An Algorithm for Finding D-e cient Equivalent-Estimation Second-Order Split-Plot Designs

Many industrial experiments involve restricted rather than complete randomization. This often leads to the use of split-plot designs, which limit the number of independent settings of some of the experimental factors. These factors, named whole-plot factors, are often, in some way, hard to change. The remaining factors, called sub-plot factors, are easier to change. Their levels are therefore independently reset for every run of the experiment. In general, model estimation from data from split-plot experiments requires the use of generalized least squares (GLS). However, for some split-plot designs, the ordinary least squares (OLS) estimator will produce the same factor-e ect estimates as the GLS estimator. These designs are called equivalent-estimation split-plot designs and o er the advantage that estimation of the factor e ects does not require estimation of the variance components in the split-plot model. While many of the equivalent-estimation second-order response surface designs presented in the literature do not perform well in terms of estimation e ciency (as measured by the D-optimality criterion), Macharia and Goos (2010) showed that, in many instances, it is possible to generate second-order equivalent-estimation split-plot designs that are highly e cient and, hence, provide precise factor-e ect estimates. In the present paper, we present an algorithm that allow us to (i) identify equivalent-estimation designs for scenarios where Macharia and Goos (2010) did not nd equivalent-estimation designs, and (ii) nd equivalent-estimation designs that outperform those of Macharia and Goos (2010) in terms of the D-optimality criterion.