What does predict.glm(, type="terms") actually do?

Question

I am confused with the way predict.glm function in R works. According to the help,

The "terms" option returns a matrix giving the fitted values of each term in the model formula on the linear predictor scale.

Thus, if my model has form f(y) = X*beta, then command

predict(model, X, type='terms')

is expected to produce the same matrix X, multiplied by beta element-wise. For example, if I train the following model

test.data = data.frame(y = c(0,0,0,1,1,1,1,1,1), x=c(1,2,3,1,2,2,3,3,3))
model = glm(y~(x==1)+(x==2), family = 'binomial', data = test.data)

the resulting coefficients are

beta <- model$coef

Design matrix is

X <- model.matrix(y~(x==1)+(x==2), data = test.data)

  (Intercept) x == 1TRUE x == 2TRUE
1           1          1          0
2           1          0          1
3           1          0          0
4           1          1          0
5           1          0          1
6           1          0          1
7           1          0          0
8           1          0          0
9           1          0          0

Then multiplied by coefficients it should look like

pred1 <- t(beta * t(X))

  (Intercept) x == 1TRUE x == 2TRUE
1    1.098612  -1.098612  0.0000000
2    1.098612   0.000000 -0.4054651
3    1.098612   0.000000  0.0000000
4    1.098612  -1.098612  0.0000000
5    1.098612   0.000000 -0.4054651
6    1.098612   0.000000 -0.4054651
7    1.098612   0.000000  0.0000000
8    1.098612   0.000000  0.0000000
9    1.098612   0.000000  0.0000000

However, actual matrix produced by predict.glm seems to be unrelated to this. The following code

pred2 <- predict(model, test.data, type = 'terms')

      x == 1     x == 2
1 -0.8544762  0.1351550
2  0.2441361 -0.2703101
3  0.2441361  0.1351550
4 -0.8544762  0.1351550
5  0.2441361 -0.2703101
6  0.2441361 -0.2703101
7  0.2441361  0.1351550
8  0.2441361  0.1351550
9  0.2441361  0.1351550
attr(,"constant")
[1] 0.7193212

How does one interpret such results?

Answer 1

I have already edited your question, to include "correct" way of getting (raw) model matrix, model coefficients, and your intended term-wise prediction. So your other question on how to get these are already solved. In the following, I shall help you understand predict.glm().

---

predict.glm() (actually, predict.lm()) has applied centring constraints for each model term when doing term-wise prediction.

Initially, you have a model matrix

X <- model.matrix(y~(x==1)+(x==2), data = test.data)

but it is centred, by dropping column means:

avx <- colMeans(X)
X1 <- sweep(X, 2L, avx)

> avx
(Intercept)  x == 1TRUE  x == 2TRUE 
  1.0000000   0.2222222   0.3333333 

> X1
  (Intercept) x == 1TRUE x == 2TRUE
1           0  0.7777778 -0.3333333
2           0 -0.2222222  0.6666667
3           0 -0.2222222 -0.3333333
4           0  0.7777778 -0.3333333
5           0 -0.2222222  0.6666667
6           0 -0.2222222  0.6666667
7           0 -0.2222222 -0.3333333
8           0 -0.2222222 -0.3333333
9           0 -0.2222222 -0.3333333

Then term-wise computation is done using this centred model matrix:

t(beta*t(X1))

  (Intercept) x == 1TRUE x == 2TRUE
1           0 -0.8544762  0.1351550
2           0  0.2441361 -0.2703101
3           0  0.2441361  0.1351550
4           0 -0.8544762  0.1351550
5           0  0.2441361 -0.2703101
6           0  0.2441361 -0.2703101
7           0  0.2441361  0.1351550
8           0  0.2441361  0.1351550
9           0  0.2441361  0.1351550

After centring, different terms are vertically shifted to have zero mean. As a result, intercept will be come 0. No worry, a new intercept is computed, by aggregating shifts of all model terms:

intercept <- as.numeric(crossprod(avx, beta))
# [1] 0.7193212

Now you should have seen what predict.glm(, type = "terms") gives you.