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

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().

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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.