Suppose I have the following code to fit a hyperbolic parabola:
# attach(mtcars)
hp_fit <- lm(mpg ~ poly(wt, disp, degree = 2), data = mtcars)
Where wt is the x variable, disp is the y variable, and mpg is the z variable. (summary(hp_fit))$coefficients outputs the following:
>(summary(hp_fit))$coefficients
Estimate Std. Error t value Pr(>|t|)
(Intercept) 22.866173 3.389734 6.7457122 3.700396e-07
poly(wt, disp, degree = 2)1.0 -13.620499 8.033068 -1.6955539 1.019151e-01
poly(wt, disp, degree = 2)2.0 15.331818 17.210260 0.8908534 3.811778e-01
poly(wt, disp, degree = 2)0.1 -9.865903 5.870741 -1.6805208 1.048332e-01
poly(wt, disp, degree = 2)1.1 -100.022013 121.159039 -0.8255431 4.165742e-01
poly(wt, disp, degree = 2)0.2 14.719928 9.874970 1.4906301 1.480918e-01
I do not understand how to interpret the varying numbers to the right of poly() under the (Intercept) column. What is the significance of these numbers and how would I construct an equation for the hyperbolic paraboloid fit from this summary?
When you compare
with(mtcars, poly(wt, disp, degree=2))
with(mtcars, poly(wt, degree=2))
with(mtcars, poly(disp, degree=2))
the 1.0 2.0 refer to the first and second degree of wt, and the 0.1 0.2 refer to the first and second degree of disp. The 1.1 is an interaction term. You may check this by comparing:
summary(lm(mpg ~ poly(wt, disp, degree=2, raw=T), data=mtcars))$coe
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 4.692786e+01 7.008139762 6.6961935 4.188891e-07
# poly(wt, disp, degree=2, raw=T)1.0 -1.062827e+01 8.311169003 -1.2787937 2.122666e-01
# poly(wt, disp, degree=2, raw=T)2.0 2.079131e+00 2.333864211 0.8908534 3.811778e-01
# poly(wt, disp, degree=2, raw=T)0.1 -3.172401e-02 0.060528241 -0.5241191 6.046355e-01
# poly(wt, disp, degree=2, raw=T)1.1 -2.660633e-02 0.032228884 -0.8255431 4.165742e-01
# poly(wt, disp, degree=2, raw=T)0.2 2.019044e-04 0.000135449 1.4906301 1.480918e-01
summary(lm(mpg ~ wt*disp + I(wt^2) + I(disp^2) , data=mtcars))$coe[c(1:2, 4:3, 6:5), ]
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 4.692786e+01 7.008139762 6.6961935 4.188891e-07
# wt -1.062827e+01 8.311169003 -1.2787937 2.122666e-01
# I(wt^2) 2.079131e+00 2.333864211 0.8908534 3.811778e-01
# disp -3.172401e-02 0.060528241 -0.5241191 6.046355e-01
# wt:disp -2.660633e-02 0.032228884 -0.8255431 4.165742e-01
# I(disp^2) 2.019044e-04 0.000135449 1.4906301 1.480918e-01
This yields the same values. Note that I used raw=TRUE for comparison purposes.