[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
Tyler
tylermw at gmail.com
Thu Nov 2 16:11:51 CET 2017
Hi Arie,
The book out of which this behavior is based does not use factor (in this
section) to refer to categorical factor. I will again point to this
sentence, from page 40, in the same section and referring to the behavior
under question, that shows F_j is not limited to categorical factors:
"Numeric variables appear in the computations as themselves, uncoded.
Therefore, the rule does not do anything special for them, and it remains
valid, in a trivial sense, whenever any of the F_j is numeric rather than
categorical."
Note the "... whenever any of the F_j is numeric rather than categorical."
Factor here is used in the more general sense of the word, not referring to
the R type "factor." The behavior of R does not match the heuristic that
it's citing.
Best regards,
Tyler
On Thu, Nov 2, 2017 at 2:51 AM, Arie ten Cate <arietencate at gmail.com> wrote:
> Hello Tyler,
>
> Thank you for searching for, and finding, the basic description of the
> behavior of R in this matter.
>
> I think your example is in agreement with the book.
>
> But let me first note the following. You write: "F_j refers to a
> factor (variable) in a model and not a categorical factor". However:
> "a factor is a vector object used to specify a discrete
> classification" (start of chapter 4 of "An Introduction to R".) You
> might also see the description of the R function factor().
>
> You note that the book says about a factor F_j:
> "... F_j is coded by contrasts if T_{i(j)} has appeared in the
> formula and by dummy variables if it has not"
>
> You find:
> "However, the example I gave demonstrated that this dummy variable
> encoding only occurs for the model where the missing term is the
> numeric-numeric interaction, ~(X1+X2+X3)^3-X1:X2."
>
> We have here T_i = X1:X2:X3. Also: F_j = X3 (the only factor). Then
> T_{i(j)} = X1:X2, which is dropped from the model. Hence the X3 in T_i
> must be encoded by dummy variables, as indeed it is.
>
> Arie
>
> On Tue, Oct 31, 2017 at 4:01 PM, Tyler <tylermw at gmail.com> wrote:
> > Hi Arie,
> >
> > Thank you for your further research into the issue.
> >
> > Regarding Stata: On the other hand, JMP gives model matrices that use the
> > main effects contrasts in computing the higher order interactions,
> without
> > the dummy variable encoding. I verified this both by analyzing the linear
> > model given in my first example and noting that JMP has one more degree
> of
> > freedom than R for the same model, as well as looking at the generated
> model
> > matrices. It's easy to find a design where JMP will allow us fit our
> model
> > with goodness-of-fit estimates and R will not due to the extra degree(s)
> of
> > freedom required. Let's keep the conversation limited to R.
> >
> > I want to refocus back onto my original bug report, which was not for a
> > missing main effects term, but rather for a missing lower-order
> interaction
> > term. The behavior of model.matrix.default() for a missing main effects
> term
> > is a nice example to demonstrate how model.matrix encodes with dummy
> > variables instead of contrasts, but doesn't demonstrate the inconsistent
> > behavior my bug report highlighted.
> >
> > I went looking for documentation on this behavior, and the issue stems
> not
> > from model.matrix.default(), but rather the terms() function in
> interpreting
> > the formula. This "clever" replacement of contrasts by dummy variables to
> > maintain marginality (presuming that's the reason) is not described
> anywhere
> > in the documentation for either the model.matrix() or the terms()
> function.
> > In order to find a description for the behavior, I had to look in the
> > underlying C code, buried above the "TermCode" function of the "model.c"
> > file, which says:
> >
> > "TermCode decides on the encoding of a model term. Returns 1 if variable
> > ``whichBit'' in ``thisTerm'' is to be encoded by contrasts and 2 if it
> is to
> > be encoded by dummy variables. This is decided using the heuristic
> > described in Statistical Models in S, page 38."
> >
> > I do not have a copy of this book, and I suspect most R users do not as
> > well. Thankfully, however, some of the pages describing this behavior
> were
> > available as part of Amazon's "Look Inside" feature--but if not for
> that, I
> > would have no idea what heuristic R was using. Since those pages could
> made
> > unavailable by Amazon at any time, at the very least we have an problem
> with
> > a lack of documentation.
> >
> > However, I still believe there is a bug when comparing R's
> implementation to
> > the heuristic described in the book. From Statistical Models in S, page
> > 38-39:
> >
> > "Suppose F_j is any factor included in term T_i. Let T_{i(j)} denote the
> > margin of T_i for factor F_j--that is, the term obtained by dropping F_j
> > from T_i. We say that T_{i(j)} has appeared in the formula if there is
> some
> > term T_i' for i' < i such that T_i' contains all the factors appearing in
> > T_{i(j)}. The usual case is that T_{i(j)} itself is one of the preceding
> > terms. Then F_j is coded by contrasts if T_{i(j)} has appeared in the
> > formula and by dummy variables if it has not"
> >
> > Here, F_j refers to a factor (variable) in a model and not a categorical
> > factor, as specified later in that section (page 40): "Numeric variables
> > appear in the computations as themselves, uncoded. Therefore, the rule
> does
> > not do anything special for them, and it remains valid, in a trivial
> sense,
> > whenever any of the F_j is numeric rather than categorical."
> >
> > Going back to my original example with three variables: X1 (numeric), X2
> > (numeric), X3 (categorical). This heuristic prescribes encoding X1:X2:X3
> > with contrasts as long as X1:X2, X1:X3, and X2:X3 exist in the formula.
> When
> > any of the preceding terms do not exist, this heuristic tells us to use
> > dummy variables to encode the interaction (e.g. "F_j [the interaction
> term]
> > is coded ... by dummy variables if it [any of the marginal terms
> obtained by
> > dropping a single factor in the interaction] has not [appeared in the
> > formula]"). However, the example I gave demonstrated that this dummy
> > variable encoding only occurs for the model where the missing term is the
> > numeric-numeric interaction, "~(X1+X2+X3)^3-X1:X2". Otherwise, the
> > interaction term X1:X2:X3 is encoded by contrasts, not dummy variables.
> This
> > is inconsistent with the description of the intended behavior given in
> the
> > book.
> >
> > Best regards,
> > Tyler
> >
> >
> > On Fri, Oct 27, 2017 at 2:18 PM, Arie ten Cate <arietencate at gmail.com>
> > wrote:
> >>
> >> Hello Tyler,
> >>
> >> I want to bring to your attention the following document: "What
> >> happens if you omit the main effect in a regression model with an
> >> interaction?"
> >> (https://stats.idre.ucla.edu/stata/faq/what-happens-if-you-
> omit-the-main-effect-in-a-regression-model-with-an-interaction).
> >> This gives a useful review of the problem. Your example is Case 2: a
> >> continuous and a categorical regressor.
> >>
> >> The numerical examples are coded in Stata, and they give the same
> >> result as in R. Hence, if this is a bug in R then it is also a bug in
> >> Stata. That seems very unlikely.
> >>
> >> Here is a simulation in R of the above mentioned Case 2 in Stata:
> >>
> >> df <- expand.grid(socst=c(-1:1),grp=c("1","2","3","4"))
> >> print("Full model")
> >> print(model.matrix(~(socst+grp)^2 ,data=df))
> >> print("Example 2.1: drop socst")
> >> print(model.matrix(~(socst+grp)^2 -socst ,data=df))
> >> print("Example 2.2: drop grp")
> >> print(model.matrix(~(socst+grp)^2 -grp ,data=df))
> >>
> >> This gives indeed the following regressors:
> >>
> >> "Full model"
> >> (Intercept) socst grp2 grp3 grp4 socst:grp2 socst:grp3 socst:grp4
> >> "Example 2.1: drop socst"
> >> (Intercept) grp2 grp3 grp4 socst:grp1 socst:grp2 socst:grp3 socst:grp4
> >> "Example 2.2: drop grp"
> >> (Intercept) socst socst:grp2 socst:grp3 socst:grp4
> >>
> >> There is a little bit of R documentation about this, based on the
> >> concept of marginality, which typically forbids a model having an
> >> interaction but not the corresponding main effects. (You might see the
> >> references in https://en.wikipedia.org/wiki/Principle_of_marginality )
> >> See "An Introduction to R", by Venables and Smith and the R Core
> >> Team. At the bottom of page 52 (PDF: 57) it says: "Although the
> >> details are complicated, model formulae in R will normally generate
> >> the models that an expert statistician would expect, provided that
> >> marginality is preserved. Fitting, for [a contrary] example, a model
> >> with an interaction but not the corresponding main effects will in
> >> general lead to surprising results ....".
> >> The Reference Manual states that the R functions dropterm() and
> >> addterm() resp. drop or add only terms such that marginality is
> >> preserved.
> >>
> >> Finally, about your singular matrix t(mm)%*%mm. This is in fact
> >> Example 2.1 in Case 2 discussed above. As discussed there, in Stata
> >> and in R the drop of the continuous variable has no effect on the
> >> degrees of freedom here: it is just a reparameterisation of the full
> >> model, protecting you against losing marginality... Hence the
> >> model.matrix 'mm' is still square and nonsingular after the drop of
> >> X1, unless of course when a row is removed from the matrix 'design'
> >> when before creating 'mm'.
> >>
> >> Arie
> >>
> >> On Sun, Oct 15, 2017 at 7:05 PM, Tyler <tylermw at gmail.com> wrote:
> >> > You could possibly try to explain away the behavior for a missing main
> >> > effects term, since without the main effects term we don't have main
> >> > effect
> >> > columns in the model matrix used to compute the interaction columns
> (At
> >> > best this is undocumented behavior--I still think it's a bug, as we
> know
> >> > how we would encode the categorical factors if they were in fact
> >> > present.
> >> > It's either specified in contrasts.arg or using the default set in
> >> > options). However, when all the main effects are present, why would
> the
> >> > three-factor interaction column not simply be the product of the main
> >> > effect columns? In my example: we know X1, we know X2, and we know X3.
> >> > Why
> >> > does the encoding of X1:X2:X3 depend on whether we specified a
> >> > two-factor
> >> > interaction, AND only changes for specific missing interactions?
> >> >
> >> > In addition, I can use a two-term example similar to yours to show how
> >> > this
> >> > behavior results in a singular covariance matrix when, given the
> desired
> >> > factor encoding, it should not be singular.
> >> >
> >> > We start with a full factorial design for a two-level continuous
> factor
> >> > and
> >> > a three-level categorical factor, and remove a single row. This design
> >> > matrix does not leave enough degrees of freedom to determine
> >> > goodness-of-fit, but should allow us to obtain parameter estimates.
> >> >
> >> >> design = expand.grid(X1=c(1,-1),X2=c("A","B","C"))
> >> >> design = design[-1,]
> >> >> design
> >> > X1 X2
> >> > 2 -1 A
> >> > 3 1 B
> >> > 4 -1 B
> >> > 5 1 C
> >> > 6 -1 C
> >> >
> >> > Here, we first calculate the model matrix for the full model, and then
> >> > manually remove the X1 column from the model matrix. This gives us the
> >> > model matrix one would expect if X1 were removed from the model. We
> then
> >> > successfully calculate the covariance matrix.
> >> >
> >> >> mm = model.matrix(~(X1+X2)^2,data=design)
> >> >> mm
> >> > (Intercept) X1 X2B X2C X1:X2B X1:X2C
> >> > 2 1 -1 0 0 0 0
> >> > 3 1 1 1 0 1 0
> >> > 4 1 -1 1 0 -1 0
> >> > 5 1 1 0 1 0 1
> >> > 6 1 -1 0 1 0 -1
> >> >
> >> >> mm = mm[,-2]
> >> >> solve(t(mm) %*% mm)
> >> > (Intercept) X2B X2C X1:X2B X1:X2C
> >> > (Intercept) 1 -1.0 -1.0 0.0 0.0
> >> > X2B -1 1.5 1.0 0.0 0.0
> >> > X2C -1 1.0 1.5 0.0 0.0
> >> > X1:X2B 0 0.0 0.0 0.5 0.0
> >> > X1:X2C 0 0.0 0.0 0.0 0.5
> >> >
> >> > Here, we see the actual behavior for model.matrix. The undesired
> >> > re-coding
> >> > of the model matrix interaction term makes the information matrix
> >> > singular.
> >> >
> >> >> mm = model.matrix(~(X1+X2)^2-X1,data=design)
> >> >> mm
> >> > (Intercept) X2B X2C X1:X2A X1:X2B X1:X2C
> >> > 2 1 0 0 -1 0 0
> >> > 3 1 1 0 0 1 0
> >> > 4 1 1 0 0 -1 0
> >> > 5 1 0 1 0 0 1
> >> > 6 1 0 1 0 0 -1
> >> >
> >> >> solve(t(mm) %*% mm)
> >> > Error in solve.default(t(mm) %*% mm) : system is computationally
> >> > singular:
> >> > reciprocal condition number = 5.55112e-18
> >> >
> >> > I still believe this is a bug.
> >> >
> >> > Best regards,
> >> > Tyler Morgan-Wall
> >> >
> >> > On Sun, Oct 15, 2017 at 1:49 AM, Arie ten Cate <arietencate at gmail.com
> >
> >> > wrote:
> >> >
> >> >> I think it is not a bug. It is a general property of interactions.
> >> >> This property is best observed if all variables are factors
> >> >> (qualitative).
> >> >>
> >> >> For example, you have three variables (factors). You ask for as many
> >> >> interactions as possible, except an interaction term between two
> >> >> particular variables. When this interaction is not a constant, it is
> >> >> different for different values of the remaining variable. More
> >> >> precisely: for all values of that variable. In other words: you have
> a
> >> >> three-way interaction, with all values of that variable.
> >> >>
> >> >> An even smaller example is the following script with only two
> >> >> variables, each being a factor:
> >> >>
> >> >> df <- expand.grid(X1=c("p","q"), X2=c("A","B","C"))
> >> >> print(model.matrix(~(X1+X2)^2 ,data=df))
> >> >> print(model.matrix(~(X1+X2)^2 -X1,data=df))
> >> >> print(model.matrix(~(X1+X2)^2 -X2,data=df))
> >> >>
> >> >> The result is:
> >> >>
> >> >> (Intercept) X1q X2B X2C X1q:X2B X1q:X2C
> >> >> 1 1 0 0 0 0 0
> >> >> 2 1 1 0 0 0 0
> >> >> 3 1 0 1 0 0 0
> >> >> 4 1 1 1 0 1 0
> >> >> 5 1 0 0 1 0 0
> >> >> 6 1 1 0 1 0 1
> >> >>
> >> >> (Intercept) X2B X2C X1q:X2A X1q:X2B X1q:X2C
> >> >> 1 1 0 0 0 0 0
> >> >> 2 1 0 0 1 0 0
> >> >> 3 1 1 0 0 0 0
> >> >> 4 1 1 0 0 1 0
> >> >> 5 1 0 1 0 0 0
> >> >> 6 1 0 1 0 0 1
> >> >>
> >> >> (Intercept) X1q X1p:X2B X1q:X2B X1p:X2C X1q:X2C
> >> >> 1 1 0 0 0 0 0
> >> >> 2 1 1 0 0 0 0
> >> >> 3 1 0 1 0 0 0
> >> >> 4 1 1 0 1 0 0
> >> >> 5 1 0 0 0 1 0
> >> >> 6 1 1 0 0 0 1
> >> >>
> >> >> Thus, in the second result, we have no main effect of X1. Instead,
> the
> >> >> effect of X1 depends on the value of X2; either A or B or C. In fact,
> >> >> this is a two-way interaction, including all three values of X2. In
> >> >> the third result, we have no main effect of X2, The effect of X2
> >> >> depends on the value of X1; either p or q.
> >> >>
> >> >> A complicating element with your example seems to be that your X1 and
> >> >> X2 are not factors.
> >> >>
> >> >> Arie
> >> >>
> >> >> On Thu, Oct 12, 2017 at 7:12 PM, Tyler <tylermw at gmail.com> wrote:
> >> >> > Hi,
> >> >> >
> >> >> > I recently ran into an inconsistency in the way
> model.matrix.default
> >> >> > handles factor encoding for higher level interactions with
> >> >> > categorical
> >> >> > variables when the full hierarchy of effects is not present.
> >> >> > Depending on
> >> >> > which lower level interactions are specified, the factor encoding
> >> >> > changes
> >> >> > for a higher level interaction. Consider the following minimal
> >> >> reproducible
> >> >> > example:
> >> >> >
> >> >> > --------------
> >> >> >
> >> >> >> runmatrix = expand.grid(X1=c(1,-1),X2=c(1,-1),X3=c("A","B","C"))>
> >> >> model.matrix(~(X1+X2+X3)^3,data=runmatrix) (Intercept) X1 X2 X3B
> X3C
> >> >> X1:X2 X1:X3B X1:X3C X2:X3B X2:X3C X1:X2:X3B X1:X2:X3C
> >> >> > 1 1 1 1 0 0 1 0 0 0 0
> >> >> > 0 0
> >> >> > 2 1 -1 1 0 0 -1 0 0 0 0
> >> >> > 0 0
> >> >> > 3 1 1 -1 0 0 -1 0 0 0 0
> >> >> > 0 0
> >> >> > 4 1 -1 -1 0 0 1 0 0 0 0
> >> >> > 0 0
> >> >> > 5 1 1 1 1 0 1 1 0 1 0
> >> >> > 1 0
> >> >> > 6 1 -1 1 1 0 -1 -1 0 1 0
> >> >> > -1 0
> >> >> > 7 1 1 -1 1 0 -1 1 0 -1 0
> >> >> > -1 0
> >> >> > 8 1 -1 -1 1 0 1 -1 0 -1 0
> >> >> > 1 0
> >> >> > 9 1 1 1 0 1 1 0 1 0 1
> >> >> > 0 1
> >> >> > 10 1 -1 1 0 1 -1 0 -1 0 1
> >> >> > 0 -1
> >> >> > 11 1 1 -1 0 1 -1 0 1 0 -1
> >> >> > 0 -1
> >> >> > 12 1 -1 -1 0 1 1 0 -1 0 -1
> >> >> > 0 1
> >> >> > attr(,"assign")
> >> >> > [1] 0 1 2 3 3 4 5 5 6 6 7 7
> >> >> > attr(,"contrasts")
> >> >> > attr(,"contrasts")$X3
> >> >> > [1] "contr.treatment"
> >> >> >
> >> >> > --------------
> >> >> >
> >> >> > Specifying the full hierarchy gives us what we expect: the
> >> >> > interaction
> >> >> > columns are simply calculated from the product of the main effect
> >> >> columns.
> >> >> > The interaction term X1:X2:X3 gives us two columns in the model
> >> >> > matrix,
> >> >> > X1:X2:X3B and X1:X2:X3C, matching the products of the main effects.
> >> >> >
> >> >> > If we remove either the X2:X3 interaction or the X1:X3 interaction,
> >> >> > we
> >> >> get
> >> >> > what we would expect for the X1:X2:X3 interaction, but when we
> remove
> >> >> > the
> >> >> > X1:X2 interaction the encoding for X1:X2:X3 changes completely:
> >> >> >
> >> >> > --------------
> >> >> >
> >> >> >> model.matrix(~(X1+X2+X3)^3-X1:X3,data=runmatrix) (Intercept)
> X1 X2
> >> >> X3B X3C X1:X2 X2:X3B X2:X3C X1:X2:X3B X1:X2:X3C
> >> >> > 1 1 1 1 0 0 1 0 0 0
> 0
> >> >> > 2 1 -1 1 0 0 -1 0 0 0
> 0
> >> >> > 3 1 1 -1 0 0 -1 0 0 0
> 0
> >> >> > 4 1 -1 -1 0 0 1 0 0 0
> 0
> >> >> > 5 1 1 1 1 0 1 1 0 1
> 0
> >> >> > 6 1 -1 1 1 0 -1 1 0 -1
> 0
> >> >> > 7 1 1 -1 1 0 -1 -1 0 -1
> 0
> >> >> > 8 1 -1 -1 1 0 1 -1 0 1
> 0
> >> >> > 9 1 1 1 0 1 1 0 1 0
> 1
> >> >> > 10 1 -1 1 0 1 -1 0 1 0
> -1
> >> >> > 11 1 1 -1 0 1 -1 0 -1 0
> -1
> >> >> > 12 1 -1 -1 0 1 1 0 -1 0
> 1
> >> >> > attr(,"assign")
> >> >> > [1] 0 1 2 3 3 4 5 5 6 6
> >> >> > attr(,"contrasts")
> >> >> > attr(,"contrasts")$X3
> >> >> > [1] "contr.treatment"
> >> >> >
> >> >> >
> >> >> >
> >> >> >> model.matrix(~(X1+X2+X3)^3-X2:X3,data=runmatrix) (Intercept)
> X1 X2
> >> >> X3B X3C X1:X2 X1:X3B X1:X3C X1:X2:X3B X1:X2:X3C
> >> >> > 1 1 1 1 0 0 1 0 0 0
> 0
> >> >> > 2 1 -1 1 0 0 -1 0 0 0
> 0
> >> >> > 3 1 1 -1 0 0 -1 0 0 0
> 0
> >> >> > 4 1 -1 -1 0 0 1 0 0 0
> 0
> >> >> > 5 1 1 1 1 0 1 1 0 1
> 0
> >> >> > 6 1 -1 1 1 0 -1 -1 0 -1
> 0
> >> >> > 7 1 1 -1 1 0 -1 1 0 -1
> 0
> >> >> > 8 1 -1 -1 1 0 1 -1 0 1
> 0
> >> >> > 9 1 1 1 0 1 1 0 1 0
> 1
> >> >> > 10 1 -1 1 0 1 -1 0 -1 0
> -1
> >> >> > 11 1 1 -1 0 1 -1 0 1 0
> -1
> >> >> > 12 1 -1 -1 0 1 1 0 -1 0
> 1
> >> >> > attr(,"assign")
> >> >> > [1] 0 1 2 3 3 4 5 5 6 6
> >> >> > attr(,"contrasts")
> >> >> > attr(,"contrasts")$X3
> >> >> > [1] "contr.treatment"
> >> >> >
> >> >> >
> >> >> >> model.matrix(~(X1+X2+X3)^3-X1:X2,data=runmatrix) (Intercept)
> X1 X2
> >> >> X3B X3C X1:X3B X1:X3C X2:X3B X2:X3C X1:X2:X3A X1:X2:X3B X1:X2:X3C
> >> >> > 1 1 1 1 0 0 0 0 0 0 1
> >> >> > 0 0
> >> >> > 2 1 -1 1 0 0 0 0 0 0 -1
> >> >> > 0 0
> >> >> > 3 1 1 -1 0 0 0 0 0 0 -1
> >> >> > 0 0
> >> >> > 4 1 -1 -1 0 0 0 0 0 0 1
> >> >> > 0 0
> >> >> > 5 1 1 1 1 0 1 0 1 0 0
> >> >> > 1 0
> >> >> > 6 1 -1 1 1 0 -1 0 1 0 0
> >> >> > -1 0
> >> >> > 7 1 1 -1 1 0 1 0 -1 0 0
> >> >> > -1 0
> >> >> > 8 1 -1 -1 1 0 -1 0 -1 0 0
> >> >> > 1 0
> >> >> > 9 1 1 1 0 1 0 1 0 1 0
> >> >> > 0 1
> >> >> > 10 1 -1 1 0 1 0 -1 0 1 0
> >> >> > 0 -1
> >> >> > 11 1 1 -1 0 1 0 1 0 -1 0
> >> >> > 0 -1
> >> >> > 12 1 -1 -1 0 1 0 -1 0 -1 0
> >> >> > 0 1
> >> >> > attr(,"assign")
> >> >> > [1] 0 1 2 3 3 4 4 5 5 6 6 6
> >> >> > attr(,"contrasts")
> >> >> > attr(,"contrasts")$X3
> >> >> > [1] "contr.treatment"
> >> >> >
> >> >> > --------------
> >> >> >
> >> >> > Here, we now see the encoding for the interaction X1:X2:X3 is now
> the
> >> >> > interaction of X1 and X2 with a new encoding for X3 where each
> factor
> >> >> level
> >> >> > is represented by its own column. I would expect, given the two
> >> >> > column
> >> >> > dummy variable encoding for X3, that the X1:X2:X3 column would also
> >> >> > be
> >> >> two
> >> >> > columns regardless of what two-factor interactions we also
> specified,
> >> >> > but
> >> >> > in this case it switches to three. If other two factor interactions
> >> >> > are
> >> >> > missing in addition to X1:X2, this issue still occurs. This also
> >> >> > happens
> >> >> > regardless of the contrast specified in contrasts.arg for X3. I
> don't
> >> >> > see
> >> >> > any reasoning for this behavior given in the documentation, so I
> >> >> > suspect
> >> >> it
> >> >> > is a bug.
> >> >> >
> >> >> > Best regards,
> >> >> > Tyler Morgan-Wall
> >> >> >
> >> >> > [[alternative HTML version deleted]]
> >> >> >
> >> >> > ______________________________________________
> >> >> > R-devel at r-project.org mailing list
> >> >> > https://stat.ethz.ch/mailman/listinfo/r-devel
> >> >>
> >> >
> >> > [[alternative HTML version deleted]]
> >> >
> >> > ______________________________________________
> >> > R-devel at r-project.org mailing list
> >> > https://stat.ethz.ch/mailman/listinfo/r-devel
> >
> >
>
[[alternative HTML version deleted]]
More information about the R-devel
mailing list