[R-sig-ME] parallel MCMCglmm, RNGstreams, starting values & priors
Ruben Arslan
rubenarslan at gmail.com
Thu Aug 28 18:45:30 CEST 2014
Hi Jarrod,
> 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().
I agree it's a weird specification, I was a bit lost and thought I could get away with just putting some random numbers in the starting value.
I didn't do R1=rpois(2,1)+1 though, I did R1=diag(rpois(2,1)+1), so I got a 2x2 matrix, but yes, bound to be integer.
I didn't know starting values should come from a conjugate distribution, though that probably means I didn't think about it much.
I'm now doing
start <- list(
liab=c(rnorm( nrow(krmh.1)*2 )),
R = list(R1 = rIW( diag(2), nrow(krmh.1)) ),
G = list(G1 = rIW( diag(2), nrow(krmh.1)) )
)
Is this what you had in mind?
I am especially unsure if I am supposed to use such a low sampling variability (my sample size is probably not even relevant for the starting values) and if I should start from diag(2).
And, I am still happily confused that this specification still doesn't lead to errors with respect to rcov = ~trait:units . Does this mean I'm doing it wrong?
> 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?
My parallel run using the above starting values isn't finished yet.
a) After applying the above starting values I get, for the location effects 1600-2000 samples for a 2000 sample chain (with thin set to 50). G and R-structure are from 369 (za_children.idParents) to 716 (and 0 for the fixed part).
Effective sample sizes were similar for my run using the starting values for G/R that I drew from rpois, and using 40 chains I of course get
b) I don't think I am getting extreme categories. I would probably be getting extreme categories if I included the forever-alones (they almost never have children), but this way no.
I wasn't sure how to examine the range of the latents separately for the za part, but for a single chain it looks okay:
> quantile(as.numeric(m1$Liab),probs = c(0,0.01,0,0.99,1))
0% 1% 0% 99% 100%
-4.934111 -1.290728 -4.934111 3.389847 7.484206
Well, all considered now that I use the above starting value specification I get slightly different estimates for all za-coefficients. Nothing major, but still leading me to
think my estimates aren't exactly independent of the starting values I use. I'll see what the parallel run yields.
Thanks a lot,
Ruben
>
> 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.
>>>
>>>
>>
>>
>
>
>
> --
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
>
>
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