lm(y ~ x1) vs. lm(y ~ x0 + x1 - 1) with x0 <- rep(1, length(y))

On Jan 21, 2011, at 9:03 PM, jochen laubrock wrote:

Dear list,

the following came up in an introductory class. Please help me understand
the -1 (or 0+) syntax in formulae: Why do the enumerator dfs, F-statisics
etc. differ between the models lm(y ~ x1) and lm(y ~ x0 + x1 - 1), if x0 is
a vector containing simply ones?

You are testing something different. In the first case you are testing the
difference between the baseline and the second level of x1 (so there is only
one d.f.), while in the second case you are testing for both of the
coefficients being zero (so the numerator has 2 d.f.). It would be easier to
see if you did print() on the fit object. The first model would give you an
estimate for an "Intercept", which is really an estimate for the first level
of x1. ?Having been taught to think of anova as just a special case of
regression is helpful here. Look at the model first ?and only then look at
the anova table.

Example:

N ?<- 40
x0 <- rep(1,N)
x1 <- 1:N
vare <- N/8
set.seed(4)
e <- rnorm(N, 0, vare^2)

X <- cbind(x0, x1)
beta <- c(.4, 1)
y <- X %*% beta + e

summary(lm(y ~ x1))
# [...]
# Residual standard error: 20.92 on 38 degrees of freedom
# Multiple R-squared: 0.1151, ? Adjusted R-squared: 0.09182
# F-statistic: 4.943 on 1 and 38 DF, ?p-value: 0.03222

summary(lm(y ~ x0 + x1 - 1)) ? ? ? ?# or summary(lm(y ~ 0 + x0 + x1))
# [...]
# Residual standard error: 20.92 on 38 degrees of freedom
# Multiple R-squared: 0.6888, ? Adjusted R-squared: 0.6724
# F-statistic: 42.05 on 2 and 38 DF, ?p-value: 2.338e-10


Thanks in advance,
Jochen


----
Jochen Laubrock, Dept. of Psychology, University of Potsdam,
Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany
phone: +49-331-977-2346, fax: +49-331-977-2793

David Winsemius, MD
West Hartford, CT