Model matrix with all pairwise interactions between columns
Let's say that I have a numeric data matrix with columns w, x, y, z and I also want to add in the columns that are equivalent to w*x, w*y, w*z, x*y, x*z, y*z since I want my covariate matrix to include all pairwise interactions.
Is there a clean and effective way to do this?
Answer
If you mean in a model formula, then the ^ operator does this.
## dummy data
set.seed(1)
dat <- data.frame(Y = rnorm(10), x = rnorm(10), y = rnorm(10), z = rnorm(10))
The formula is
form <- Y ~ (x + y + z)^2
which gives (using model.matrix() - which is used internally by the standard model fitting functions)
model.matrix(form, data = dat)
R> form <- Y ~ (x + y + z)^2
R> form
Y ~ (x + y + z)^2
R> model.matrix(form, data = dat)
(Intercept) x y z x:y x:z y:z
1 1 1.51178 0.91898 1.35868 1.389293 2.054026 1.24860
2 1 0.38984 0.78214 -0.10279 0.304911 -0.040071 -0.08039
3 1 -0.62124 0.07456 0.38767 -0.046323 -0.240837 0.02891
4 1 -2.21470 -1.98935 -0.05381 4.405817 0.119162 0.10704
5 1 1.12493 0.61983 -1.37706 0.697261 -1.549097 -0.85354
6 1 -0.04493 -0.05613 -0.41499 0.002522 0.018647 0.02329
7 1 -0.01619 -0.15580 -0.39429 0.002522 0.006384 0.06143
8 1 0.94384 -1.47075 -0.05931 -1.388149 -0.055982 0.08724
9 1 0.82122 -0.47815 1.10003 -0.392667 0.903364 -0.52598
10 1 0.59390 0.41794 0.76318 0.248216 0.453251 0.31896
attr(,"assign")
[1] 0 1 2 3 4 5 6
If you don't know how many variables you have, or it is tedious to write out all of them, use the . notation too
R> form <- Y ~ .^2
R> model.matrix(form, data = dat)
(Intercept) x y z x:y x:z y:z
1 1 1.51178 0.91898 1.35868 1.389293 2.054026 1.24860
2 1 0.38984 0.78214 -0.10279 0.304911 -0.040071 -0.08039
3 1 -0.62124 0.07456 0.38767 -0.046323 -0.240837 0.02891
4 1 -2.21470 -1.98935 -0.05381 4.405817 0.119162 0.10704
5 1 1.12493 0.61983 -1.37706 0.697261 -1.549097 -0.85354
6 1 -0.04493 -0.05613 -0.41499 0.002522 0.018647 0.02329
7 1 -0.01619 -0.15580 -0.39429 0.002522 0.006384 0.06143
8 1 0.94384 -1.47075 -0.05931 -1.388149 -0.055982 0.08724
9 1 0.82122 -0.47815 1.10003 -0.392667 0.903364 -0.52598
10 1 0.59390 0.41794 0.76318 0.248216 0.453251 0.31896
attr(,"assign")
[1] 0 1 2 3 4 5 6
The "power" in the ^ operator, here 2, controls the order of interactions. With ^2 we get second order interactions of all pairs of variables considered by the ^ operator. If you want up to 3rd-order interactions, then use ^3.
R> form <- Y ~ .^3
R> head(model.matrix(form, data = dat))
(Intercept) x y z x:y x:z y:z x:y:z
1 1 1.51178 0.91898 1.35868 1.389293 2.05403 1.24860 1.887604
2 1 0.38984 0.78214 -0.10279 0.304911 -0.04007 -0.08039 -0.031341
3 1 -0.62124 0.07456 0.38767 -0.046323 -0.24084 0.02891 -0.017958
4 1 -2.21470 -1.98935 -0.05381 4.405817 0.11916 0.10704 -0.237055
5 1 1.12493 0.61983 -1.37706 0.697261 -1.54910 -0.85354 -0.960170
6 1 -0.04493 -0.05613 -0.41499 0.002522 0.01865 0.02329 -0.001047