Response-Surface Designs for Quantitative and Qualitative Variables

TLDR

In response‑surface studies with mixed quantitative and qualitative variables, standard designs often fail, and choosing an appropriate design requires careful use of available degrees of freedom while accounting for model‑assumption uncertainty. The paper presents and illustrates sound guidelines for selecting designs in such mixed‑variable response‑surface studies. The authors explain that an n‑point design yields n linearly independent contrasts, of which p estimate model parameters and the remaining n‑p can assess pure error and lack‑of‑fit. The study clarifies why standard designs are inadequate for mixed quantitative‑qualitative response‑surface problems.

Abstract

When some variables are quantitative and some qualitative in a response-surface context, standard designs may not be suitable. The reasons for this are illuminated. Some alternative designs are discussed. Every n-point design provides n linearly independent estimation contrasts. Some of these, p say, are needed to estimate the p parameters of the postulated model. The remaining n — p linearly independent estimation contrasts are available to estimate pure error (if used) and to test for lack of fit, either overall or in particular ways. The key to choosing a good design is to use the available degrees of freedom well, given certain assumptions about the model to be fitted. When there is also uncertainty about the model assumptions, dogmatic design advice is not possible. Sound guidelines are available, however, and these are presented and illustrated.