The columns of the model matrix are all orthogonal. So the problem
lies with poly(), not with lm().
> x = rep(1:5,3)
y = rnorm(15)
z <- model.matrix(lm(y ~ poly(x, 12)))
x = rep(1:5,3)
> y = rnorm(15)
> z <- model.matrix(lm(y ~ poly(x, 12)))
> round(crossprod(z),15)
(Intercept) poly(x, 12)1 poly(x, 12)2 poly(x, 12)3
poly(x, 12)4
(Intercept) 15 0 0
0 0
poly(x, 12)1 0 1 0
0 0
poly(x, 12)2 0 0 1
0 0
poly(x, 12)3 0 0 0
1 0
poly(x, 12)4 0 0 0
0 1
poly(x, 12)5 0 0 0
0 0
poly(x, 12)6 0 0 0
0 0
poly(x, 12)7 0 0 0
0 0
poly(x, 12)8 0 0 0
0 0
poly(x, 12)9 0 0 0
0 0
poly(x, 12)10 0 0 0
0 0
poly(x, 12)11 0 0 0
0 0
poly(x, 12)12 0 0 0
0 0
poly(x, 12)5 poly(x, 12)6 poly(x, 12)7 poly(x, 12)8
poly(x, 12)9
(Intercept) 0 0 0
0 0
poly(x, 12)1 0 0 0
0 0
poly(x, 12)2 0 0 0
0 0
poly(x, 12)3 0 0 0
0 0
poly(x, 12)4 0 0 0
0 0
poly(x, 12)5 1 0 0
0 0
poly(x, 12)6 0 1 0
0 0
poly(x, 12)7 0 0 1
0 0
poly(x, 12)8 0 0 0
1 0
poly(x, 12)9 0 0 0
0 1
poly(x, 12)10 0 0 0
0 0
poly(x, 12)11 0 0 0
0 0
poly(x, 12)12 0 0 0
0 0
poly(x, 12)10 poly(x, 12)11 poly(x, 12)12
(Intercept) 0 0 0
poly(x, 12)1 0 0 0
poly(x, 12)2 0 0 0
poly(x, 12)3 0 0 0
poly(x, 12)4 0 0 0
poly(x, 12)5 0 0 0
poly(x, 12)6 0 0 0
poly(x, 12)7 0 0 0
poly(x, 12)8 0 0 0
poly(x, 12)9 0 0 0
poly(x, 12)10 1 0 0
poly(x, 12)11 0 1 0
poly(x, 12)12 0 0 1
John Maindonald email: john.maindonald@anu.edu.au
phone : +61 2 (6125)3473 fax : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
On 24 Apr 2008, at 8:00 PM, r-devel-request@r-project.org
wrote:> From: russell-lenth@uiowa.edu
> Date: 24 April 2008 3:05:28 AM
> To: r-devel@stat.math.ethz.ch
> Cc: R-bugs@biostat.ku.dk
> Subject: [Rd] poly() can exceed degree k - 1 for k distinct points
> (PR#11251)
>
>
> The poly() function can create more variables than can be fitted when
> there are replicated values. In the example below, 'x' has only 5
> distinct values, but I can apparently fit a 12th-degree polynomial
> with
> no error messages or even nonzero coefficients:
>
> R> x = rep(1:5,3)
> R> y = rnorm(15)
> R> lm(y ~ poly(x, 12))
>
> Call:
> lm(formula = y ~ poly(x, 12))
>
> Coefficients:
> (Intercept) poly(x, 12)1 poly(x, 12)2 poly(x, 12)3
> -0.27442 0.35822 -0.26412 2.11780
> poly(x, 12)4 poly(x, 12)5 poly(x, 12)6 poly(x, 12)7
> 1.83117 -0.09260 -0.48572 1.94030
> poly(x, 12)8 poly(x, 12)9 poly(x, 12)10 poly(x, 12)11
> -0.88297 -1.04556 0.74289 -0.01422
> poly(x, 12)12
> -0.46548
>
>
>
>
> If I try the same with raw=TRUE, only a 4th-degree polynomial is
> obtained:
>
> R> lm(y ~ poly(x, 12, raw=TRUE))
>
> Call:
> lm(formula = y ~ poly(x, 12, raw = TRUE))
>
> Coefficients:
> (Intercept) poly(x, 12, raw = TRUE)1
> 9.7527 -22.0971
> poly(x, 12, raw = TRUE)2 poly(x, 12, raw = TRUE)3
> 15.3293 -4.1005
> poly(x, 12, raw = TRUE)4 poly(x, 12, raw = TRUE)5
> 0.3686 NA
> poly(x, 12, raw = TRUE)6 poly(x, 12, raw = TRUE)7
> NA NA
> poly(x, 12, raw = TRUE)8 poly(x, 12, raw = TRUE)9
> NA NA
> poly(x, 12, raw = TRUE)10 poly(x, 12, raw = TRUE)11
> NA NA
> poly(x, 12, raw = TRUE)12
> NA
>
> I believe that what is needed is a check on the 'rank' result after
> poly() calls the qr() function.
>
>
>
> System info:
> R version: 2.6.2
> Windows XP Pro; also get same results on Linux x_64 dual-core system.
>
>
>
> [I thought I submitted this via the website yesterday, but I can
> find no
> trace of it. I apologize if this is a duplicate, but I don't think
> it is.]
> --
> Russell V. Lenth, Professor
> Department of Statistics
> & Actuarial Science (319)335-0814 FAX (319)335-3017
> The University of Iowa russell-lenth@uiowa.edu
> Iowa City, IA 52242 USA http://www.stat.uiowa.edu/~rlenth/
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Actually, this may be a useful feature! It allows calculation of a basis for the orthogonal complement of the space spanned by model.matrix(lm(y ~ poly(x,12)). However, the default ought surely to be to disallow df > k-1 in poly(x,df), where k = length(unique(x)). John Maindonald email: john.maindonald@anu.edu.au phone : +61 2 (6125)3473 fax : +61 2(6125)5549 Centre for Mathematics & Its Applications, Room 1194, John Dedman Mathematical Sciences Building (Building 27) Australian National University, Canberra ACT 0200. On 24 Apr 2008, at 8:00 PM, r-devel-request@r-project.org wrote:> From: russell-lenth@uiowa.edu > Date: 24 April 2008 3:05:28 AM > To: r-devel@stat.math.ethz.ch > Cc: R-bugs@biostat.ku.dk > Subject: [Rd] poly() can exceed degree k - 1 for k distinct points > (PR#11251) > > > The poly() function can create more variables than can be fitted when > there are replicated values. In the example below, 'x' has only 5 > distinct values, but I can apparently fit a 12th-degree polynomial > with > no error messages or even nonzero coefficients: > > R> x = rep(1:5,3) > R> y = rnorm(15) > R> lm(y ~ poly(x, 12)) > > Call: > lm(formula = y ~ poly(x, 12)) > > Coefficients: > (Intercept) poly(x, 12)1 poly(x, 12)2 poly(x, 12)3 > -0.27442 0.35822 -0.26412 2.11780 > poly(x, 12)4 poly(x, 12)5 poly(x, 12)6 poly(x, 12)7 > 1.83117 -0.09260 -0.48572 1.94030 > poly(x, 12)8 poly(x, 12)9 poly(x, 12)10 poly(x, 12)11 > -0.88297 -1.04556 0.74289 -0.01422 > poly(x, 12)12 > -0.46548 ><snip> <snip>> [I thought I submitted this via the website yesterday, but I can > find no > trace of it. I apologize if this is a duplicate, but I don't think > it is.] > -- > Russell V. Lenth, Professor > Department of Statistics > & Actuarial Science (319)335-0814 FAX (319)335-3017 > The University of Iowa russell-lenth@uiowa.edu > Iowa City, IA 52242 USA http://www.stat.uiowa.edu/~rlenth/[[alternative HTML version deleted]]