[R-sig-ME] parallel MCMCglmm, RNGstreams, starting values & priors
Jarrod Hadfield
j.hadfield at ed.ac.uk
Wed Aug 27 19:39:59 CEST 2014
Hi Ruben,
1) it did not return an error with rcov = ~trait:units because you
used R1=rpois(2,1)+1 and yet this specification only fits a single
variance (not a 2x2 covariance matrix). R1=rpois(2,1)+1 is a bit of a
weird specification since it has to be integer. I would obtain
starting values using rIW().
2) yes, that will prevent convergence
3) a) how many effective samples do you have for each parameter? and
b) are you getting extreme category problems/numerical issues? If you
store the latent variables (pl=TUE) what is their range for the Zi/za
part?
Cheers,
Jarrod
Quoting Ruben Arslan <rubenarslan at gmail.com> on Wed, 27 Aug 2014
19:23:42 +0200:
> Hi Jarrod,
>
> thanks again. I was able to get it running with your advice.
> Some points of confusion remain:
>
> - You wrote that zi/za models would return an error with rcov =
> ~trait:units + starting values. This did not happen in my case, so I
> didn't build MCMCglmm myself with your suggested edits. Also, have
> you considered putting your own MCMCglmm repo on Github? Your users
> would be able to install pre-releases and I'd think you'd get some
> time-saving pull requests too.
> - In my attempts to get my models to run properly, I messed up a
> prior and did not use fix=2 in my prior specification for my za
> models. This led to crappy convergence, it's much better now and for
> some of my simpler models I think I won't need parallel chains. I'm
> reminded of Gelman's folk theorem of statistical computing.
> - I followed your advice, but of course I could not set the true
> values as starting values, but wanted to set random, bad starting
> values. I pasted below what I arrived at, I'm especially unsure
> whether I specified the starting values for G and R properly (I
> think not).
> start <- list(
> liab=c(rnorm( nrow(krmh.1)*2 )),
> R = list(R1 = diag(rpois(2, 1)+1)),
> G = list(G1 = diag(rpois(2, 1)+1))
> )
>
>
> However, even though I may not need multiple chains for some of my
> simpler models, I've now run into conflicting diagnostics. The
> geweke.diag for each chain (and examination of the traces) gives
> satisfactory diagnostics. Comparing multiple chains using
> gelman.diag, however, leads to one bad guy, namely the
> traitza_children:spouses interaction.
> I think this implies that I've got some starting value dependence
> for this parameter, that won't be easily rectified through longer
> burnin?
> Do you have any ideas how to rectify this?
> I am currently doing sequential analyses on episodes of selection
> and in historical human data only those who marry have a chance at
> having kids. I exclude the unmarried
> from my sample where I predict number of children, because I examine
> that in a previous model and the zero-inflation (65% zeros, median
> w/o unmarried = 4) when including the unmarried is so excessive.
> Number of spouses is easily the strongest predictor in the model,
> but only serves as a covariate here. Since my other estimates are
> stable across chains and runs and agree well across models and with
> theory, I'm
> inclined to shrug this off. But probably I shouldn't ignore this
> sign of non-convergence?
>
>> gelman.diag(mcmc_1)
> Potential scale reduction factors:
>
> Point est. Upper C.I.
> (Intercept) 1.00 1.00
> traitza_children 1.27 1.39
> male 1.00 1.00
> spouses 1.00 1.00
> paternalage.mean 1.00 1.00
> paternalage.factor(25,30] 1.00 1.00
> paternalage.factor(30,35] 1.00 1.00
> paternalage.factor(35,40] 1.00 1.00
> paternalage.factor(40,45] 1.00 1.00
> paternalage.factor(45,50] 1.00 1.00
> paternalage.factor(50,55] 1.00 1.00
> paternalage.factor(55,90] 1.00 1.00
> traitza_children:male 1.22 1.32
> traitza_children:spouses 1.83 2.13
> traitza_children:paternalage.mean 1.02 1.02
> traitza_children:paternalage.factor(25,30] 1.03 1.05
> traitza_children:paternalage.factor(30,35] 1.05 1.08
> traitza_children:paternalage.factor(35,40] 1.10 1.15
> traitza_children:paternalage.factor(40,45] 1.12 1.17
> traitza_children:paternalage.factor(45,50] 1.19 1.28
> traitza_children:paternalage.factor(50,55] 1.12 1.18
> traitza_children:paternalage.factor(55,90] 1.11 1.17
>
> Multivariate psrf
>
> 7.27
>
>
> Best regards,
>
> Ruben
>
>
> On 26 Aug 2014, at 13:04, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote:
>
>> Hi Ruben,
>>
>> There are 400 liabilities in a zapoisson model (2 per datum). This
>> code should work:
>>
>> g <-sample(letters[1:10], size = 200, replace = T)
>> pred <- rnorm(200)
>>
>> l1<-rnorm(200, -1, sqrt(1))
>> l2<-rnorm(200, -1, sqrt(1))
>>
>> y<-VGAM::rzapois(200, exp(l1), exp(-exp(l2)))
>>
>> # generate zero-altered data with an intercept of -1 (because the
>> intercept and variance are the same for both processes this is just
>> standard Poisson)
>>
>> dat<-data.frame(y=y, g = g, pred = pred)
>>
>>
>> start.1<-list(liab=c(l1,l2), R = list(R1=diag(2)), G=list(G1=diag(2)))
>> prior.1<-list(R=list(V=diag(2), nu=1.002, fix=2),
>> G=list(G1=list(V=diag(2), nu=2, alpha.mu=c(0,0),
>> alpha.V=diag(2)*1000)))
>>
>> m1<-MCMCglmm(y~trait + pred:trait, random=~us(trait):g,
>> family="zapoisson",rcov=~idh(trait):units, data=dat, prior=prior.1,
>> start= start.1)
>>
>> However, there are 2 bugs in the current version of MCMCglmm that
>> return an error message when the documentation implies it should be
>> fine:
>>
>> a) it should be possible to have R=diag(2) rather than R =
>> list(R1=diag(2)). This bug cropped up when I implemented
>> block-diagonal R structures. It can be fixed by inserting:
>>
>> if(!is.list(start$R)){
>> start$R<-list(R1=start$R)
>> }
>>
>> on L514 of MCMCglmm.R below
>>
>> if(!is.list(prior$R[[1]])){
>> prior$R<-list(R1=prior$R)
>> }
>>
>> b) rcov=~trait:units models for zi/za models will return an error
>> when passing starting values. To fix this insert
>>
>> if(diagR==3){
>> if(dim(start)[1]!=1){
>> stop("V is the wrong dimension for some
>> strart$G/start$R elements")
>> }
>> start<-diag(sum(nfl))*start[1]
>> }
>>
>> on L90 of priorfromat.R below
>>
>> if(is.matrix(start)==FALSE){
>> start<-as.matrix(start)
>> }
>>
>> I will put these in the new version.
>>
>> Cheers,
>>
>> Jarrod
>>
>>
>>
>>
>>
>>
>>
>> Quoting Ruben Arslan <rubenarslan at gmail.com> on Mon, 25 Aug 2014
>> 21:52:30 +0200:
>>
>>> Hi Jarrod,
>>>
>>> thanks for these pointers.
>>>
>>>>> You will need to provide over-dispersed starting values for
>>>>> multiple-chain convergence diagnostics to be useful (GLMM are so
>>>>> simple I am generally happy if the output of a single run looks
>>>>> reasonable).
>>>
>>> Oh, I would be happy with single chains, but since computation
>>> would take weeks this way, I wanted to parallelise and I would use
>>> the multi-chain convergence as a criterion that my parallelisation
>>> was proper
>>> and is as informative as a single long chain. There don't seem to
>>> be any such checks built-in – I was analysing my 40 chains for a
>>> bit longer than I like to admit until I noticed they were
>>> identical (effectiveSize
>>> and summary.mcmc.list did not yell at me for this).
>>>
>>>>> # use some very bad starting values
>>> I get that these values are bad, but that is the goal for my
>>> multi-chain aim, right?
>>>
>>> I can apply this to my zero-truncated model, but am again getting
>>> stuck with the zero-altered one.
>>> Maybe I need only specify the Liab values for this?
>>> At least I'm getting nowhere with specifying R and G starting
>>> values here. When I got an error, I always
>>> went to the MCMCglmm source to understand why the checks failed,
>>> but I didn't always understand
>>> what was being checked and couldn't get it to work.
>>>
>>> Here's a failing example:
>>>
>>> l<-rnorm(200, -1, sqrt(1))
>>> t<-(-log(1-runif(200)*(1-exp(-exp(l)))))
>>> g = sample(letters[1:10], size = 200, replace = T)
>>> pred = rnorm(200)
>>> y<-rpois(200,exp(l)-t)
>>> y[1:40] = 0
>>> # generate zero-altered data with an intercept of -1
>>>
>>> dat<-data.frame(y=y, g = g, pred = pred)
>>> set.seed(1)
>>> start_true = list(Liab=l, R = 1, G = 1 )
>>> m1<-MCMCglmm(y~1 + pred,random = ~ g,
>>> family="zapoisson",rcov=~us(trait):units, data=dat, start=
>>> start_true)
>>>
>>> # use true latent variable as starting values
>>> set.seed(1)
>>> # use some very bad starting values
>>> start_rand = list(Liab=rnorm(200), R = rpois(1, 1)+1, G = rpois(1, 1)+1 )
>>> m2<-MCMCglmm(y~1 + pred,random = ~ g,rcov=~us(trait):units,
>>> family="zapoisson", data=dat, start = start_rand)
>>>
>>> Best,
>>>
>>> Ruben
>>>
>>> On 25 Aug 2014, at 18:29, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote:
>>>
>>>> Hi Ruben,
>>>>
>>>> Sorry - I was wrong when I said that everything is Gibbs sampled
>>>> conditional on the latent variables. The location effects (fixed
>>>> and random effects) are also sampled conditional on the
>>>> (co)variance components so you should add them to the starting
>>>> values. In the case where the true values are used:
>>>>
>>>> m1<-MCMCglmm(y~1, family="ztpoisson", data=dat, start=list(Liab=l,R=1))
>>>>
>>>> Cheers,
>>>>
>>>> Jarrod
>>>>
>>>>
>>>>
>>>> Quoting Jarrod Hadfield <j.hadfield at ed.ac.uk> on Mon, 25 Aug 2014
>>>> 17:14:14 +0100:
>>>>
>>>>> Hi Ruben,
>>>>>
>>>>> You will need to provide over-dispersed starting values for
>>>>> multiple-chain convergence diagnostics to be useful (GLMM are so
>>>>> simple I am generally happy if the output of a single run looks
>>>>> reasonable).
>>>>>
>>>>> With non-Gaussian data everything is Gibbs sampled conditional
>>>>> on the latent variables, so you only need to pass them:
>>>>>
>>>>> l<-rnorm(200, -1, sqrt(1))
>>>>> t<-(-log(1-runif(200)*(1-exp(-exp(l)))))
>>>>> y<-rpois(200,exp(l)-t)+1
>>>>> # generate zero-truncated data with an intercept of -1
>>>>>
>>>>> dat<-data.frame(y=y)
>>>>> set.seed(1)
>>>>> m1<-MCMCglmm(y~1, family="ztpoisson", data=dat, start=list(Liab=l))
>>>>> # use true latent variable as starting values
>>>>> set.seed(1)
>>>>> m2<-MCMCglmm(y~1, family="ztpoisson", data=dat,
>>>>> start=list(Liab=rnorm(200)))
>>>>> # use some very bad starting values
>>>>>
>>>>> plot(mcmc.list(m1$Sol, m2$Sol))
>>>>> # not identical despite the same seed because of different
>>>>> starting values but clearly sampling the same posterior
>>>>> distribution:
>>>>>
>>>>> gelman.diag(mcmc.list(m1$Sol, m2$Sol))
>>>>>
>>>>> Cheers,
>>>>>
>>>>> Jarrod
>>>>>
>>>>> Quoting Ruben Arslan <rubenarslan at gmail.com> on Mon, 25 Aug 2014
>>>>> 18:00:08 +0200:
>>>>>
>>>>>> Dear Jarrod,
>>>>>>
>>>>>> thanks for the quick reply. Please, don't waste time looking
>>>>>> into doMPI – I am happy that I
>>>>>> get the expected result, when I specify that reproducible seed,
>>>>>> whyever that may be.
>>>>>> I'm pretty sure that is the deciding factor, because I tested
>>>>>> it explicitly, I just have no idea
>>>>>> how/why it interacts with the choice of family.
>>>>>>
>>>>>> That said, is setting up different RNG streams for my workers
>>>>>> (now that it works) __sufficient__
>>>>>> so that I get independent chains and can use gelman.diag() for
>>>>>> convergence diagnostics?
>>>>>> Or should I still tinker with the starting values myself?
>>>>>> I've never found a worked example of supplying starting values
>>>>>> and am thus a bit lost.
>>>>>>
>>>>>> Sorry for sending further questions, I hope someone else takes
>>>>>> pity while
>>>>>> you're busy with lectures.
>>>>>>
>>>>>> Best wishes
>>>>>>
>>>>>> Ruben
>>>>>>
>>>>>>
>>>>>>
>>>>>> On 25 Aug 2014, at 17:29, Jarrod Hadfield <j.hadfield at ed.ac.uk> wrote:
>>>>>>
>>>>>>> Hi Ruben,
>>>>>>>
>>>>>>> I do not think the issue is with the starting values, because
>>>>>>> even if the same starting values were used the chains would
>>>>>>> still differ because of the randomness in the Markov Chain (if
>>>>>>> I interpret your `identical' test correctly). This just
>>>>>>> involves a call to GetRNGstate() in the C++ code (L 871
>>>>>>> ofMCMCglmm.cc) so I think for some reason doMPI/foreach is not
>>>>>>> doing what you expect. I am not familiar with doMPI and am in
>>>>>>> the middle of writing lectures so haven't got time to look
>>>>>>> into it carefully. Outside of the context of doMPI I get the
>>>>>>> behaviour I expect:
>>>>>>>
>>>>>>>
>>>>>>> l<-rnorm(200, -1, sqrt(1))
>>>>>>> t<-(-log(1-runif(200)*(1-exp(-exp(l)))))
>>>>>>> y<-rpois(200,exp(l)-t)+1
>>>>>>> # generate zero-truncated data with an intercept of -1
>>>>>>>
>>>>>>> dat<-data.frame(y=y)
>>>>>>> set.seed(1)
>>>>>>> m1<-MCMCglmm(y~1, family="ztpoisson", data=dat)
>>>>>>> set.seed(2)
>>>>>>> m2<-MCMCglmm(y~1, family="ztpoisson", data=dat)
>>>>>>> set.seed(2)
>>>>>>> m3<-MCMCglmm(y~1, family="ztpoisson", data=dat)
>>>>>>>
>>>>>>> plot(mcmc.list(m1$Sol, m2$Sol))
>>>>>>> # different, as expected
>>>>>>> plot(mcmc.list(m2$Sol, m3$Sol))
>>>>>>> # the same, as expected
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Quoting Ruben Arslan <rubenarslan at gmail.com> on Mon, 25 Aug
>>>>>>> 2014 16:58:06 +0200:
>>>>>>>
>>>>>>>> Dear list,
>>>>>>>>
>>>>>>>> sorry for bumping my old post, I hope to elicit a response
>>>>>>>> with a more focused question:
>>>>>>>>
>>>>>>>> When does MCMCglmm automatically start from different values
>>>>>>>> when using doMPI/foreach?
>>>>>>>>
>>>>>>>> I have done some tests with models of varying complexity. For
>>>>>>>> example, the script in my last
>>>>>>>> post (using "zapoisson") yielded 40 identical chains:
>>>>>>>>> identical(mcmclist[1], mcmclist[30])
>>>>>>>> TRUE
>>>>>>>>
>>>>>>>> A simpler (?) model (using "ztpoisson" and no specified
>>>>>>>> prior), however, yielded different chains
>>>>>>>> and I could use them to calculate gelman.diag()
>>>>>>>>
>>>>>>>> Changing my script to the version below, i.e. seeding foreach
>>>>>>>> using .options.mpi=list( seed= 1337)
>>>>>>>> so as to make RNGstreams reproducible (or so I thought), led
>>>>>>>> to different chains even for the
>>>>>>>> "zapoisson" model.
>>>>>>>>
>>>>>>>> In no case have I (successfully) tried to supplant the
>>>>>>>> default of MCMCglmm's "start" argument.
>>>>>>>> Is starting my models from different RNGsubstreams inadequate
>>>>>>>> compared to manipulating
>>>>>>>> the start argument explicitly? If so, is there any worked
>>>>>>>> example of explicit starting value manipulation
>>>>>>>> in parallel computation?
>>>>>>>> I've browsed the MCMCglmm source to understand how the
>>>>>>>> default starting values are generated,
>>>>>>>> but didn't find any differences with respect to RNG for the
>>>>>>>> two families "ztpoisson" and "zapoisson"
>>>>>>>> (granted, I did not dig very deep).
>>>>>>>>
>>>>>>>> Best regards,
>>>>>>>>
>>>>>>>> Ruben Arslan
>>>>>>>>
>>>>>>>>
>>>>>>>> # bsub -q mpi -W 12:00 -n 41 -R np20 mpirun -H localhost -n
>>>>>>>> 41 R --slave -f
>>>>>>>> "/usr/users/rarslan/rpqa/rpqa_main/rpqa_children_parallel.R"
>>>>>>>>
>>>>>>>> library(doMPI)
>>>>>>>> cl <-
>>>>>>>> startMPIcluster(verbose=T,workdir="/usr/users/rarslan/rpqa/rpqa_main/")
>>>>>>>> registerDoMPI(cl)
>>>>>>>> Children_mcmc1 = foreach(i=1:clusterSize(cl),.options.mpi =
>>>>>>>> list(seed=1337) ) %dopar% {
>>>>>>>> library(MCMCglmm);library(data.table)
>>>>>>>> load("/usr/users/rarslan/rpqa/rpqa1.rdata")
>>>>>>>>
>>>>>>>> nitt = 130000; thin = 100; burnin = 30000
>>>>>>>> prior.m5d.2 = list(
>>>>>>>> R = list(V = diag(c(1,1)), nu = 0.002),
>>>>>>>> G=list(list(V=diag(c(1,1e-6)),nu=0.002))
>>>>>>>> )
>>>>>>>>
>>>>>>>> rpqa.1 = na.omit(rpqa.1[spouses>0, list(idParents, children,
>>>>>>>> male, urban, spouses, paternalage.mean, paternalage.factor)])
>>>>>>>> (m1 = MCMCglmm( children ~ trait * (male + urban + spouses +
>>>>>>>> paternalage.mean + paternalage.factor),
>>>>>>>> rcov=~us(trait):units,
>>>>>>>> random=~us(trait):idParents,
>>>>>>>> family="zapoisson",
>>>>>>>> prior = prior.m5d.2,
>>>>>>>> data=rpqa.1,
>>>>>>>> pr = F, saveX = F, saveZ = F,
>>>>>>>> nitt=nitt,thin=thin,burnin=burnin))
>>>>>>>> }
>>>>>>>>
>>>>>>>> library(coda)
>>>>>>>> mcmclist = mcmc.list(lapply(Children_mcmc1,FUN=function(x) { x$Sol}))
>>>>>>>> save(Children_mcmc1,mcmclist, file =
>>>>>>>> "/usr/users/rarslan/rpqa/rpqa_main/rpqa_mcmc_kids_za.rdata")
>>>>>>>> closeCluster(cl)
>>>>>>>> mpi.quit()
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> On 04 Aug 2014, at 20:25, Ruben Arslan <rubenarslan at gmail.com> wrote:
>>>>>>>>
>>>>>>>>> Dear list,
>>>>>>>>>
>>>>>>>>> would someone be willing to share her or his efforts in
>>>>>>>>> parallelising a MCMCglmm analysis?
>>>>>>>>>
>>>>>>>>> I had something viable using harvestr that seemed to
>>>>>>>>> properly initialise
>>>>>>>>> the starting values from different random number streams
>>>>>>>>> (which is desirable,
>>>>>>>>> as far as I could find out), but I ended up being unable to
>>>>>>>>> use harvestr, because
>>>>>>>>> it uses an old version of plyr, where parallelisation works
>>>>>>>>> only for multicore, not for
>>>>>>>>> MPI.
>>>>>>>>>
>>>>>>>>> I pasted my working version, that does not do anything about
>>>>>>>>> starting values or RNG
>>>>>>>>> at the end of this email. I can try to fumble further in the
>>>>>>>>> dark or try to update harvestr,
>>>>>>>>> but maybe someone has gone through all this already.
>>>>>>>>>
>>>>>>>>> I'd also appreciate any tips for elegantly post-processing
>>>>>>>>> such parallel data, as some of my usual
>>>>>>>>> extraction functions and routines are hampered by the fact
>>>>>>>>> that some coda functions
>>>>>>>>> do not aggregate results over chains. (What I get from a
>>>>>>>>> single-chain summary in MCMCglmm
>>>>>>>>> is a bit more comprehensive, than what I managed to cobble
>>>>>>>>> together with my own extraction
>>>>>>>>> functions).
>>>>>>>>>
>>>>>>>>> The reason I'm parallelising my analyses is that I'm having
>>>>>>>>> trouble getting a good effective
>>>>>>>>> sample size for any parameter having to do with the many
>>>>>>>>> zeroes in my data.
>>>>>>>>> Any pointers are very appreciated, I'm quite inexperienced
>>>>>>>>> with MCMCglmm.
>>>>>>>>>
>>>>>>>>> Best wishes
>>>>>>>>>
>>>>>>>>> Ruben
>>>>>>>>>
>>>>>>>>> # bsub -q mpi-short -W 2:00 -n 42 -R np20 mpirun -H
>>>>>>>>> localhost -n 41 R --slave -f
>>>>>>>>> "rpqa/rpqa_main/rpqa_children_parallel.r"
>>>>>>>>> library(doMPI)
>>>>>>>>> cl <- startMPIcluster()
>>>>>>>>> registerDoMPI(cl)
>>>>>>>>> Children_mcmc1 = foreach(i=1:40) %dopar% {
>>>>>>>>> library(MCMCglmm)
>>>>>>>>> load("rpqa1.rdata")
>>>>>>>>>
>>>>>>>>> nitt = 40000; thin = 100; burnin = 10000
>>>>>>>>> prior = list(
>>>>>>>>> R = list(V = diag(c(1,1)), nu = 0.002),
>>>>>>>>> G=list(list(V=diag(c(1,1e-6)),nu=0.002))
>>>>>>>>> )
>>>>>>>>>
>>>>>>>>> MCMCglmm( children ~ trait -1 + at.level(trait,1):male +
>>>>>>>>> at.level(trait,1):urban + at.level(trait,1):spouses +
>>>>>>>>> at.level(trait,1):paternalage.mean +
>>>>>>>>> at.level(trait,1):paternalage.factor,
>>>>>>>>> rcov=~us(trait):units,
>>>>>>>>> random=~us(trait):idParents,
>>>>>>>>> family="zapoisson",
>>>>>>>>> prior = prior,
>>>>>>>>> data=rpqa.1,
>>>>>>>>> pr = F, saveX = T, saveZ = T,
>>>>>>>>> nitt=nitt,thin=thin,burnin=burnin)
>>>>>>>>> }
>>>>>>>>>
>>>>>>>>> library(coda)
>>>>>>>>> mcmclist = mcmc.list(lapply(Children_mcmc1,FUN=function(x) { x$Sol}))
>>>>>>>>> save(Children_mcmc1,mcmclist, file = "rpqa_mcmc_kids_za.rdata")
>>>>>>>>> closeCluster(cl)
>>>>>>>>> mpi.quit()
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> --
>>>>>>>>> Ruben C. Arslan
>>>>>>>>>
>>>>>>>>> Georg August University G�ttingen
>>>>>>>>> Biological Personality Psychology and Psychological Assessment
>>>>>>>>> Georg Elias M�ller Institute of Psychology
>>>>>>>>> Go�lerstr. 14
>>>>>>>>> 37073 G�ttingen
>>>>>>>>> Germany
>>>>>>>>> Tel.: +49 551 3920704
>>>>>>>>> https://psych.uni-goettingen.de/en/biopers/team/arslan
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> [[alternative HTML version deleted]]
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> --
>>>>>>> The University of Edinburgh is a charitable body, registered in
>>>>>>> Scotland, with registration number SC005336.
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> The University of Edinburgh is a charitable body, registered in
>>>>> Scotland, with registration number SC005336.
>>>>>
>>>>> _______________________________________________
>>>>> R-sig-mixed-models at r-project.org mailing list
>>>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>>>>
>>>>
>>>>
>>>>
>>>> --
>>>> The University of Edinburgh is a charitable body, registered in
>>>> Scotland, with registration number SC005336.
>>>>
>>>>
>>>
>>>
>>>
>>
>>
>>
>> --
>> The University of Edinburgh is a charitable body, registered in
>> Scotland, with registration number SC005336.
>>
>>
>
>
--
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