In a formula, what is the interaction of the intercept and a factor?

model.matrix(y ~ 1:group, data = dat)

model.matrix(y ~ (1+x):group, data = dat)

model.matrix(y ~ (1+x)*group, data = dat)

For a pedagogical purpose, I was trying to show how the formula for a simple
regression line (~1+x) could be crossed with a factor (~1:group + x:group)
to fit separate regressions by group.  For example:

set.seed(201108)
dat <- data.frame(x=1:15, y=1:15+rnorm(15),
           group = sample(c('A','B'), size=15,
                   replace=TRUE))

m1 <- lm(y~ 1 + x, data=dat)
m2 <- lm(y ~ group + x:group, data=dat)
m3 <- lm(y ~ 1:group + x:group, data=dat)
m4 <- lm(y ~ 1 + x:group, data=dat)

The simple regression is model m1.

The usual way to write the by-group regression is model m2.

In model m3 was trying to be explicitly clear and interact "1+x" with
"group".

Looking only at the coefficients, it appears that model m3 is simplified to
model m4.

R> coef(m3)
(Intercept)    groupA:x    groupB:x
 0.3775140   0.9213835   0.9879690

R> coef(m4)
(Intercept)    x:groupA    x:groupB
 0.3775140   0.9213835   0.9879690

I wonder if anyone can shed some light on what R is doing with the "1:group"
term.

Kevin

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