[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
Arie ten Cate
arietencate at gmail.com
Sat Nov 4 11:50:17 CET 2017
Hello Tyler,
I rephrase my previous mail, as follows:
In your example, T_i = X1:X2:X3. Let F_j = X3. (The numerical
variables X1 and X2 are not encoded at all.) Then T_{i(j)} = X1:X2,
which in the example is dropped from the model. Hence the X3 in T_i
must be encoded by dummy variables, as indeed it is.
Arie
On Thu, Nov 2, 2017 at 4:11 PM, Tyler <tylermw at gmail.com> wrote:
> 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]]
>> >> >> >
>> >> >> > ______________________________________________
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