R中的x ^ 2和I（x ^ 2）有什么区别？

Question

What is the difference between these two models in R?

model1 <- glm(y~ x + x^2, family=binomial(link=logit), weights=numbers))
model2 <- glm(y~ x + I(x^2),family=binomial(link=logit), weights=numbers))

Also what is the equvalent of I(x^2) in SAS?

Answer 1

The I() function means 'as is' whereas the ^n (to the power of n) operator means 'include these variables and all interactions up to n way'

This means:

I(X^2) is literally regressing Y against X squared and

X^2 means include X and the 2 way interaction of X but since it is only one variable there is no interaction so it returns only itself ie X. Note that in your formula you say X + X^2 which translates to X + X which in the formula syntax is only taken into account once. Ie one of the two Xs will be removed.

Demonstration:

Y <- runif(100)
X2 <- runif(100)
df <- data.frame(Y,X1,X2)

b <- lm( Y ~ X2 + X2^2 + X2,data=df)

> b

Call:
lm(formula = Y ~ X2 + X2^2 + X2, data = df)

Coefficients:
(Intercept)           X2  
    0.48470      0.05098  


a <- lm( Y ~ X2 + I(X2^2),data=df)

> a

Call:
lm(formula = Y ~ X2 + I(X2^2), data = df)

Coefficients:
(Intercept)           X2      I(X2^2)  
    0.47545      0.11339     -0.06682  

Hope it helps!