[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
Tyler
tylermw at gmail.com
Tue Oct 31 16:01:31 CET 2017
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-o
> mit-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
>
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