When dealing with quantitative factors, typically we assume some kind of continuity across varying levels. A typical research hypothesis assumes that the data can be approximated by a polynomial equation of a certain degree with a certain accuracy. The mathematical approximation theory often provides the proper framework, and the statistical approach only follows a typically involved analytical theory.
The commonly used approximation is by orthogonal polynomials. In this article we provide basic information on orthogonal polynomials, as used in statistics. We also assemble a body of related mathematical facts which help in generalizing the approach and constructing one's own orthogonal contrasts adapted to specific problems.
See the attached documents for specific information.