[R] Intercept Coefficient in a Model with Orthogonal Polynomials

[Chuck Cleland]

  This very likely falls in the category of an unexpected result due to
user ignorance.  I generated the following data:

time <- 0:10

set.seed(4302007)

y <- 268 + -9*time + .4*(time^2) + rnorm(11, 0, .1)

  I then fit models using both orthogonal and raw polynomials:

fit1 <- lm(y ~ poly(time, 2))
fit2 <- lm(y ~ poly(time, degree=2, raw=TRUE))

> predict(fit1, data.frame(time = 0:10))
       1        2        3        4        5        6        7
268.1339 259.4912 251.6542 244.6230 238.3976 232.9780 228.3642
       8        9       10       11
224.5562 221.5540 219.3575 217.9669

> predict(fit2, data.frame(time = 0:10))
       1        2        3        4        5        6        7
268.1339 259.4912 251.6542 244.6230 238.3976 232.9780 228.3642
       8        9       10       11
224.5562 221.5540 219.3575 217.9669

> coef(fit1)
   (Intercept) poly(time, 2)1 poly(time, 2)2
     237.00698      -52.61565       11.80144

> coef(fit2)
                        (Intercept)
                        268.1339235
poly(time, degree = 2, raw = TRUE)1
                         -9.0456491
poly(time, degree = 2, raw = TRUE)2
                          0.4028944

  I expected the intercept coefficient in the model with orthogonal
polynomials to correspond to the predicted value of y when time=5.
Instead, it seems to correspond to y at time between time=4 and time=5.
 Is there a way of figuring out what time the intercept corresponds to
on the original time scale (0:10 here)?  Any comments and pointers to
references would be greatly appreciated.

thanks,

Chuck Cleland

-- 
Chuck Cleland, Ph.D.
NDRI, Inc.
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