Fitting a model with poly(x,2) in R gives me different result

Question

this is a rather simple question but I noticed that the linear regression on 1 variable fitted with poly() gives different results if poly is not used. If I were to use poly() how can I convert back to the original variables?

For example:

First, define data

x = seq(50, 275, 25)
y = c(335, 326, 316, 313, 311, 314, 318, 328, 337, 345)

Now fit $y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \varepsilon_i$

Fit it on $x$ and $x^2$

m1 = lm(y~x+I(x^2))
s1 = summary(m1)

Now fit using the poly() function

m2 = lm(y~poly(x,2))
s2 = summary(m2)

s1 is

Call:
lm(formula = y ~ x + I(x^2))

Residuals:
     Min       1Q   Median       3Q      Max 
-2.75455 -1.20341 -0.00076  1.13182  2.78333 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.622e+02  3.268e+00  110.83 1.28e-12 ***
x           -6.674e-01  4.501e-02  -14.83 1.52e-06 ***
I(x^2)       2.236e-03  1.359e-04   16.45 7.48e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.952 on 7 degrees of freedom
Multiple R-squared:  0.9785,    Adjusted R-squared:  0.9723 
F-statistic: 159.2 on 2 and 7 DF,  p-value: 1.461e-06

s2 is

Call:
lm(formula = y ~ poly(x, 2))

Residuals:
     Min       1Q   Median       3Q      Max 
-2.75455 -1.20341 -0.00076  1.13182  2.78333 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 324.3000     0.6174 525.262  < 2e-16 ***
poly(x, 2)1  13.4868     1.9524   6.908  0.00023 ***
poly(x, 2)2  32.1173     1.9524  16.450 7.48e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.952 on 7 degrees of freedom
Multiple R-squared:  0.9785,    Adjusted R-squared:  0.9723 
F-statistic: 159.2 on 2 and 7 DF,  p-value: 1.461e-06

How can I convert s2 to s1?

Answer 1

The apparent discrepancy is due to not using the raw polynomials.

There is a difference between raw and orthogonal polynomials; check this CV post for a quick reference: Raw or orthogonal polynomial regression?.

Use m2 = lm(y ~ poly(x, 2, raw = TRUE)).

By default, raw = FALSE; for resorting to raw polynomials, explicitly make it TRUE; the result will match with that of what was done manually.