The decomposition of treatment degrees of freedom in quantitative factorial experiments

In an experiment in which every treatment factor has the same number, p, of levels, where p is prime, there is a classical breakdown of the treatment degrees of freedom into components such as AB, AB^2, AB^2C, ..., each of (p-1) degrees of freedom. The disadvantages of this system are (i) the lack of canonical single degrees of freedom and (ii) the fact that the breakdown bears no relation to the contrasts typically of interest if all treatment factors are quantitative. In this note we decompose the treatment degrees of freedom into single degrees of freedom,
and use these to show how appropriate choice of levels of quantitative factors enables quantitative contrasts to be completely unconfounded in classical designs.