[R-sig-ME] What is the lmer/nlme equivalent of the REPEATED subcommand in SPSS's MIXED procedure?
Maarten Jung
Maarten.Jung at mailbox.tu-dresden.de
Thu Mar 22 20:16:36 CET 2018
Thanks for giving this nice example - it makes perfect sense! I'll have a
closer look at these models.
Regards,
Maarten
On Thu, Mar 22, 2018, 17:54 Rune Haubo <rune.haubo at gmail.com> wrote:
> On 22 March 2018 at 16:11, Maarten Jung
> <Maarten.Jung at mailbox.tu-dresden.de> wrote:
> > Dear Rune,
> >
> > thanks for making clear what /REPEATED stands for!
> >
> > Independently, does it make sense to fit a model via
> >
> > gls(value ~ factor1, data, correlation = corSymm(form = ~ 1|person),
> weights
> > = varIdent(form = ~1|factor1))
> >
> > when there is only one observation per subject-factor1-combination?
>
> Yes, I'd say so. And Ben Pelzer's email from yesterday has a data+code
> example of exactly that if I'm not mistaken.
>
> If factor1 is 'time' this model is quite similar to the famous MMRM*
> often used in pharma to model clinical trials (for purposes of
> handling/'imputation' of missing values) in which measurements are
> made on subjects at, say, a handful of timepoints over the trial
> period. This model essentially specifies that each subject-profile
> follows a multivariate normal distribution with the so-called
> 'unstructured' variance-covariance matrix with dimensions equal to the
> number of timepoints, further, all the subject profiles are
> independent. Differences between treatment groups are then estimated
> at each timepoint and usually the differences at the last timepoint
> are of primary interest. Long story short: Yes it makes sense and it's
> done all the time in the pharmaceutical world.
>
> Cheers
> Rune
>
> *The MMRM is a very specific model not just any 'mixed model for
> repeated measurements', in fact, it is not really a mixed model (the
> lack of random effects give it away...).
>
> >
> > Regards,
> > Maarten
> >
> > On Thu, Mar 22, 2018 at 11:38 AM, Rune Haubo <rune.haubo at gmail.com>
> wrote:
> >>
> >> On 22 March 2018 at 10:05, Maarten Jung
> >> <Maarten.Jung at mailbox.tu-dresden.de> wrote:
> >> > I think the problem is that there is only one observation per
> >> > subject-occasion-combination in this example.
> >> > In this case the random slopes are confounded with the residual
> >> > variation
> >> > (see [1]).
> >>
> >> I agree.
> >>
> >> >
> >> > One *can* fit this model using lmer(test ~ 1 + occ2 + occ3 + (1 +
> occ2 +
> >> > occ3|person), data = mydata, control = lmerControl(check.nobs.vs.nRE
> =
> >> > "ignore")) or
> >> > lme(test ~ 1 + occ2 + occ3, mydata, random = ~ occ2 + occ3|person).
> >> >
> >> > However, I don't know if the gls() fit ist more trustworthy than the
> >> > lmer/lme fit here.
> >> > I would be grateful if somebody more experienced in mixed models could
> >> > comment on this.
> >>
> >> If you are trying to replicate a model that is specified with a
> >> 'repeated' statement (and 'repeated' in SPSS means the same as in SAS)
> >> then we are talking about specifying a structure in the residual
> >> variance-covariance matrix (cf.
> >>
> >>
> https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_mixed_sect019.htm
> ).
> >> If we write the model as y = X beta + Z b + e with b ~ N(0, G) and e ~
> >> N(0, R), then 'repeated' is specifying the structure in R. This is
> >> exactly what nlme::gls and nlme::lme do via the weights and
> >> correlation arguments; you need lme if you have 'true' random effects
> >> on top of the structure in R.
> >>
> >> If we write the marginal distribution of y as y ~ N(X beta, V), with V
> >> = ZGZ' + R then the structure in V can be obtained by appropriate
> >> specification of either Z and G or R, or both. This means that in some
> >> cases there is more than one 'natural' way to specify the same
> >> marginal model (compound symmetry is the classical example*). Now,
> >> lmer has the structure of R fixed at sigma^2 I, i.e. a multiple of the
> >> identity matrix, but with appropriate random-effect specification you
> >> can sometimes obtain the same (marginal) likelihood as if the
> >> structure was specified in the R matrix. My advice, is however, that
> >> if you want to fit a model with a particular structure in R, then
> >> don't use lmer; you really have to know all details of what is going
> >> to be sure that it works as intended.
> >>
> >> Cheers,
> >> Rune
> >>
> >> *Actually the compound symmetry and random intercept models are _not_
> >> the same (though often claimed to be) because the parameter spaces
> >> differ. This means that for some data sets the model fits are the same
> >> (i.e. same likelihood) and for other data sets the model fits are
> >> different.
> >>
> >> >
> >> > Best regards,
> >> > Maarten
> >> >
> >> > [1]
> >> >
> >> >
> https://stackoverflow.com/questions/26465215/random-slope-for-time-in-subject-not-working-in-lme4?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa
> >> >
> >> > On Wed, Mar 21, 2018 at 9:27 PM, Ben Pelzer <b.pelzer at maw.ru.nl>
> wrote:
> >> >
> >> >> Hi Maarten,
> >> >>
> >> >> Here is an example which shows the unstructured model with gls and
> the
> >> >> not converging model with lmer. In this example, we have three
> >> >> occasions
> >> >> on which the dependent variable "test" was observed, for each of 20
> >> >> persons. In total then we have 60 observations, with the "occasion"
> >> >> variable taking values 1, 2, 3. The data also contain the person id
> >> >> variable "person" and dummy variables "occ1", "occ2", "occ3" as (0 or
> >> >> 1)
> >> >> indicators of the occasion. In the syntax below, a factor variable
> >> >> "factor1" is created also, to be in line with your question.
> >> >>
> >> >> I used two different specifications for the unstructured model with
> >> >> gls,
> >> >> depending on whether dummies or factor1 was used. For lmer, I used
> >> >> these
> >> >> three different specifications, none of which converges.
> >> >>
> >> >> The lmer syntax was added only to show the problem which lmer has
> with
> >> >> estimating an unstructured correlation pattern.
> >> >>
> >> >>
> >> >> #-----------------------------------------------------------
> >> >> ------------------------------------------------------------
> >> >> -------------------------------------
> >> >> mydata <-
> >> >> read.table(url("https://surfdrive.surf.nl/files/index.
> >> >> php/s/XfE3mtbFCTUejIz/download"),
> >> >> header=TRUE)
> >> >>
> >> >>
> >> >> #------------------- unstructured correlation matrix
> >> >> -----------------------
> >> >>
> >> >>
> >> >> # Before applying a model, let's first examine the variances and
> >> >> correlations
> >> >> # for the three occasions. We have a strong violation of the
> >> >> assumptions
> >> >> # of homoscedasticity and compound symmetry.
> >> >> test1 <- mydata[mydata$occasion==1,"test"]
> >> >> test2 <- mydata[mydata$occasion==2,"test"]
> >> >> test3 <- mydata[mydata$occasion==3,"test"]
> >> >> cor(cbind(test1, test2, test3))
> >> >> var(cbind(test1, test2, test3))
> >> >>
> >> >> # Unstructured model using gls from package nlme and dummies for
> >> >> occasion.
> >> >> # This model exactly reproduces the observed correlations between
> >> >> occasions.
> >> >> unstruc.gls1 <- gls(test ~ 1+ occ2 + occ3,
> >> >> method="REML", data=mydata,
> >> >> correlation=corSymm(form = ~ 1 |person),
> >> >> weights = varIdent(form = ~1|occasion))
> >> >> summary(unstruc.gls1)
> >> >>
> >> >>
> >> >> # Unstructured model using factor1 for occasion instead of dummies.
> >> >> # The results are exactly the same as those above, as should be.
> >> >> mydata$factor1 <- as.factor(mydata$occasion)
> >> >> unstruc.gls2 <- gls(test ~ factor1,
> >> >> method="REML", data=mydata,
> >> >> correlation=corSymm(form = ~ 1|person),
> >> >> weights = varIdent(form = ~1|factor1))
> >> >> summary(unstruc.gls2)
> >> >>
> >> >>
> >> >> # Unstructured model using lmer and dummies for occasion: does not
> >> >> converge.
> >> >> unstruc.lmer <- lmer(test ~ 1+ occ2 + occ3 + (1+occ2+occ3|person),
> >> >> data=mydata, REML=TRUE)
> >> >> summary(unstruc.lmer)
> >> >>
> >> >>
> >> >> # Unstructured model using lmer and factor1 for occasion: does not
> >> >> converge.
> >> >> unstruc.lmer <- lmer(test ~ 1+ factor1 + (1+factor1|person),
> >> >> data=mydata, REML=TRUE)
> >> >> summary(unstruc.lmer)
> >> >>
> >> >>
> >> >> # Unstructured model using lmer and factor1 for occasion, no
> intercept
> >> >> specified: does not converge.
> >> >> unstruc.lmer <- lmer(test ~ factor1 + (factor1|person),
> >> >> data=mydata, REML=TRUE)
> >> >> summary(unstruc.lmer)
> >> >>
> >> >>
> >> >>
> >> >> On 21/03/2018 13:07, Maarten Jung wrote:
> >> >> > Dear Ben,
> >> >> >
> >> >> > I am a bit puzzled.
> >> >> >
> >> >> > Do you mean that
> >> >> >
> >> >> > m1 <- gls(value ~ factor1, data, correlation = corSymm(form = ~
> >> >> > 1|participant), weights = varIdent(form = ~ 1|factor))
> >> >> >
> >> >> > would be equivalent to
> >> >> >
> >> >> > m2 <- lmer(value ~ factor1 + (factor1|participant), data)
> >> >> >
> >> >> > and one should use gls() because it allows for the same covariance
> >> >> > structures as /REPEATED does?
> >> >> >
> >> >>
> >> >>
> >> >> the two specifications are not equivalent in the sense that lmer also
> >> >> tries to estimate residual variance. However, with the given lmer
> model
> >> >> specification, the random factor1 effects capture all variance there
> is
> >> >> and no residual variance remains.
> >> >>
> >> >>
> >> >> > And, if so, why should m2 cause an identification problem and m1
> >> >> > doesn't?
> >> >> >
> >> >> > Regards,
> >> >> > Maarten
> >> >> >
> >> >> Regards, Ben.
> >> >>
> >> >>
> >> >>
> >> >> >
> >> >> > On Wed, Mar 21, 2018 at 12:03 PM, Ben Pelzer <b.pelzer at maw.ru.nl
> >> >> > <mailto:b.pelzer at maw.ru.nl>> wrote:
> >> >> >
> >> >> > Dear all,
> >> >> >
> >> >> > As far as I know, the specification for lmer using
> >> >> >
> >> >> > value ~ factor1 + (factor1 | participant)
> >> >> >
> >> >> > causes an identification problem, because the residual variance
> >> >> > is
> >> >> not
> >> >> > excluded from the estimations. It would indeed work (e.g. in
> >> >> > MlWin
> >> >> > this
> >> >> > can be done) if we could constrain that residual variance to
> >> >> > zero.
> >> >> > There
> >> >> > have been some mails in this list about whether or not
> >> >> > constraining
> >> >> > residual variance to zero is possible in lmer, but I believe
> this
> >> >> > is not
> >> >> > possible. Would be nice if we could do this in lmer!
> >> >> >
> >> >> > Best regards, Ben.
> >> >> >
> >> >> >
> >> >> > On 20-3-2018 18:34, Douglas Bates wrote:
> >> >> > > Kind of looks like SPSS went for bug-for-bug compatibility
> with
> >> >> > SAS on
> >> >> > > this one. In SAS PROC MIXED, "REPEATED" and "RANDOM" are two
> >> >> > ways of
> >> >> > > specifying the random effects variance structure but they
> often
> >> >> boil
> >> >> > > down to the same model.
> >> >> > >
> >> >> > > I believe the model can be specified in lme4 as
> >> >> > >
> >> >> > > value ~ factor1 + (factor1 | participant)
> >> >> > >
> >> >> > > This is what the mis-named* "UNSTRUCTURED" covariance type
> >> >> > means
> >> >> > >
> >> >> > > * Old-guy, get off my lawn rant about terminology *
> >> >> > > As a recovering mathematician I find the name "unstructured"
> >> >> > being
> >> >> > > used to denote a positive-definite symmetric matrix to be,
> >> >> > well,
> >> >> > > inaccurate.
> >> >> > >
> >> >> > > On Tue, Mar 20, 2018 at 12:19 PM Mollie Brooks
> >> >> > > <mollieebrooks at gmail.com <mailto:mollieebrooks at gmail.com>
> >> >> > <mailto:mollieebrooks at gmail.com
> >> >> > <mailto:mollieebrooks at gmail.com>>>
> >> >> > wrote:
> >> >> > >
> >> >> > > I don’t know anything about spss, but if you basically
> want
> >> >> lme4
> >> >> > > with more correlation structures, you could look at the
> >> >> > structures
> >> >> > > available with glmmTMB.
> >> >> > >
> >> >> > https://cran.r-project.org/web/packages/glmmTMB/
> >> >> vignettes/covstruct.html
> >> >> > <https://cran.r-project.org/web/packages/glmmTMB/
> >> >> vignettes/covstruct.html>
> >> >> > >
> >> >> > > cheers,
> >> >> > > Mollie
> >> >> > >
> >> >> > > > On 20Mar 2018, at 18:11, Ben Pelzer <
> b.pelzer at maw.ru.nl
> >> >> > <mailto:b.pelzer at maw.ru.nl>
> >> >> > > <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>>>
> >> >> wrote:
> >> >> > > >
> >> >> > > > Hi Maarten,
> >> >> > > >
> >> >> > > > You are right: you need nlme and NOT lme4 to specify
> >> >> > particular
> >> >> > > > correlation structures. Also, in nlme you would need
> gls
> >> >> > to make it
> >> >> > > > similar to mixed in spss. The repeated command in spss
> >> >> > gives the
> >> >> > > same
> >> >> > > > results as gls does for any of the covariance
> structures.
> >> >> > > >
> >> >> > > > Regards, Ben.
> >> >> > > >
> >> >> > > >
> >> >> > > > On 20/03/2018 17:30, Maarten Jung wrote:
> >> >> > > >> Dear Ben, dear Phillip,
> >> >> > > >>
> >> >> > > >> comparing [1] with [2] I think the /REPEATED command
> >> >> > specifies
> >> >> > > >> the error (co)variance structure of the model. Would
> you
> >> >> > agree
> >> >> > > with that?
> >> >> > > >> If so, AFAIK this is not possible with lmer and thus
> the
> >> >> > answer on
> >> >> > > >> Stack Overflow [3] would be wrong.
> >> >> > > >>
> >> >> > > >> [1]
> >> >> > > >>
> >> >> > >
> >> >> > https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/>
> >> >> > > >> [2]
> >> >> > > >>
> >> >> > >
> >> >> > https://stats.idre.ucla.edu/spss/examples/alda/chapter7/
> >> >> applied-longitudinal-data-analysis-modeling-change-and-
> >> >> event-occurrenceby-judith-d-singer-and-john-b-willett-
> >> >> chapter-7-examining-the-multilevel-model-s-erro/
> >> >> > <https://stats.idre.ucla.edu/spss/examples/alda/chapter7/
> >> >> applied-longitudinal-data-analysis-modeling-change-and-
> >> >> event-occurrenceby-judith-d-singer-and-john-b-willett-
> >> >> chapter-7-examining-the-multilevel-model-s-erro/>
> >> >> > > >> [3]
> >> >> > > >>
> >> >> > >
> >> >> > https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc>
> >> >> > > >>
> >> >> > > >> Regards,
> >> >> > > >> Maarten
> >> >> > > >>
> >> >> > > >> On Tue, Mar 20, 2018 at 2:10 PM, Ben Pelzer
> >> >> > <b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>
> >> >> > > <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>>
> >> >> > > >> <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl
> >
> >> >> > <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>>
> >> >> > > <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>
> >> >> > <mailto:b.pelzer at maw.ru.nl <mailto:b.pelzer at maw.ru.nl>>>>>
> wrote:
> >> >> > > >>
> >> >> > > >> Dear Maarten,
> >> >> > > >>
> >> >> > > >> Take a look at
> >> >> > > >>
> >> >> > > >>
> >> >> > >
> >> >> > https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/>
> >> >> > >
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/>>
> >> >> > > >>
> >> >> > >
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/>
> >> >> > >
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/
> >> >> > <https://stats.idre.ucla.edu/r/examples/alda/r-applied-
> >> >> longitudinal-data-analysis-ch-7/>>>
> >> >> > > >>
> >> >> > > >> which shows you a number of covariance structures,
> >> >> > among
> >> >> > > which is
> >> >> > > >> the unstructured matrix, for repeated measures in R
> >> >> > with lme. It
> >> >> > > >> refers to chapter 7 of Singer and Willett where
> they
> >> >> > discuss all
> >> >> > > >> these different structures and how to choose among
> >> >> > them.
> >> >> > > Regards,
> >> >> > > >>
> >> >> > > >> Ben.
> >> >> > > >>
> >> >> > > >> On 20-3-2018 9:00, Maarten Jung wrote:
> >> >> > > >>
> >> >> > > >> Dear list,
> >> >> > > >> I came across a SPSS syntax like this
> >> >> > > >>
> >> >> > > >> MIXED value BY factor1
> >> >> > > >> /CRITERIA=CIN(95) MXITER(100) MXSTEP(10)
> >> >> > SCORING(1)
> >> >> > > >> SINGULAR(0.000000000001)
> >> >> > > >> HCONVERGE(0, ABSOLUTE) LCONVERGE(0,
> >> >> > ABSOLUTE)
> >> >> > > >> PCONVERGE(0.000001,
> >> >> > > >> ABSOLUTE)
> >> >> > > >> /FIXED=factor1 | SSTYPE(3)
> >> >> > > >> /METHOD=REML
> >> >> > > >> /REPEATED=factor1 | SUBJECT(participant)
> >> >> > COVTYPE(UN).
> >> >> > > >>
> >> >> > > >> and struggle to find an equivalent lmer/nlme
> (or
> >> >> > R in
> >> >> > > general)
> >> >> > > >> formulation
> >> >> > > >> for this kind of models.
> >> >> > > >> Does anybody know how to convert the REPEATED
> >> >> > subcommand
> >> >> > > into
> >> >> > > >> R code?
> >> >> > > >>
> >> >> > > >> Please note that I asked the question on Stack
> >> >> > Overflow
> >> >> > > about
> >> >> > > >> two month ago:
> >> >> > > >>
> >> >> > >
> >> >> > https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc>
> >> >> > >
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc>>
> >> >> > > >>
> >> >> > >
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc>
> >> >> > >
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc
> >> >> > <https://stackoverflow.com/questions/48518514/what-is-
> >> >>
> >> >>
> the-lmer-nlme-equivalent-of-the-repeated-subcommand-in-spsss-mixed-proc>>>
> >> >> > > >>
> >> >> > > >> Best regards,
> >> >> > > >> Maarten
> >> >> > > >>
> >> >> > > >> [[alternative HTML version deleted]]
> >> >> > > >>
> >> >> > > >> _______________________________________________
> >> >> > > >> 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>>>> mailing list
> >> >> > > >>
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >> >> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>
> >> >> > > <
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >> >> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>
> >> >> > > >>
> >> >> > > <
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >> >> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>
> >> >> > > <
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >> >> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>>
> >> >> > > >>
> >> >> > > >>
> >> >> > > >> _______________________________________________
> >> >> > > >> 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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> >> >> > > >
> >> >> > > > _______________________________________________
> >> >> > > > 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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> >> >> > > _______________________________________________
> >> >> > > R-sig-mixed-models at r-project.org
> >> >> > <mailto:R-sig-mixed-models at r-project.org>
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