Abstract
Many industrial experiments involve some factors whose levels are harder to set than others. The best way to deal with these is to plan the experiment carefully as a split-plot, or more generally a multistratum, design. Several different approaches for constructing split-plot type response surface designs have been proposed in the literature since 2001, which has allowed experimenters to make better use of their resources by using more efficient designs than the classical balanced ones. One of these approaches, the stratum-by-stratum strategy has been shown to produce designs that are less efficient than locally D-optimal designs. An improved stratum-by-stratum algorithm is given, which, though more computationally intensive than the old one, makes better use of the advantages of this approach, that is, it can be used for any structure and does not depend on prior estimates of the variance components. This is shown to be almost as good as the locally optimal designs in terms of their own criteria and more robust across a range of criteria. Supplementary materials for this article are available online.
Highlights
Fractional factorial and response surface designs are widely used in industrial and laboratory based experiments. It has been increasingly recognized in recent years that many, perhaps most, industrial experiments and many laboratory experiments involve some factors whose levels are harder to set than others
Each level of hardness-toset in factors which is taken account of in the design defines a stratum, as does each level of blocking, and, following Trinca and Gilmour (2001), we refer to designs with factors in at least two strata as multistratum designs
For the highest stratum i (i ∈ {1, 2, . . . , s}) for which there are factors to be applied, proceed as follows: 1. If i = 1 choose the treatment design for the factors to be applied to the units in stratum i considering the efficiency for estimating the model parameters involving the factors in this stratum only
Summary
Fractional factorial and response surface designs are widely used in industrial and laboratory based experiments. Letsinger, Myers, and Lentner (1996) recommended analyzing the data using residual maximum likelihood (REML) to estimate the variance components and generalized least squares (GLS) to estimate the fixed (treatment) effects This is accepted as the standard analysis method, Gilmour and Goos (2009) showed that it can be unreliable when there are small numbers of units in the higher strata. The first article to recommend choosing designs with a specific split-plot or other multistratum structure in mind was by Trinca and Gilmour (2001) They suggested a stratumby-stratum strategy for building designs and combining the designs from the different strata to optimize particular criteria for each step in the procedure.
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