Bug in model.matrix.default for higher-order interaction encoding when specific model terms are missing
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
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