[R] use of poly()

[Bill.Venables at csiro.au]

You ask

	When using continuous data in both Y and X, does the 
	difference between "raw" and "orthagonal" polynomials 
	have any practical meaning?

Yes, indeed it does, even if X is not 'continuous'.  There are (at
least) two practical differences:

	1. With orthogonal polynomials you are using an orthogonal
basis, so the estimates of the regression coefficients are statistically
independent.  This makes it much easier in model building to get an idea
of the degree of polynomial warranted by the data.  You can usually do
it from a single model fit.

	2. With an orthogonal polynomial basis your model matrix has, in
principle, an optimal condition number and the numerical properties of
the least squares fitting algorithm can be much better.  If you really
want the raw coefficients and their standard errors, &c, you unravel
this a bit, but why would you want to? 

If all you are interested in is the fitted curve, though, (and this is
indeed the key thing, not the coefficients), then what kind of basis you
use is pretty irrelevant. 

Regards,
W.

Bill Venables
CSIRO Laboratories
PO Box 120, Cleveland, 4163
AUSTRALIA
Office Phone (email preferred): +61 7 3826 7251
Fax (if absolutely necessary):  +61 7 3826 7304
Mobile:                         +61 4 8819 4402
Home Phone:                     +61 7 3286 7700
mailto:Bill.Venables at csiro.au
http://www.cmis.csiro.au/bill.venables/ 

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Dylan Beaudette
Sent: Thursday, 14 February 2008 6:42 AM
To: r-help at r-project.org
Subject: [R] use of poly()

Hi,

I am curious about how to interpret the results of a polynomial
regression-- 
using poly(raw=TRUE) vs. poly(raw=FALSE).

set.seed(123456)
x <- rnorm(100)
y <- jitter(1*x + 2*x^2 + 3*x^3 , 250)
plot(y ~ x)

l.poly <- lm(y ~ poly(x, 3))
l.poly.raw <- lm(y ~ poly(x, 3, raw=TRUE))

s <- seq(-3, 3, by=0.1)

lines(s, predict(l.poly, data.frame(x=s)), col=1)
lines(s, predict(l.poly.raw, data.frame(x=s)), col=2)

The results are the same, but the regression coeficients are different:

as.vector(coef(l.poly))
1.806618 88.078858 16.194423 58.051642

as.vector(coef(l.poly.raw))
-0.1025114  1.5265248  2.0617970  2.7393995


When using continuous data in both Y and X, does the difference between
"raw" 
and "orthagonal" polynomials have any practical meaning?

Thanks,

Dylan



-- 
Dylan Beaudette
Soil Resource Laboratory
http://casoilresource.lawr.ucdavis.edu/
University of California at Davis
530.754.7341

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