[R-sig-ME] What is the lmer/nlme equivalent of the REPEATED subcommand in SPSS's MIXED procedure?
Phillip Alday
phillip.alday at mpi.nl
Thu Mar 22 11:35:56 CET 2018
I'm also somewhat curious whether these really count as "unstructured"
(subject to the terminological issues pointed out by D. Bates)
covariance matrices. The gls examples seem to be imposing additional
constraints (such as setting the residual variance to 0 via varIdent())
to overcome these issues.
(As a sidebar, I'm not sure replicating pairwise correlation is what we
should aim for in the covariance structure of mixed-effects models.
Forcing us to match pairwise correlation/regression structure seems to
be forcing us to reduce a lot of the advantages of things like partial
pooling, shrinkage, etc. in mixed-effects models.)
Going back to the original question, is this really the same thing as
the "unstructured" covariance in SPSS? If so, how does SPSS allow for
"unstructured" covariance that has non-zero residual variance? Or in
terms of R, how would one distinguish terminologically between the lme4
"unstructured" and the gls "unstructured but residual constrained to zero"?
Best,
Phillip
On 22/03/18 11:03, Maarten Jung wrote:
> Hi Ben,
>
> I'm aware of this problem for lmer.
> But how does gls() overcome the problem?
>
> Regards,
> Maarten
>
> On Thu, Mar 22, 2018, 10:44 Ben Pelzer <b.pelzer at maw.ru.nl> wrote:
>
>> Hi Maarten,
>>
>> Notice that with the syntax for lmer, 6 random-effect (co)variances must
>> be estimated and 1 residual variance, so in total 7
>> (co)variance-parameters. However, there are only 6 observed covariances,
>> meaning that the model is over-specified. Many solutions are possible
>> all having the same loglikelihood. Ignoring the nobs.vs.nRE rule leads
>> to just one of the many solutions. I would not be surprised if you would
>> find another solution after manipulating the starting values for the
>> covariances or other criteria for convergence. Best regards,
>>
>> Ben.
>>
>>
>> On 22/03/2018 10:05, Maarten Jung 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]).
>>>
>>> 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.
>>>
>>> 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
>>> <mailto: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
>>> <
>> 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>
>>> > <mailto: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>>
>>> > <mailto: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>
>>> >
>>> <
>> 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>>
>>> > > <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:
>>> > > >
>>> > > > 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/
>>>
>>> >
>>> <
>> 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/
>>>
>>> >
>>> <
>> 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
>>>
>>> >
>>> <
>> 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>>
>>> > <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 <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/
>>>>>
>>> > > >>
>>> > >
>>> >
>>> <
>> 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
>>>>>
>>> > > >>
>>> > >
>>> >
>>> <
>> 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>>>>
>>> > > >> <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>>>>> 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>>>
>>> > > >>
>>> > >
>>> <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
>>> <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>>>>> mailing list
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>>> > > >>
>>> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>> <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>
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>>> <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>>>>
>>> > > >>
>>> > > >>
>>> > > >
>>> > > >
>>> > > > [[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
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>>> > <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>> <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>>
>>> > >
>>> > > [[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>>> mailing list
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>>> > [[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
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>>> <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>>
>>> >
>>> >
>>>
>>>
>>> [[alternative HTML version deleted]]
>>>
>>> _______________________________________________
>>> R-sig-mixed-models at r-project.org
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>>> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>>> <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>
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