Puzzling coefficients for linear fitting to polynom

Hi,
I can not comprehend the linear fitting results of polynoms.
For example, given the following data (representing y = x^2):

x <- 1:3
y <- c(1, 4, 9)

performing a linear fit

f <- lm(y ~ poly(x, 2))

gives weird coefficients:

coefficients(f)

(Intercept) poly(x, 2)1 poly(x, 2)2 
  4.6666667   5.6568542   0.8164966 

However the fitted() result makes sense:

fitted(f)

1 2 3 
1 4 9 

This is very confusing. How should one understand the result of
coefficients()?

Thanks for any tips,
Firas.

Have a look at the values returned by poly(x,2).
The coefficients you are getting are the results of fitting

  y = a + b1*poly(x,2)[,1] + b2*poly(x,2)[,2]

where

  poly(x, 2)[,1]
# [1] -7.071068e-01 -9.073264e-17  7.071068e-01
  poly(x, 2)[,2]
# [1]  0.4082483 -0.8164966  0.4082483

which is probably not what you may have thought you were doing!

It is certainly not the same as fitting

  y = a + b1*x + b2*(x^2)

though of course the fitted values will be the same.

Best wishes,
Ted.

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Date: 07-Mar-08                                       Time: 08:40:46
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