[R] Confidence intervals and polynomial fits

[Ben Haller]

  Hi all!  I'm getting a model fit from glm() (a binary logistic regression fit, but I don't think that's important) for a formula that contains powers of the explanatory variable up to fourth.  So the fit looks something like this (typing into mail; the actual fit code is complicated because it involves step-down and so forth):

x_sq <- x * x
x_cb <- x * x * x
x_qt <- x * x * x * x
model <- glm(outcome ~ x + x_sq + x_cb + x_qt, binomial)

  This works great, and I get a nice fit.  My question regards the confidence intervals on that fit.  I am calling:

cis <- confint.default(model, level=0.95)

to get the confidence intervals for each coefficient.  This confint.default() call gives me a table like this (where "dispersal" is the real name of what I'm calling "x" above):

                             2.5 %      97.5 %
(Intercept)              8.7214297   8.9842533
dispersal              -37.5621412 -35.6629981
dispersal_sq            66.8077669  74.4490123
dispersal_cb           -74.5963861 -62.8347057
dispersal_qt            22.6590159  28.6506186

  A side note: calling confint() instead of confint.default() is much, much slower -- my model has many other terms I'm not discussing here, and is fit to 300,000 data points -- and a check indicates that the intervals returned for my model by confint() are not virtually identical, so this is not the source of my problem:

                             2.5 %      97.5 %
(Intercept)              8.7216693   8.9844958
dispersal              -37.5643922 -35.6652276
dispersal_sq            66.8121519  74.4534753
dispersal_cb           -74.5995820 -62.8377766
dispersal_qt            22.6592724  28.6509494

  These tables show the problem: the 95% confidence limits for the quartic term are every bit as wide as the limits for the other terms.  Since the quartic term coefficient gets multiplied by the fourth power of x, this means that the width of the confidence band starts out nice and narrow (when x is close to zero, where the width of the confidence band is pretty much just determined by the confidence limits on the intercept) but balloons out to be tremendously wide for larger values of x.  The width of it looks quite unreasonable to me, given that my fit is constrained by 300,000 data points.

  Intuitively, I would have expected the confidence limits for the quartic term to be much narrower than for, say, the linear term, because the effect of variation in the quartic term is so much larger.  But the way confidence intervals are calculated in R, at least, does not seem to follow my instincts here.  In other words, predicted values using the 2.5% value of the linear coefficient and the 97.5% value of the linear coefficient are really not very different, but predicted values using the 2.5% value of the quadratic coefficient and the 97.5% value of the quadratic coefficient are enormously, wildly divergent -- far beyond the range of variability in the original data, in fact.  I feel quite confident, in a seat-of-the-pants sense, that the actual 95% limits of that quartic coefficient are much, much narrower than what R is giving me.

  Looking at it another way: if I just exclude the quartic term from the glm() fit, I get a dramatically narrower confidence band, even though the fit to the data is not nearly as good (I'm keeping the fourth power for good reasons).  This again seems to me to indicate that the confidence band with the quartic term included is artificially, and incorrectly, wide.  If a fit with reasonably narrow confidence limits can be produced for the model with terms only up to cubic (or, taking this reductio even further, for a quadratic or a linear model, also), why does adding the quadratic term, which improves the quality of the fit to the data, cause the confidence limits to nevertheless balloon outwards?  That seems fundamentally illogical to me, but maybe my instincts are bad.

  So what's going on here?  Is this just a limitation of the way confint() computes confidence intervals?  Is there a better function to use, that is compatible with binomial glm() fits?  Or am I misinterpreting the meaning of the confidence limits in some way?  Or what?

  Thanks for any advice.  I've had trouble finding examples of polynomial fits like this; people sometimes fit the square of the explanatory variable, of course, but going up to fourth powers seems to be quite uncommon, so I've had trouble finding out how others have dealt with these sorts of issues that crop up only with higher powers.

Ben Haller
McGill University

http://biology.mcgill.ca/grad/ben/