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# An Algorithm for Finding D-Effcient Equivalent-Estimation Second-Order Split-Plot Designs

Summary: 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 subplot 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-effect estimates as the GLS estimator. These designs are called equivalent-estimation split-plot designs and offer the advantage that estimation of the factor effects 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 efficiency (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 efficient and, hence, provide precise factor-effect estimates. In this work, we present an algorithm that allows us to (i) identify equivalent-estimation designs for scenarios where Macharia and Goos (2010) did not find equivalent-estimation designs and (ii) find equivalent-estimation designs that outperform those of Macharia and Goos (2010) in terms of the D-optimality criterion. We also study the performance of equivalent-estimation designs when it comes to estimating the variance components in the split-plot model and observe that they outperform D-optimal designs in this respect.

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**Topics:**Statistics**Keywords:**Algorithm, D-optimality, Equivalency, Estimation, Least squares, Maximum likelihood estimate (MLE), Second-order designs, Split-plot design, Variance components**Author:**Jones, Bradley; Goos, Peter**Journal:**Journal of Quality Technology