# Optimal designs for *K*-factor two-level models with first-order interactions on a symmetrically restricted design region

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## Abstract

We develop *D*-optimal designs for linear models with first-order interactions on a subset of the \(2^K\) full factorial design region, when both the number of factors set to the higher level and the number of factors set to the lower level are simultaneously bounded by the same threshold. It turns out that in the case of narrow margins the optimal design is concentrated only on those design points, for which either the threshold is attained or the numbers of high and low levels are as equal as possible. In the case of wider margins the settings are more spread and the resulting optimal designs are as efficient as a full factorial design. These findings also apply to other optimality criteria.

## Keywords

D-optimality Restricted design region Invariant design criterion Two-level factorial designs Interactions## Mathematics Subject Classification

62K05 62J05## Notes

## References

- Bhatia R, Davis C (2000) A better bound on the variance. Am Math Mon 107:353–357MathSciNetCrossRefzbMATHGoogle Scholar
- Filová L, Harman R, Klein T (2011) Approximate E-optimal designs for the model of spring balance weighing with a constant bias. J Stat Plan Inference 141:2480–2488MathSciNetCrossRefzbMATHGoogle Scholar
- Freise F, Holling H, Schwabe R (2018) Optimal designs for two-level main effects models on a restricted design region. arXiv:1808.06901 [math.ST]
- Harman R (2008) Equivalence theorem for Schur optimality of experimental designs. J Stat Plan Inference 138:1201–1209MathSciNetCrossRefzbMATHGoogle Scholar
- Kiefer J, Wolfowitz J (1960) The equivalence of two extremum problems. Can J Math 12:363–366MathSciNetCrossRefzbMATHGoogle Scholar
- Muilwijk J (1966) Note on a Theorem of M. N. Murthy and V. K. Sethi, Sankhyā. Indian J Stat Ser B 28:183Google Scholar
- Pukelsheim F (1993) Optimal design of experiments. Wiley, New YorkzbMATHGoogle Scholar
- R Core Team (2018) R: a language and environment for statistical computing. https://www.R-project.org/
- Schwabe R (1996) Optimum designs for multi-factor models. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Silvey SD (1980) Optimal design. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
- Venables B, Hornik K, Maechler M (2016) polynom: a collection of functions to implement a class for univariate polynomial manipulations. https://CRAN.R-project.org/package=polynom