Response surface designs within a split-plot structure

In many industrial experiments, time and/or cost constraints often force certain factors in a designed experiment to be much harder to change than others. An appropriate approach to such an experiment restricts the randomization, which leads to a split-plot structure. This paper first establishes how one can modify the common central-composite design to efficiently accommodate a split-plot structure. The proposed designs allow pure-error estimates of the two variance components. We next discuss the conditions on the design that make ordinary least squares and weighted least squares estimates equivalent. These conditions are easy to obtain in practice. An important consequence is that people can use standard experimental design software to analyze second-order response surfaces. We give examples of central composite and Box-Behnken designs that meet the conditions. We conclude with an example.