Different result from the same regression

Question

Why do I get different results from

summary(lm(mpg~horsepower + I(horsepower^2),data = Auto))$coef

and

summary(lm(mpg∼poly(horsepower,2) ,data=Auto))$coef

PS: I'm practicing the labs of ISLR

Answer 1

poly uses orthogonal polynomials by default. If you use poly(..., 2, raw = TRUE) it will use raw polynomials in which case the results are the same.

If you use the default orthogonal polynomials then although it parameterizes the model differently the model still gives the same predictions. That is, fitted(lm(...)) will be the same for both your models.

library(ISLR)

fo1 <- mpg ~ horsepower + I(horsepower ^ 2)
fo2 <- mpg ~ poly(horsepower, 2)
fo3 <- mpg ~ poly(horsepower, 2, raw = TRUE)

fm1 <- lm(fo1, Auto)
fm2 <- lm(fo2, Auto)
fm3 <- lm(fo3, Auto)

all.equal(coef(summary(fm1)), coef(summary(fm3)), check.attributes = FALSE)
## [1] TRUE

all.equal(fitted(fm1), fitted(fm2))
## [1] TRUE

all.equal(fitted(fm1), fitted(fm3))
## [1] TRUE

The discussion here may be helpful: What does the R function `poly` really do?