R help - May 2011 - In a formula, what is the interaction of the intercept and a factor?

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

	[[alternative HTML version deleted]]

With some guessing: does lm(formula = y ~ -1 + group + x:group, data = dat)
do what you want? I'm not sure now 1:group is treated, if at all.

Kenn

On Tue, May 31, 2011 at 11:35 PM, Kevin Wright <kw.stat at gmail.com>
wrote:
> 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
>
> ? ? ? ?[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

Try it for yourself:
> model.matrix(y ~ 1:group, data = dat)
    (Intercept)
1            1
...

Or to do

'interact "1+x" with "group"'
> model.matrix(y ~ (1+x):group, data = dat)
    (Intercept) x:groupA x:groupB
1            1        0        1
...

Note that you usually want to do '*' when you say 'interact
with':
> model.matrix(y ~ (1+x)*group, data = dat)
    (Intercept)  x groupB x:groupB
1            1  1      1        1
...


On Tue, 31 May 2011, Kevin Wright wrote:
> 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
>
> 	[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595