example, may make the term in x non-significant, because of collinearity, in an
equation containing both, but x may become significant after x has been removed.
Monomial • When the highest monomials (monomial: each term of a polynomial expression) have been eliminated, use one of the standard selection procedures described above to find the most effective subset of monomials (highest R2) in which all terms are statistically significant. The best procedure, of course, is one that tests all possible models with fewer terms. Short of that, backward, forward, or stepwise procedures may be used, with caution. The final equation does not necessarily possess all successive monomials from degree 1 up.
Polynomial regression procedures are directly available in some statistical packages (beware: in most packages, polynomial regression procedures do not allow one to remove the monomials of degree lower than k if xk is retained in the equation).
When no such packaged procedure is available, one can easily construct a series of variables corresponding to the successive monomials. Starting with a variable x, it is easy to square it to construct x2, and so on, creating as many new variables as deemed necessary. These are then introduced in a multiple regression procedure. See also the last paragraph of Section 2.8.
One must be aware of the fact that the successive terms of an ordinary polynomial expression are not linearly independent of one another. Starting for instance with a
variable x made of the successive integers 1 to 10, variables x2, x3, and x4 computed from it display the following correlations:
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