>>>>> simulation procedure that Thierry
described seems to recommended
>>>>>>>> more
>>>>>>>> often.
>>>>>>>>
>>>>>>>> https://github.com/glmmTMB/glmmTMB
>>>>>>>> http://glmmadmb.r-forge.r-project.org
>>>>>>>>
>>>>>>>> glmmTMB is still in the development
phase, but we?ve done a lot of
>>>>>>>> testing.
>>>>>>>>
>>>>>>>> cheers, Mollie
>>>>>>>>
>>>>>>>> ------------------------ Mollie Brooks,
PhD Postdoctoral Researcher,
>>>>>>>> Population Ecology Research Group
Department of Evolutionary Biology
>>>>>>>> & Environmental Studies, University of
Z?rich
>>>>>>>> http://www.popecol.org/team/mollie-brooks/ >>>>>>>> >>>>>>>> >>>>>>>> On 23Jun 2016, at 8:22, Philipp Singer
<killver at gmail.com <mailto:killver at gmail.com>> wrote:
>>>>>>>>>
>>>>>>>>> Thanks, great information, that is
really helpful.
>>>>>>>>>
>>>>>>>>> I agree that those are different
things, however when using a
>>>>>>>>> random effect for overdispersion, I can
simulate the same number of
>>>>>>>>> zero outcomes (~95%).
>>>>>>>>>
>>>>>>>>> On 23.06.2016 15:50, Thierry Onkelinx
wrote:
>>>>>>>>>
>>>>>>>>>> Be careful when using overdispersion
to model zero-inflation.
>>>>>>>>>> Those are two different things.
>>>>>>>>>>
>>>>>>>>>> I've put some information together in
>>>>>>>>>> http://rpubs.com/INBOstats/zeroinflation
>>>>>>>>>>
>>>>>>>>>> ir. Thierry Onkelinx Instituut voor
natuur- en bosonderzoek /
>>>>>>>>>> Research Institute for Nature and
Forest team Biometrie &
>>>>>>>>>> Kwaliteitszorg / team Biometrics &
Quality Assurance
>>>>>>>>>> Kliniekstraat 25 1070 Anderlecht Belgium
>>>>>>>>>>
>>>>>>>>>> To call in the statistician after the
experiment is done may be
>>>>>>>>>> no more than asking him to perform a
post-mortem examination: he
>>>>>>>>>> may be able to say what the experiment
died of. ~ Sir Ronald
>>>>>>>>>> Aylmer Fisher The plural of anecdote
is not data. ~ Roger
>>>>>>>>>> Brinner The combination of some data
and an aching desire for an
>>>>>>>>>> answer does not ensure that a
reasonable answer can be extracted
>>>>>>>>>> from a given body of data. ~ John Tukey
>>>>>>>>>>
>>>>>>>>>> 2016-06-23 12:42 GMT+02:00 Philipp
Singer <killver at gmail.com <mailto:killver at gmail.com>
>>>>>>>>>> <mailto:killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com
<mailto:killver at gmail.com>>>>:
>>>>>>>>>>
>>>>>>>>>> Thanks! Actually, accounting for
overdispersion is super
>>>>>>>>>> important as it seems, then the zeros
can be captured well.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On 23.06.2016 11:50, Thierry Onkelinx
wrote:
>>>>>>>>>>
>>>>>>>>>>> Dear Philipp,
>>>>>>>>>>>
>>>>>>>>>>> 1. Fit a Poisson model to the data.
2. Simulate a new response
>>>>>>>>>>> vector for the dataset according to
the model. 3. Count the
>>>>>>>>>>> number of zero's in the simulated
response vector. 4. Repeat
>>>>>>>>>>> step 2 and 3 a decent number of time
and plot a histogram of
>>>>>>>>>>> the number of zero's in the
simulation. If the number of zero's
>>>>>>>>>>> in the original dataset is larger
than those in the
>>>>>>>>>>> simulations, then the model can't
capture all zero's. In such
>>>>>>>>>>> case, first try to update the model
and repeat the procedure.
>>>>>>>>>>> If that fails, look for zero-inflated
models.
>>>>>>>>>>>
>>>>>>>>>>> Best regards,
>>>>>>>>>>>
>>>>>>>>>>> ir. Thierry Onkelinx Instituut voor
natuur- en bosonderzoek /
>>>>>>>>>>> Research Institute for Nature and
Forest team Biometrie &
>>>>>>>>>>> Kwaliteitszorg / team Biometrics &
Quality Assurance
>>>>>>>>>>> Kliniekstraat 25 1070 Anderlecht Belgium
>>>>>>>>>>>
>>>>>>>>>>> To call in the statistician after the
experiment is done may
>>>>>>>>>>> be no more than asking him to perform
a post-mortem
>>>>>>>>>>> examination: he may be able to say
what the experiment died of.
>>>>>>>>>>> ~ Sir Ronald Aylmer Fisher The plural
of anecdote is not data.
>>>>>>>>>>> ~ Roger Brinner The combination of
some data and an aching
>>>>>>>>>>> desire for an answer does not ensure
that a reasonable answer
>>>>>>>>>>> can be extracted from a given body of
data. ~ John Tukey
>>>>>>>>>>>
>>>>>>>>>>> 2016-06-23 11:27 GMT+02:00 Philipp
Singer <killver at gmail.com <mailto:killver at gmail.com>
>>>>>>>>>>> <mailto:killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com
<mailto:killver at gmail.com>>>>:
>>>>>>>>>>>
>>>>>>>>>>> Thanks Thierry - That totally makes
sense. Is there some way of
>>>>>>>>>>> formally checking that, except
thinking about the setting and
>>>>>>>>>>> underlying processes?
>>>>>>>>>>>
>>>>>>>>>>> On 23.06.2016 11:04, Thierry Onkelinx
wrote:
>>>>>>>>>>>
>>>>>>>>>>>> Dear Philipp,
>>>>>>>>>>>>
>>>>>>>>>>>> Do you have just lots of zero's, or
more zero's than the
>>>>>>>>>>>>
>>>>>>>>>>> Poisson
>>>>>>>>>>>
>>>>>>>>>>>> distribution can explain? Those are
two different things.
>>>>>>>>>>>>
>>>>>>>>>>> The example
>>>>>>>>>>>
>>>>>>>>>>>> below generates data from a Poisson
distribution and has
>>>>>>>>>>>>
>>>>>>>>>>> 99% zero's
>>>>>>>>>>>
>>>>>>>>>>>> but no zero-inflation. The second
example has only 1%
>>>>>>>>>>>>
>>>>>>>>>>> zero's but is
>>>>>>>>>>>
>>>>>>>>>>>> clearly zero-inflated.
>>>>>>>>>>>>
>>>>>>>>>>>> set.seed(1) n <- 1e8 sim <- rpois(n,
lambda = 0.01) mean(sim
>>>>>>>>>>>> == 0) hist(sim)
>>>>>>>>>>>>
>>>>>>>>>>>> sim.infl <- rbinom(n, size = 1, prob
= 0.99) * rpois(n,
>>>>>>>>>>>>
>>>>>>>>>>> lambda = 1000)
>>>>>>>>>>>
>>>>>>>>>>>> mean(sim.infl == 0) hist(sim.infl)
>>>>>>>>>>>>
>>>>>>>>>>>> So before looking for zero-inflated
models, try to model
>>>>>>>>>>>>
>>>>>>>>>>> the zero's.
>>>>>>>>>>>
>>>>>>>>>>>> Best regards,
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> ir. Thierry Onkelinx Instituut voor
natuur- en bosonderzoek /
>>>>>>>>>>>> Research Institute
>>>>>>>>>>>>
>>>>>>>>>>> for Nature
>>>>>>>>>>>
>>>>>>>>>>>> and Forest team Biometrie &
Kwaliteitszorg / team Biometrics
>>>>>>>>>>>> & Quality
>>>>>>>>>>>>
>>>>>>>>>>> Assurance
>>>>>>>>>>>
>>>>>>>>>>>> Kliniekstraat 25 1070 Anderlecht Belgium
>>>>>>>>>>>>
>>>>>>>>>>>> To call in the statistician after
the experiment is done
>>>>>>>>>>>>
>>>>>>>>>>> may be no
>>>>>>>>>>>
>>>>>>>>>>>> more than asking him to perform a
post-mortem examination:
>>>>>>>>>>>>
>>>>>>>>>>> he may be
>>>>>>>>>>>
>>>>>>>>>>>> able to say what the experiment died
of. ~ Sir Ronald
>>>>>>>>>>>>
>>>>>>>>>>> Aylmer Fisher
>>>>>>>>>>>
>>>>>>>>>>>> The plural of anecdote is not data.
~ Roger Brinner The
>>>>>>>>>>>> combination of some data and an
aching desire for an
>>>>>>>>>>>>
>>>>>>>>>>> answer does
>>>>>>>>>>>
>>>>>>>>>>>> not ensure that a reasonable answer
can be extracted from a
>>>>>>>>>>>>
>>>>>>>>>>> given body
>>>>>>>>>>>
>>>>>>>>>>>> of data. ~ John Tukey
>>>>>>>>>>>>
>>>>>>>>>>>> 2016-06-23 10:07 GMT+02:00 Philipp
Singer
>>>>>>>>>>>>
>>>>>>>>>>> <killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com <mailto:killver at gmail.com>>
>>>>>>>>>>> <mailto:killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com
<mailto:killver at gmail.com>>>
>>>>>>>>>>>
>>>>>>>>>>>> <mailto:killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com <mailto:killver at gmail.com>>
>>>>>>>>>>>> <mailto:killver at gmail.com
<mailto:killver at gmail.com>
<mailto:killver at gmail.com
<mailto:killver at gmail.com>>>>>:
>>>>>>>>>>>>
>>>>>>>>>>>> Dear group - I am currently fitting
a Poisson glmer
>>>>>>>>>>>>
>>>>>>>>>>> where I have
>>>>>>>>>>>
>>>>>>>>>>>> an excess of outcomes that are zero
(>95%). I am now
>>>>>>>>>>>>
>>>>>>>>>>> debating on
>>>>>>>>>>>
>>>>>>>>>>>> how to proceed and came up with
three options:
>>>>>>>>>>>>
>>>>>>>>>>>> 1.) Just fit a regular glmer to the
complete data. I am
>>>>>>>>>>>>
>>>>>>>>>>> not fully
>>>>>>>>>>>
>>>>>>>>>>>> sure how interpret the coefficients
then, are they more
>>>>>>>>>>>>
>>>>>>>>>>> optimizing
>>>>>>>>>>>
>>>>>>>>>>>> towards distinguishing zero and
non-zero, or also
>>>>>>>>>>>>
>>>>>>>>>>> capturing the
>>>>>>>>>>>
>>>>>>>>>>>> differences in those outcomes that
are non-zero?
>>>>>>>>>>>>
>>>>>>>>>>>> 2.) Leave all zeros out of the data
and fit a glmer to
>>>>>>>>>>>>
>>>>>>>>>>> only those
>>>>>>>>>>>
>>>>>>>>>>>> outcomes that are non-zero. Then, I
would only learn about
>>>>>>>>>>>> differences in the non-zero outcomes
though.
>>>>>>>>>>>>
>>>>>>>>>>>> 3.) Use a zero-inflated Poisson
model. My data is quite
>>>>>>>>>>>> large-scale, so I am currently
playing around with the EM
>>>>>>>>>>>> implementation of Bolker et al. that
alternates between
>>>>>>>>>>>>
>>>>>>>>>>> fitting a
>>>>>>>>>>>
>>>>>>>>>>>> glmer with data that are weighted
according to their zero
>>>>>>>>>>>> probability, and fitting a logistic
regression for the
>>>>>>>>>>>>
>>>>>>>>>>> probability
>>>>>>>>>>>
>>>>>>>>>>>> that a data point is zero. The
method is elaborated for
>>>>>>>>>>>>
>>>>>>>>>>> the OWL
>>>>>>>>>>>
>>>>>>>>>>>> data in:
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> <
>>>>>>>>>>>
>>>>>>>>>>> >
>>>>>>>>>>>
>>>>>>>>>> I am not fully sure how to interpret
the results for the
>>>>>>>
>>>>>>>> zero-inflated version though. Would I
need to interpret the
>>>>>>>>>>>> coefficients for the result of the
glmer similar to as
>>>>>>>>>>>>
>>>>>>>>>>> I would do
>>>>>>>>>>>
>>>>>>>>>>>> for my idea of 2)? And then on top
of that interpret the
>>>>>>>>>>>> coefficients for the logistic
regression regarding whether
>>>>>>>>>>>> something is in the perfect or
imperfect state? I am
>>>>>>>>>>>>
>>>>>>>>>>> also not
>>>>>>>>>>>
>>>>>>>>>>>> quite sure what the common approach
for the zformula is
>>>>>>>>>>>>
>>>>>>>>>>> here. The
>>>>>>>>>>>
>>>>>>>>>>>> OWL elaborations only use
zformula=z~1, so no random
>>>>>>>>>>>>
>>>>>>>>>>> effect; I
>>>>>>>>>>>
>>>>>>>>>>>> would use the same formula as for
the glmer.
>>>>>>>>>>>>
>>>>>>>>>>>> I am appreciating some help and
pointers.
>>>>>>>>>>>>
>>>>>>>>>>>> Thanks! Philipp
>>>>>>>>>>>>
>>>>>>>>>>>>
_______________________________________________
>>>>>>>>>>>> R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>
>>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>>
>>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>>
>>>>>>>>>>>>
>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>
>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>>>
>>>>>>>>>>>
>>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>
>>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>>
>>>>>>>>>>>>
>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>
>>>>>>>>>>>
<mailto:R-sig-mixed-models at r-project.org
<mailto:R-sig-mixed-models at r-project.org>>>>
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>>>>>>>>>>>
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>>>>>>>>>>>>
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>>>>>>>>>>>>
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>>>>>>>>>>>
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>>>>>>>>>>>
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>>>>>>>>>>>
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>>>>>>>>>>>
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>>>>>>>>>>>
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>>>>>>>>>
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<mailto:R-sig-mixed-models at r-project.org>>
>>>>>>>>>
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>>>>>>>>>
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>>>>>>>> R-sig-mixed-models at r-project.org
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>>>>>>>>
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>>>>>>>
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>>>>>> R-sig-mixed-models at r-project.org
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