×

zbMATH — the first resource for mathematics

Model-robust designs for split-plot experiments. (English) Zbl 1255.62222
Summary: Split-plot experiments are appropriate when some factors are more difficult and/or expensive to change than others. They require two levels of randomization resulting in a non-independent error structure. The design of such experiments has gained much recent attention, including work on exact D-optimal split-plot designs. However, many of these procedures rely on the a priori assumption that the form of the regression function is known. We relax this assumption by allowing a set of model forms to be specified, and use a scaled product criterion along with an exchange algorithm to produce designs that account for all models in the set. We include also a generalization which allows weights to be assigned to each model, though they appear to have only a slight effect. We present two examples from the literature, and compare the scaled product designs with designs optimal for a single model. We also discuss a maximin alternative.
Reviewer: Reviewer (Berlin)

MSC:
62K05 Optimal statistical designs
62K25 Robust parameter designs
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anbari, F.T., Lucas, J.M., 1994. Super-efficient designs: how to run your experiment for higher efficiency and lower cost. In: ASQC 48th Annual Quality Congress Proceedings.
[2] Anbari, F.T.; Lucas, J.M., Design and running super-efficient experiments: optimum blocking with one hard-to-change factor, Journal of quality technology, 40, 1, 31-45, (2008)
[3] Arnouts, H.; Goos, P., Update formulas for split-plot and block designs, Computational statistics & data analysis, 54, 12, 3381-3391, (2010) · Zbl 1284.62485
[4] Atkinson, A.C.; Donev, A.N.; Tobias, R.D., Optimum experimental designs, with SAS, (2007), Oxford University Press New York · Zbl 1183.62129
[5] Bingham, D.R.; Schoen, E.D.; Sitter, R.R., Designing fractional factorial split-plot experiments with few whole-plot factors, The journal of the royal statistical society, series C (applied statistics), 53, 325-339, (2004) · Zbl 1111.62316
[6] Bingham, D.; Sitter, R.R., Minimum-aberration two-level fractional factorial split-plot designs, Technometrics, 41, 1, 62-70, (1999)
[7] Bingham, D.R.; Sitter, R.R., Design issues in fractional factorial split-plot experiments, Journal of quality technology, 33, 1, 2-15, (2001)
[8] Bisgaard, S., The design and analysis of \(2^{k - p} \times 2^{q - r}\) split plot experiments, Journal of quality technology, 32, 1, 39-56, (2000)
[9] Box, G.E.P.; Draper, N.R., A basis for the selection of a response surface design, Journal of the American statistical association, 54, 287, 622-654, (1959) · Zbl 0116.36804
[10] Cook, R.D., Nachtsheim, C.J., 1982. An analysis of the \(k\)-exchange algorithm. Tech. Rep., University of Minnesota, Dept. of Operations and Management Science.
[11] Cornell, J.A., Experiments with mixtures: designs, models, and the analysis of mixture data, (1990), Wiley-Interscience · Zbl 0732.62069
[12] Dette, H., A generalization of \(\mathcal{D}\)- and \(\mathcal{D}_1\)-optimal design in polynomial regression, Annals of statistics, 18, 1784-1804, (1990) · Zbl 0714.62068
[13] Dette, H.; Franke, T., Robust designs for polynomial regression by maximizing a minimum of \(\mathcal{D}\)- and \(\mathcal{D}_1\)-efficiencies, The annals of statistics, 29, 4, 1024-1049, (2001) · Zbl 1012.62080
[14] Donev, A.N.; Atkinson, A.C., An adjustment algorithm for the construction of exact \(D\)-optimum experimental designs, Technometrics, 30, 429-433, (1988) · Zbl 0668.62049
[15] Draper, N.R.; John, J.A., Response surface designs where levels of some factors are difficult to change, Australian and New Zealand journal of statistics, 40, 4, 487-495, (1998) · Zbl 0923.62083
[16] DuMouchel, W.; Jones, B., A simple Bayesian modification of \(D\)-optimal designs to reduce dependence on an assumed model, Technometrics, 36, 1, 37-47, (1994) · Zbl 0800.62472
[17] Fang, Z.; Wiens, D.P., Robust regression designs for approximate polynomial models, Journal of statistical planning and inference, 117, 305-321, (2003) · Zbl 1021.62058
[18] Gilmour, S.G.; Trinca, L.A., Optimum design of experiments for statistical inference, The journal of the royal statistical society, series C (applied statistics), (2012)
[19] Goos, P., The optimal design of blocked and split-plot experiments, (2002), Springer · Zbl 1008.62068
[20] Goos, P., Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments, Statistica neerlandica, 60, 361-378, (2006) · Zbl 1108.62073
[21] Goos, P.; Donev, A.N., Tailor-made split-plot designs for mixture and process variables, Journal of quality technology, 39, 4, 326-339, (2007)
[22] Goos, P.; Kobilinsky, A.; O’Brien, T.E.; Vandebroek, M., Model-robust and model-sensitive designs, Computational statistics & data analysis, 49, 201-216, (2005) · Zbl 1429.62339
[23] Goos, P.; Vandebroek, M., Optimal split-plot designs, The journal of quality technology, 33, 4, 436-450, (2001)
[24] Goos, P.; Vandebroek, M., \(D\)-optimal split-plot designs with given numbers and sizes of whole plots, Technometrics, 45, 3, 235-245, (2003)
[25] Goos, P.; Vandebroek, M., Outperforming completely randomized designs, Journal of quality technology, 36, 1, 12-26, (2004)
[26] Heredia-Langner, A.; Montgomery, D.C.; Carlyle, W.M.; Borror, C.M., Model-robust optimal designs: a genetic algorithm approach, Journal of quality technology, 36, 3, 263-279, (2004)
[27] Huang, P.; Chen, D.; Voelkel, J.O., Minimum-aberration two-level split-plot designs, Technometrics, 40, 4, 314-326, (1998) · Zbl 1064.62552
[28] Jones, B.; Goos, P., A candidate-set-free algorithm for generating \(D\)-optimal split-plot designs, The journal of the royal statistical society, series C (applied statistics), 56, 347-364, (2007)
[29] Jones, B., Goos, P., 2012. \(I\)-optimal versus \(D\)-optimal split-plot response surface designs. Working Paper 2012002. University of Antwerp, Faculty of Applied Economics. URL http://ideas.repec.org/p/ant/wpaper/2012002.html.
[30] Jones, B.; Nachtsheim, C.J., Split-plot designs: what, why, and how, Journal of quality technology, 41, 4, 340-361, (2009)
[31] Kowalski, S.M.; Cornell, J.A.; Vining, G.G., Split-plot designs and estimation methods for mixture experiments with process variables, Technometrics, 44, 1, 72-79, (2002)
[32] Läuter, E., Experimental design in a class of models, Mathematische operationsforschung und statistik, 5, 379-398, (1974) · Zbl 0297.62056
[33] Letsinger, J.D.; Myers, R.H.; Lentner, M., Response surface methods for bi-randomization structures, Journal of quality technology, 28, 4, 381-397, (1996)
[34] Li, W.; Nachtsheim, C.J., Model-robust factorial designs, Technometrics, 42, 4, 345-352, (2000)
[35] Lucas, J.M., Optimum composite designs, Technometrics, 16, 561-567, (1974) · Zbl 0294.62099
[36] Lucas, J.M., Ju, H.L., 1992. Split plotting and randomization in industrial experiments. In: ASQC Quality Congress Transactions.
[37] Parker, P.A.; Kowalski, S.M.; Vining, G.G., Classes of split-plot response surface designs for equivalent estimation, Quality and reliability engineering international, 22, 3, 291-305, (2006)
[38] Parker, P.A.; Kowalski, S.M.; Vining, G.G., Construction of balanced equivalent estimation second-order split-plot designs, Technometrics, 49, 1, 56-65, (2007)
[39] Piepel, G.F.; Cooley, S.K.; Jones, B., Construction of a 21-component layered mixture experiment design using a new mixture coordinate-exchange algorithm, Quality engineering, 17, 579-594, (2005)
[40] Smucker, B.J.; del Castillo, E.; Rosenberger, J.L., Exchange algorithms for constructing model-robust experimental designs, Journal of quality technology, 43, 1, (2011)
[41] Smucker, B.J., del Castillo, E., Rosenberger, J.L., 2012. Model-robust two-level designs using coordinate exchange algorithms and a maximin criterion. Tentatively accepted to Technometrics.
[42] Stigler, S.M., Optimal experimental design for polynomial regression, Journal of the American statistical association, 66, 334, 311-318, (1971) · Zbl 0217.51701
[43] Studden, W.J., Some robust-type \(D\)-optimal designs in polynomial regression, Journal of the American statistical association, 77, 380, 916-921, (1982) · Zbl 0505.62062
[44] Trinca, L.A.; Gilmour, S.G., Multistratum response surface designs, Technometrics, 43, 1, 25-33, (2001) · Zbl 1072.62623
[45] Tsai, P.-W.; Gilmour, S.G., A general criterion for factorial designs under model uncertainty, Technometrics, 52, 2, 231-242, (2010)
[46] Vining, G.G.; Kowalski, S.M.; Montgomery, D.C., Response surface designs within a split-plot structure, Journal of quality technology, 37, 2, 115-129, (2005)
[47] Welch, W.J., A Mean squared error criterion for the design of experiments, Biometrika, 70, 1, 205-213, (1983) · Zbl 0517.62066
[48] Yates, F., Complex experiments, with discussion, Journal of the royal statistical society, series B, 2, 181-223, (1935)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.