Polynomial regression

- uses in econometrics: cost and production functions.  marginal cost
  curves and total cost functions.

- example of ice breakup trends on the Tenana River in Alaska.
  Sagarin ("Climate Change in Non-Traditional Datasets", Science 2001)
  models ice break as a function of year using a cubic polynomial.

- issues: influence of large values from large powers of x;
  multicollinearity and unstable coefficient estimates.

- Application to Gene Frequency and Distance (handout from class)

How do alleles and genotypes change according to distance?  Can we
detect geneflow according to distance?  Do alleles and genotypes
differ in performance as a function of environmental variables?  Are
there geographically separate population structures?

- Steps:  forward selection and residual analysis to detect curved pattern.

Koehn chooses an arcsin square root transform of frequency (variance
stabilizing transformation) as a function of a 3rd order polynomial to
model gene frequency as a function of distance.

- Risk of overfit: modeling random variation rather than the slope.
  (A possible solution is the use of orthogonal polynomials.)