On Sat, 1 Jul 2000, Paul Velleman wrote:
> I'm not real comfortable with a polynomial model that takes nearly
> half the available degrees of freedom and offers no theoretical
> motivation.
"Comfortable" is not a word that much occurs to mind in the context of
polynomial models. From the point of view of teaching about modelling,
polynomial models permit one to show a number of things, here mentioned
in no particular order:
+ Whatever functions may be theoretically justified as models, one
can find a polynomial that will more or less adequately describe
any empirical shape of function. Even if one has no idea what
kind(s) of function may be theoretically appropriate.
+ Any polynomial will rather rapidly zoom off toward infinity (or
negative infinity) if you try to extrapolate beyond the range
of the data; as Bob Hayden illustrated with one data set.
Interpolation within that range may even have some problems,
as Paul and I have both noted in the chicks data.
+ Although polynomial shapes are often described in stereotypical
terms (quadratic = parabola = 1 bend in the function; cubic
= 2 bends; etc.), particular polynomials may not appear to
display the stereotypical shape (1 bend "implies" quadratic,
2 bends "imply" cubic, etc.). The chicks data exhibit only one
bend, but a quadratic fit is not satisfactory, a cubic fit does
not show two bends unless you look carefully (or extrapolate to
the left), etc.
+ Using orthogonal polynomial components as predictors in developing
an empirical model has certain conveniences (well described in,
e.g., Draper & Smith, so I won't go into detail here).
> I'd rather fit log(wt) on day.
Agreed. (Any day!-)
This has the further virtue of permitting "doubling time" to be
defined and estimated, for the range in which the exponential growth
function appears to be an adequate description. (Exponential growth
always eventually comes to be dominated by some limiting factor(s)
inherent either in the system exhibiting such growth or in the
environment in which it takes place. Extrapolation is no more to be
trusted for such a model than for a polynomial.)
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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