Andrew Robinsons example merely illustrates that for numerical variables, the
term x1:x2
is the same as I(x1 * x2), and x1*x2 is different from x3*x4; try:
set.seed(8675309)
example <- data.frame(x1 = rnorm(100),
x2 = rnorm(100))
example$x3 <- example$x1 - 1
example$x4 <- example$x2 - 1
example$y <- with(example,
2 * x1 + 4 * x2 + x1 * x2 + rnorm(100) * 2)
with(example, plot(x1,x3))
with(example, plot(x2,x4))
## In the following code, the statistical information about the ## interaction
term is the same across the two scalings
summary(lm(y ~ x1 * x2, data = example))
summary(lm(y ~ x3 * x4, data = example))
## In the following code, the statistical information about the ## interaction
term is the not same across the two scalings
summary(lm(y ~ x1 + x1:x2, data = example))
summary(lm(y ~ x3 + x3:x4, data = example))
summary(lm(y ~ x3 + I(x3*x4), data = example))
summary(lm(y ~ x1 + I(x3*x4), data = example))
with(example, plot(x1*x2, x3*x4))
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