[R-lang] poly() and polynomials

[Sharon Goldwater]

I'm trying to build a mixed logit model using lmer, and I have some
questions about poly() and the use of quadratic terms in general.  My
understanding is that, by default, poly() creates orthogonal
polynomials, so the coefficients are not easily interpretable.  On the
other hand, using m ~ poly(x, raw=T) should be equivalent to m ~ x +
xsq, where xsq is precomputed as x^2.  I have verified the second
statement by building one model using raw poly() and another one by
precomputing all quadratic terms, and the coefficients are indeed
virtually identical.  Now I have a couple of questions:

1.  When I try to build a model using raw=F, I get the following error:

> m22 <- lmer(is_err ~ poly(pmean_sex,2)  +(1|speaker)+(1|ref), x=T,y=T,family="binomial")
Error in x * raw^2 : non-conformable arrays

but
>  m22 <- lmer(is_err ~ poly(pmean_sex,2,raw=T) +(1|speaker)+(1|ref), x=T,y=T,family="binomial")
works fine.

Normally my model has far more predictors, but this is the only one
that seems to cause the problem.  This doesn't seem to be a problem
with lmer specifically, since it's reproducible using lrm and lm as
well.  Does anyone know what this means?   I can't seem to find an
answer in R help archives that makes any sense.  More generally,
should I care about trying to fix it?  That is, is there a good reason
to prefer orthogonal polynomials rather than raw ones, aside from
reducing collinearity slightly? Since I would like to be able to
interpret the coefficients (i.e., determine the relative importance of
each variable as a predictor of error rates), I would tend towards
using raw polynomials anyway.

2.  I always hear people say that if you have both a linear and
quadratic term for a particular predictor, you shouldn't get rid of
the linear term (even if it shows up as not significant) while
retaining the quadratic term.  In fact, if you were using poly(), it
wouldn't even be possible to do that.  But suppose you instead
precompute all the quadratic terms and treat them as separate
variables.  It seems to me that retaining a quadratic term for
variable x while deleting the linear term is essentially just
performing a transformation on the variable, no different from taking
the log of x as the predictor, which we do all the time for linguistic
variables.  So is there really any reason not to do it?

--AuH2O

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