Re: [R] {Spam?} RE: {Spam?} RE: {Spam?} Re: mgcv, testing gamm vs lme, which degrees of freedom?

From: Carlo Fezzi <c.fezzi_at_uea.ac.uk>
Date: Tue, 22 Jun 2010 00:31:57 +0100 (BST)

Hi Christos,

thanks for your kind reply, I agree entirely with your interpreation.

In the first model comparison, however, "anova" does seem to work according to our interpretation, i.e. the "df" are equal in the two model. My intuition is that the "anova" command does a fixed-effect test rather than a random effect one. This is the results I get:

anova(f1$lme,f2$lme)

       Model df      AIC      BIC    logLik
f1$lme     1  5 466.6232 479.6491 -228.3116
f2$lme     2  5 347.6293 360.6551 -168.8146

Hence I was not sure our interpretation was correct.

On your second regarding mode point I totally agree about the appealing of GAMs... howver, I am working in a specific application where the quadratic function is the established benchmark and I think that testing against it will show even more strongly the appeal of a gamm approach. Any idea of which bases could work?

Finally thansk for the tip regarding gamm4, unfortunately I need to fit a bivariate smooth so I cannot use it.

Best wishes,

Carlo

>From my understanding, GAMs as well as GAMMs can be represented as
> (generalized) mixed models (Simon discusses this in the last chapter of
> his book; a condensed overview from a Bayesian perspective is given in
> the following paper by Y. Zhao, J. Staudenmayer, B. A. Coull and M. P.
> Wand : http://arxiv.org/pdf/math/0606491) .
>
> In this representation the un-penalized part appears as part of the fixed
> effects, whereas the penalized part appears in the random effect design
> matrix. The fixed effects part is determined by the basis (and the penalty
> order for TP splines), so that a comparison of the following:
>
> f1 <- gamm(y ~ s(x, k=2, bs="cr"), random =
> list(id=~1), method="ML" )
> f2 <- gamm(y ~ s(x,
> k=10, bs="cr"), random = list(id=~1), method="ML")
>
> can be undertaken via ANOVA tests since the 2 models have the same fixed
> effects.
> (If I remember correctly, the fixed effects for the cubic regression
> splines in GAM(M)s involves a design matrix with 2 columns: the intercept
> and the linear function. )
> The difference in degrees of freedom of the two models arises from the
> additional "knots" that the larger model has relative to the smaller one.

> Now when you try to compare the following models:
>
> f3 <- gamm(y ~ x + I(x^2), random = list(id=~1),
> method="ML" )
> f2 <- gamm(y ~ s(x, k=10,
> bs="cr"), random = list(id=~1), method="ML"
>
> the I(x^2) is not part of the (implicit) fixed effects design matrix of
> the cubic regression smoother and the models are not nested. However if
> you were to use a smooth term, whose fixed effects decomposition included
> the constant, linear and quadratic functions, then one could resort to an
> ANOVA type of test (or at least so I think).
>
> Having used GAM(M)s extensively over the last 14 months, I have to point
> out that the appeal of the methodology to the analyst is precisely its
> ability to let the data distinguish between f2 and f3 without fitting the
> 2 models. Rather than try to distinguish between a quadratic and a more
> general relationship, it might be easier to fit the "f3' model and note
> the resulting degrees of freedom; if it is close to 2, then the data have
> essentially told you that a quadratic function is appropriate for the
> problem at hand and you can comment about this when summarizing
> inferences. Note that I do not recommend fitting the quadratic after a
> GAMM has suggested this relationship :)
>
> One last thing you should be aware concerns the numerical performance of
> gamm (versus its cousin gamm4); the lmer package is much much faster and
> numerically more stable for large problems so that you should prefer the
> second interface for large problems.
>
>
> Christos Argyropoulos
>
>> Date: Mon, 21 Jun 2010 23:05:55 +0100
>> From: c.fezzi_at_uea.ac.uk
>> To: gunter.berton_at_gene.com
>> CC: r-help_at_r-project.org
>> Subject: Re: [R] {Spam?} RE: {Spam?} Re: mgcv, testing gamm vs lme,
>> which degrees of freedom?
>>
>> Thanks for your help, Bert,
>>
>> on your points:
>>
>> 1. I was not aware of such a list, I will check it out.
>>
>> 2. As you said Pinheiro and Bates (2000) advocate anova LR test for
>> random
>> effect with the SAME fixed effect. My question was exactly related to
>> this
>> point... are the fixed effect the same in this gamm example? My
>> understanding is that their are not.
>>
>> 3. I read Wood (2006) book and belive it is really a great source.
>> However, I do not have access to it at the moment cause I am away from
>> the
>> office for a visting period overseas. If I recall correctly there is not
>> an example such as the comparison of f2 vs f3, hence my question on the
>> list... but I may well be mistaken and I will double check as soon as I
>> am
>> back. Hopefully this does not sound too unreasonable.
>>
>> Best wishes,
>>
>> Carlo
>>
>> >
>> >
>> > Bert Gunter
>> > Genentech Nonclinical Statistics
>> >
>> >> -----Original Message-----
>> >> From: r-help-bounces_at_r-project.org
>> [mailto:r-help-bounces_at_r-project.org]
>> >> On Behalf Of Joris Meys
>> >> Sent: Monday, June 21, 2010 12:09 PM
>> >> To: Carlo Fezzi
>> >> Cc: r-help_at_r-project.org
>> >> Subject: Re: [R] {Spam?} Re: mgcv, testing gamm vs lme,which degrees
>> of
>> >> freedom?
>> >>
>> >> Hi Carlo,
>> >>
>> >> You should get the book of Simon Wood and read it thoroughly. It's
>> all
>> >> explained in there, but it will lead me too far to copy it all in a
>> >> mail. In short : random effects are part of the error structure of
>> the
>> >> model, not of the model itself. They're added to correct the error on
>> >> the parameters of the fixed model, and are inherent to the data
>> >> structure and not to the hypotheses. Hence, you rarely test their
>> >> significance.
>> >
>> > 1. This discussion probably belongs on the sig-mixed-models list.
>> >
>> > 2. Your claim is incorrect, I think. The structure of the random
>> errors =
>> > model covariance can be parameterized in various ways, and one can try
>> to
>> > test significance of nested parameterizations (for a particular fixed
>> > effects parameterizaton). Whether you can do so meaningfully
>> especially in
>> > the gamm context -- is another issue, but if you check e.g. Bates and
>> > Pinheiro, anova for different random effects parameterizations is
>> > advocated,
>> > and is implemented in the anova.lme() nlme function.
>> >
>> > 3. But I strongly endorse your suggestion to consult an authoritative
>> > resource; I believe it inherently unreasonable (but alas not unusual)
>> that
>> > posters somehow expect brief explanations on this list to illuminate
>> and
>> > resolve complex statistical issues.
>> >
>> > -- Bert
>> >
>> >
>> >>
>> >> Cheers
>> >> Joris
>> >>
>> >>
>> >> On Mon, Jun 21, 2010 at 6:54 PM, Carlo Fezzi <c.fezzi_at_uea.ac.uk>
>> wrote:
>> >> > Hi Joris (CC Simon),
>> >> >
>> >> > Thanks for your kind replies and for being so responsive.
>> >> >
>> >> > I think this post boils down to two main questions (which I feel
>> are
>> >> very
>> >> > important for gams modelling):
>> >> >
>> >> > 1- Is it appropriate to use LR tests in "gamm" to test model
>> >> reduction?
>> >> > 2- If yes, which degrees of freedom should be used?
>> >> >
>> >> > I do not think we should always use the df from "model$lme". For
>> >> example,
>> >> > compare the two models (again my first example for the data
>> >> generation):
>> >> >
>> >> > f1 <- gamm(y ~ s(x, k=2, bs="cr"), random = list(id=~1),
>> method="ML" )
>> >> > f2 <- gamm(y ~ s(x, k=10, bs="cr"), random = list(id=~1),
>> method="ML"
>> >> )
>> >> >
>> >> > The difference between the two models is in the random effects.
>> Model
>> >> "f2"
>> >> > has, if I interpreet correctly the output, 7 random effects more
>> than
>> >> the
>> >> > model "f1", but the fixed effects are the same. So the H0 = "the 7
>> >> random
>> >> > effect are not significant". In this case the (app.) likelihood
>> ratio
>> >> test
>> >> > should have 7 df... is my interpretation correct?
>> >> >
>> >> > On the other hand, to compare the following models:
>> >> >
>> >> > f3 <- gamm(y ~ x + I(x^2), random = list(id=~1), method="ML" )
>> >> > f2 <- gamm(y ~ s(x, k=10, bs="cr"), random = list(id=~1),
>> method="ML"
>> >> )
>> >> >
>> >> > Model "f3" has 1 more fixed effect than model "f2", but model "f2"
>> has
>> >> 7
>> >> > more random effects... again, if I understand correctly the output.
>> In
>> >> > this case I don't know if we can do a LR test, the model are not
>> >> strictly
>> >> > nested I think...
>> >> >
>> >> > What do you think?
>> >> >
>> >> > Again many thanks,
>> >> >
>> >> > Carlo
>> >> >
>> >> >
>> >> >
>> >> >> I don't use an LR test for non-nested models, as I fail to
>> formulate
>> >> a
>> >> >> sensible null hypothesis for such tests. Again, everything I write
>> is
>> >> >> a personal opinion, and inference in the case of these models is
>> >> still
>> >> >> subject of discussion to date. If you find a plausible way for
>> >> >> explaining the result, by all means use the LR test.
>> >> >>
>> >> >> Personally, I'd go for the AIC / BIC, but these are based on the
>> >> >> likelihood themselves. So in the case where the effective
>> complexity
>> >> >> of the model appears the same, they're completely equivalent to
>> the
>> >> >> likelihood. It's just the inference (i.e. the p-value) I don't
>> trust.
>> >> >> But then again, I'm a cautious statistician. If I'm not sure about
>> a
>> >> >> method, I'd rather don't use it and go with what I know. In my
>> view,
>> >> >> there is not one correct method for a particular problem and/or
>> >> >> dataset. Every method makes assumptions and has shortcomings. Only
>> if
>> >> >> I know which ones, I can take them into account when interpreting
>> the
>> >> >> results.
>> >> >>
>> >> >> It also depends on the focus as well. If the focus is prediction,
>> you
>> >> >> might even want to consider testing whether the variance of the
>> >> >> residuals differs significantly with a simple F-test. This
>> indicates
>> >> >> whether the predictive power differs significantly between the
>> >> models.
>> >> >> But these tests tend to get very sensitive when you have many
>> >> >> datapoints, rendering them practically useless again.
>> >> >>
>> >> >> So in the end, it always boils down to interpretation.
>> >> >>
>> >> >> Cheers
>> >> >> Joris
>> >> >>
>> >> >> On Fri, Jun 18, 2010 at 10:29 PM, Carlo Fezzi <c.fezzi_at_uea.ac.uk>
>> >> wrote:
>> >> >>> Thanks Joris,
>> >> >>>
>> >> >>> I understand your point regarding the need for the two models to
>> be
>> >> >>> nested. So, according to your in the example case the LR test is
>> not
>> >> >>> appropriate and the two model should be compared with other
>> criteria
>> >> >>> such
>> >> >>> as AIC or BIC for example.
>> >> >>>
>> >> >>> On the other hand, Simon Wood indicated that such a LR test is
>> >> >>> (approximately) correct in his previous reply... a am bit
>> confused,
>> >> >>> which
>> >> >>> is the correct approach to test the two models? Is the LR test
>> >> correct
>> >> >>> only if the parametric model is linear in the x variables maybe?
>> In
>> >> this
>> >> >>> case, which is the best appraoch to compare a "gamm" vs a "lme"
>> with
>> >> >>> quadratic specification?
>> >> >>>
>> >> >>> Best wishes,
>> >> >>>
>> >> >>> Carlo
>> >> >>>
>> >> >>>> Just realized something: You should take into account that the
>> LR
>> >> test
>> >> >>>> is actually only valid for _nested_ models. Your models are not
>> >> >>>> nested. Hence, you shouldn't use the anova function to compare
>> >> them,
>> >> >>>> and you shouldn't compare the df. In fact, if you're interested
>> in
>> >> the
>> >> >>>> contribution of a term, then using anova to compare the model
>> with
>> >> >>>> that term and without that term gives you an answer on the
>> >> hypothesis
>> >> >>>> whether that term with spline contributes significantly to the
>> >> model.
>> >> >>>>
>> >> >>>>> f2 <- gamm(y ~ s(x), random = list(id=~1), method="ML")
>> >> >>>>
>> >> >>>>> f3 <- gamm(y ~ x, random = list(id=~1), method="ML" )
>> >> >>>>
>> >> >>>>> f4 <- gamm(y ~ 1, random = list(id=~1), method="ML" )
>> >> >>>>
>> >> >>>>> anova(f3$lme,f2$lme)
>> >> >>>> Model df AIC BIC logLik Test L.Ratio p-value
>> >> >>>> f3$lme 1 4 760 770 -376
>> >> >>>> f2$lme 2 5 381 394 -186 1 vs 2 380 <.0001
>> >> >>>>
>> >> >>>>> anova(f4$lme,f2$lme)
>> >> >>>> Model df AIC BIC logLik Test L.Ratio p-value
>> >> >>>> f4$lme 1 3 945 953 -470
>> >> >>>> f2$lme 2 5 381 394 -186 1 vs 2 568 <.0001
>> >> >>>>
>> >> >>>>> anova(f3$lme,f4$lme)
>> >> >>>> Model df AIC BIC logLik Test L.Ratio p-value
>> >> >>>> f3$lme 1 4 760 770 -376
>> >> >>>> f4$lme 2 3 945 953 -470 1 vs 2 188 <.0001
>> >> >>>>
>> >> >>>> This is the correct application of a likelihood ratio test. You
>> see
>> >> >>>> that adding the spline increases the df with 1 compared to the
>> >> linear
>> >> >>>> model, as part of the spline gets into the random component.
>> Notice
>> >> as
>> >> >>>> well that the interpretation of a test in case of a random
>> >> component
>> >> >>>> is not the same as in case of a fixed component. If I understood
>> >> >>>> correctly, this LR test specifically says something over the
>> effect
>> >> of
>> >> >>>> X, without being interested in the shape of the spline. The
>> >> >>>> "significance of a spline" is a difficult concept anyway, as a
>> >> spline
>> >> >>>> can be seen as a form of local regression. It's exactly the use
>> of
>> >> the
>> >> >>>> randomization that allows for a general hypothesis about the
>> added
>> >> >>>> value of the spline, without focusing on its actual shape. Hence
>> >> the
>> >> >>>> "freedom" connected to that actual shape should not be used in
>> the
>> >> df
>> >> >>>> used to test the general hypothesis.
>> >> >>>>
>> >> >>>> Hope this makes sense someway...
>> >> >>>>
>> >> >>>> Cheers
>> >> >>>> Joris
>> >> >>>>
>> >> >>>>
>> >> >>>> On Fri, Jun 18, 2010 at 6:27 PM, Carlo Fezzi <c.fezzi_at_uea.ac.uk>
>> >> wrote:
>> >> >>>>> Dear Simon,
>> >> >>>>>
>> >> >>>>> thanks a lot for your prompt reply.
>> >> >>>>>
>> >> >>>>> Unfortunately I am still confused about which is the correct
>> way
>> >> to
>> >> >>>>> test
>> >> >>>>> the two models... as you point out: why in my example the two
>> >> models
>> >> >>>>> have
>> >> >>>>> the same degrees of freedom?
>> >> >>>>>
>> >> >>>>> Intuitively it seems to me the gamm model is more flexible
>> since,
>> >> as
>> >> I
>> >> >>>>> understand also from you response, it should contain more
>> random
>> >> >>>>> effects
>> >> >>>>> than the other model because some of the smooth function
>> >> parameters
>> >> >>>>> are
>> >> >>>>> represented as such. This should not be taken into account when
>> >> >>>>> testing
>> >> >>>>> one model vs the other?
>> >> >>>>>
>> >> >>>>> Continuing with my example, the two models:
>> >> >>>>>
>> >> >>>>> f2 <- gamm(y ~ s(x), random = list(id=~1), method="ML")
>> >> >>>>> f3 <- gamm(y ~ x + I(x^2), random = list(id=~1), method="ML" )
>> >> >>>>>
>> >> >>>>> Can be tested with:
>> >> >>>>>
>> >> >>>>> anova(f3$lme,f2$lme)
>> >> >>>>>
>> >> >>>>> But why are the df the same? Model f2 appears to be more
>> flexible
>> >> and,
>> >> >>>>> as
>> >> >>>>> such, should have more (random) parameters. Should not a test
>> of
>> >> one
>> >> >>>>> model
>> >> >>>>> vs the other take this into account?
>> >> >>>>>
>> >> >>>>> Sorry if this may sound dull, many thanks for your help,
>> >> >>>>>
>> >> >>>>> Carlo
>> >> >>>>>
>> >> >>>>>
>> >> >>>>>
>> >> >>>>>> On Wednesday 16 June 2010 20:33, Carlo Fezzi wrote:
>> >> >>>>>>> Dear all,
>> >> >>>>>>>
>> >> >>>>>>> I am using the "mgcv" package by Simon Wood to estimate an
>> >> additive
>> >> >>>>>>> mixed
>> >> >>>>>>> model in which I assume normal distribution for the
>> residuals. I
>> >> >>>>>>> would
>> >> >>>>>>> like to test this model vs a standard parametric mixed model,
>> >> such
>> >> >>>>>>> as
>> >> >>>>>>> the
>> >> >>>>>>> ones which are possible to estimate with "lme".
>> >> >>>>>>>
>> >> >>>>>>> Since the smoothing splines can be written as random effects,
>> is
>> >> it
>> >> >>>>>>> correct to use an (approximate) likelihood ratio test for
>> this?
>> >> >>>>>> -- yes this is ok (subject to the usual caveats about testing
>> on
>> >> the
>> >> >>>>>> boundary
>> >> >>>>>> of the parameter space) but your 2 example models below will
>> have
>> >> >>>>>> the
>> >> >>>>>> same
>> >> >>>>>> number of degrees of freedom!
>> >> >>>>>>
>> >> >>>>>>> If so,
>> >> >>>>>>> which is the correct number of degrees of freedom?
>> >> >>>>>> --- The edf from the lme object, if you are testing using the
>> log
>> >> >>>>>> likelihood
>> >> >>>>>> returned by the lme representation of the model.
>> >> >>>>>>
>> >> >>>>>>> Sometime the function
>> >> >>>>>>> LogLik() seems to provide strange results regarding the
>> number
>> >> of
>> >> >>>>>>> degrees
>> >> >>>>>>> of freedom (df) for the gam, for instance in the example I
>> >> copied
>> >> >>>>>>> below
>> >> >>>>>>> the df for the "gamm" are equal to the ones for the "lme",
>> but
>> >> the
>> >> >>>>>>> summary(model.gam) seems to indicate a much higher edf for
>> the
>> >> gamm.
>> >> >>>>>> --- the edf for the lme representation of the model counts
>> only
>> >> the
>> >> >>>>>> fixed
>> >> >>>>>> effects + the variance parameters (which includes smoothing
>> >> >>>>>> parameters).
>> >> >>>>>> Each
>> >> >>>>>> smooth typically contributes only one or two fixed effect
>> >> parameters,
>> >> >>>>>> with
>> >> >>>>>> the rest of the coefficients for the smooth treated as random
>> >> >>>>>> effects.
>> >> >>>>>>
>> >> >>>>>> --- the edf for the gam representation of the same model
>> differs
>> >> in
>> >> >>>>>> that
>> >> >>>>>> it
>> >> >>>>>> also counts the *effective* number of parameters used to
>> >> represent
>> >> >>>>>> each
>> >> >>>>>> smooth: this includes contributions from all those
>> coefficients
>> >> that
>> >> >>>>>> the
>> >> >>>>>> lme
>> >> >>>>>> representation treated as strictly random.
>> >> >>>>>>
>> >> >>>>>> best,
>> >> >>>>>> Simon
>> >> >>>>>>
>> >> >>>>>>
>> >> >>>>>>> I would be very grateful to anybody who could point out a
>> >> solution,
>> >> >>>>>>>
>> >> >>>>>>> Best wishes,
>> >> >>>>>>>
>> >> >>>>>>> Carlo
>> >> >>>>>>>
>> >> >>>>>>> Example below:
>> >> >>>>>>>
>> >> >>>>>>> ----
>> >> >>>>>>>
>> >> >>>>>>> rm(list = ls())
>> >> >>>>>>> library(mgcv)
>> >> >>>>>>> library(nlme)
>> >> >>>>>>>
>> >> >>>>>>> set.seed(123)
>> >> >>>>>>>
>> >> >>>>>>> x <- runif(100,1,10) #
>> regressor
>> >> >>>>>>> b0 <- rep(rnorm(10,mean=1,sd=2),each=10) # random
>> intercept
>> >> >>>>>>> id <- rep(1:10, each=10) # identifier
>> >> >>>>>>>
>> >> >>>>>>> y <- b0 + x - 0.1 * x^3 + rnorm(100,0,1) # dependent
>> variable
>> >> >>>>>>>
>> >> >>>>>>> f1 <- lme(y ~ x + I(x^2), random = list(id=~1) , method="ML"
>> )
>> >> #
>> >> >>>>>>> lme
>> >> >>>>>>> model
>> >> >>>>>>>
>> >> >>>>>>> f2 <- gamm(y ~ s(x), random = list(id=~1), method="ML" ) #
>> >> gamm
>> >> >>>>>>>
>> >> >>>>>>> ## same number of "df" according to logLik:
>> >> >>>>>>> logLik(f1)
>> >> >>>>>>> logLik(f2$lme)
>> >> >>>>>>>
>> >> >>>>>>> ## much higher edf according to summary:
>> >> >>>>>>> summary(f2$gam)
>> >> >>>>>>>
>> >> >>>>>>> -----------
>> >> >>>>>>>
>> >> >>>>>>> ______________________________________________
>> >> >>>>>>> R-help_at_r-project.org mailing list
>> >> >>>>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> >>>>>>> PLEASE do read the posting guide
>> >> >>>>>>> http://www.R-project.org/posting-guide.html and provide
>> >> commented,
>> >> >>>>>>> minimal,
>> >> >>>>>>> self-contained, reproducible code.
>> >> >>>>>>
>> >> >>>>>> --
>> >> >>>>>>> Simon Wood, Mathematical Sciences, University of Bath, Bath,
>> BA2
>> >> 7AY
>> >> >>>>>>> UK
>> >> >>>>>>> +44 1225 386603 www.maths.bath.ac.uk/~sw283
>> >> >>>>>>
>> >> >>>>>
>> >> >>>>> ______________________________________________
>> >> >>>>> R-help_at_r-project.org mailing list
>> >> >>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> >>>>> PLEASE do read the posting guide
>> >> >>>>> http://www.R-project.org/posting-guide.html
>> >> >>>>> and provide commented, minimal, self-contained, reproducible
>> code.
>> >> >>>>>
>> >> >>>>
>> >> >>>>
>> >> >>>>
>> >> >>>> --
>> >> >>>> Joris Meys
>> >> >>>> Statistical consultant
>> >> >>>>
>> >> >>>> Ghent University
>> >> >>>> Faculty of Bioscience Engineering
>> >> >>>> Department of Applied mathematics, biometrics and process
>> control
>> >> >>>>
>> >> >>>> tel : +32 9 264 59 87
>> >> >>>> Joris.Meys_at_Ugent.be
>> >> >>>> -------------------------------
>> >> >>>> Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php
>> >> >>>>
>> >> >>>
>> >> >>>
>> >> >>>
>> >> >>
>> >> >>
>> >> >>
>> >> >> --
>> >> >> Joris Meys
>> >> >> Statistical consultant
>> >> >>
>> >> >> Ghent University
>> >> >> Faculty of Bioscience Engineering
>> >> >> Department of Applied mathematics, biometrics and process control
>> >> >>
>> >> >> tel : +32 9 264 59 87
>> >> >> Joris.Meys_at_Ugent.be
>> >> >> -------------------------------
>> >> >> Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php
>> >> >>
>> >> >
>> >> >
>> >> >
>> >>
>> >>
>> >>
>> >> --
>> >> Joris Meys
>> >> Statistical consultant
>> >>
>> >> Ghent University
>> >> Faculty of Bioscience Engineering
>> >> Department of Applied mathematics, biometrics and process control
>> >>
>> >> tel : +32 9 264 59 87
>> >> Joris.Meys_at_Ugent.be
>> >> -------------------------------
>> >> Disclaimer : http://helpdesk.ugent.be/e-maildisclaimer.php
>> >>
>> >> ______________________________________________
>> >> R-help_at_r-project.org mailing list
>> >> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> PLEASE do read the posting guide http://www.R-project.org/posting-
>> >> guide.html
>> >> and provide commented, minimal, self-contained, reproducible code.
>> >
>> >
>>
>> ______________________________________________
>> R-help_at_r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
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