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 <tyle...@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 <arietenc...@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 <tyle...@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 <arietenc...@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 <tyle...@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@r-project.org mailing list >> >> > https://stat.ethz.ch/mailman/listinfo/r-devel >> >> >> > >> > [[alternative HTML version deleted]] >> > >> > ______________________________________________ >> > R-devel@r-project.org mailing list >> > https://stat.ethz.ch/mailman/listinfo/r-devel > > ______________________________________________ R-devel@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-devel