Re the chicks data posted by Don Burrill...
A reference is Snedecor and Cochran, Statistical Methods (7th ed.),
Iowa State University Press, Ames, IA, 1980, pp.394-406, which gives
an extensive analysis, including fitting an exponential and orthogonal
polynomials.
There is more on orthogonality in the ANOVA sections of S&C, and also
in an analysis of the pulse data by Don Burrill that's on the Minitab
website. The IDEA of orthogonality is the main thing -- the details
in any particular application tend to be very messy.
Re the girl's heights data, I've been asked to interpret the models,
and I want to warn that NONE of the models is much good for
extrapolation (as is the case for the chicks data).
The straight line model for the girls' heights says that girls are born
76.641 cm "tall" and grow at a constant rate of 6.3661 cm per year
thereafter. This is easy to understand, but not consistent with our
experience of how children grow. The individual coefficients do not
have such simple interpretations for the higher order models, but then
they never had such simple interpretations in algebra class either.
As a math. major, I like to think of polynomials as Taylor
approximations to an unknown function, but not everyone will like that
interpretation.
The quadratic model might better be understood in terms of the rate of
growth (first derivative), which is 8.6706-0.35454*age. In other
words, instead of a constant growth rate, the growth rate starts
"high" at 8.6706 at birth and then steadily decines at a rate of
0.35454 (cm/yr)/yr. It reaches 0 at age 24.46, which is not too bad,
but after that the girls start shrinking, to about -8 m by age 100,
considerably more than six feet under. Also, this model predicts
lower growth rates in adolescence than in the years prior to
adolescence, again contrary to our experience.
For the cubic, the growth rate is a quadratic,
11.2662-1.25758*age+0.069465age^2. This starts even higher at
11.2662 at birth, then gradually delines to less than 6 at around 9
years of age, and then begins to increase again as the girls enter
adolescence. By about 18 they are growing as fast as they were at
birth, which is unreasonable, but at least this model captures the
growth spurt during adolescence. (The data go only to age 11.)
None of the models support much extrapolation, which is reasonable as
we know growth levels off again as the girl reaches adulthood. The
line predicts steady growth forever so 100-year-oles are more than 7 m
tall. The cubic predicts even greater great grannies, and, as already
mentioned, the quadratic has them shrinking away to nothingness and
beyond.
_
| | Robert W. Hayden
| | Work: Department of Mathematics
/ | Plymouth State College MSC#29
| | Plymouth, New Hampshire 03264 USA
| * | fax (603) 535-2943
/ | Home: 82 River Street (use this in the summer)
| ) Ashland, NH 03217
L_____/ (603) 968-9914 (use this year-round)
Map of New [EMAIL PROTECTED] (works year-round)
Hampshire http://mathpc04.plymouth.edu (works year-round)
===========================================================================
This list is open to everyone. Occasionally, less thoughtful
people send inappropriate messages. Please DO NOT COMPLAIN TO
THE POSTMASTER about these messages because the postmaster has no
way of controlling them, and excessive complaints will result in
termination of the list.
For information about this list, including information about the
problem of inappropriate messages and information about how to
unsubscribe, please see the web page at
http://jse.stat.ncsu.edu/
===========================================================================
