Arie ten Cate
2017-Nov-04 10:50 UTC
[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
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]]
>> >> >> >
>> >> >> > ______________________________________________
Tyler
2017-Nov-04 16:33 UTC
[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
Hi Arie, I understand what you're saying. The following excerpt out of the book shows that F_j does not refer exclusively to categorical factors: "...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." Since F_j refers to both categorical and numeric variables, the behavior of model.matrix is not consistent with the heuristic. Best regards, Tyler On Sat, Nov 4, 2017 at 6:50 AM, Arie ten Cate <arietencate at gmail.com> wrote:> 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-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]] > >> >> >> > > >> >> >> > ______________________________________________ >[[alternative HTML version deleted]]
Arie ten Cate
2017-Nov-06 11:45 UTC
[Rd] Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
Hello Tyler,
You write that you understand what I am saying. However, I am now at
loss about what exactly is the problem with the behavior of R. Here
is a script which reproduces your experiments with three variables
(excluding the full model):
m=expand.grid(X1=c(1,-1),X2=c(1,-1),X3=c("A","B","C"))
model.matrix(~(X1+X2+X3)^3-X1:X3,data=m)
model.matrix(~(X1+X2+X3)^3-X2:X3,data=m)
model.matrix(~(X1+X2+X3)^3-X1:X2,data=m)
Below are the three results, similar to your first mail. (The first
two are basically the same, of course.) Please pick one result which
you think is not consistent with the heuristic and please give what
you think is the correct result:
model.matrix(~(X1+X2+X3)^3-X1:X3)
(Intercept)
X1 X2 X3B X3C
X1:X2 X2:X3B X2:X3C
X1:X2:X3B X1:X2:X3C
model.matrix(~(X1+X2+X3)^3-X2:X3)
(Intercept)
X1 X2 X3B X3C
X1:X2 X1:X3B X1:X3C
X1:X2:X3B X1:X2:X3C
model.matrix(~(X1+X2+X3)^3-X1:X2)
(Intercept)
X1 X2 X3B X3C
X1:X3B X1:X3C X2:X3B X2:X3C
X1:X2:X3A X1:X2:X3B X1:X2:X3C
(I take it that the combination of X3A and X3B and X3C implies dummy
encoding, and the combination of only X3B and X3C implies contrasts
encoding, with respect to X3A.)
Thanks in advance,
Arie
On Sat, Nov 4, 2017 at 5:33 PM, Tyler <tylermw at gmail.com>
wrote:> Hi Arie,
>
> I understand what you're saying. The following excerpt out of the book
shows
> that F_j does not refer exclusively to categorical factors: "...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."
Since
> F_j refers to both categorical and numeric variables, the behavior of
> model.matrix is not consistent with the heuristic.
>
> Best regards,
> Tyler
>
> On Sat, Nov 4, 2017 at 6:50 AM, Arie ten Cate <arietencate at
gmail.com> wrote:
>>
>> 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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