lm() in R and linear regression equation

Question

I have two variables for which I used the following lm() formula:

lm(y ~ x1*x2)

I used this instead of

lm(y ~ x1 + x2)

because I wanted to see the main effect of each IV (x1, x2), and their interaction.

Is the formula I used the same as y = b0 + b1 x1 + b2 x2?

Or, y = b0 + (b1 x1) (b2*x2)?

Answer 1

Using the builtin anscombe data set these two have the same model matrix so they are fitting the same model. Examine the model matrix to understand exactly what model is being used.

fm1 <- lm(y1 ~ x1 + x2 + x1:x2, anscombe)
fm2 <- lm(y1 ~ x1 * x2, anscombe)

m1 <- model.matrix(fm1)
m2 <- model.matrix(fm2)
identical(m1, m2)
## [1] TRUE

# since m1 and m2 are identical we can just display one of them
m1
##    (Intercept) x1 x2 x1:x2
## 1            1 10 10   100
## 2            1  8  8    64
## 3            1 13 13   169
## 4            1  9  9    81
## 5            1 11 11   121
## 6            1 14 14   196
## 7            1  6  6    36
## 8            1  4  4    16
## 9            1 12 12   144
## 10           1  7  7    49
## 11           1  5  5    25
## attr(,"assign")
## [1] 0 1 2 3