Construction and properties of Box-Behnken designs

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dc.contributor.authorJo, Jinnamen
dc.contributor.committeechairHinkelmann, Klaus H.en
dc.contributor.committeememberMyers, Raymond H.en
dc.contributor.committeememberLentner, Marvin M.en
dc.contributor.committeememberReynolds, Marion R. Jr.en
dc.contributor.committeememberFoutz, Roberten
dc.contributor.departmentStatisticsen
dc.date.accessioned2014-03-14T21:08:59Zen
dc.date.adate2006-02-01en
dc.date.available2014-03-14T21:08:59Zen
dc.date.issued1992-10-05en
dc.date.rdate2006-02-01en
dc.date.sdate2006-02-01en
dc.description.abstractBox-Behnken designs are used to estimate parameters in a second-order response surface model (Box and Behnken, 1960). These designs are formed by combining ideas from incomplete block designs (BIBD or PBIBD) and factorial experiments, specifically 2<sup>k</sup> full or 2<sup>k-1</sup> fractional factorials. In this dissertation, a more general mathematical formulation of the Box-Behnken method is provided, a general expression for the coefficient matrix in the least squares analysis for estimating the parameters in the second order model is derived, and the properties of Box-Behnken designs with respect to the estimability of all parameters in a second-order model are investigated when 2<sup>k</sup>full factorials are used. The results show that for all pure quadratic coefficients to be estimable, the PBIB(m) design has to be chosen such that its incidence matrix is of full rank, and for all mixed quadratic coefficients to be estimable the PBIB(m) design has to be chosen such that the parameters λ₁, λ₂, ...,λ<sub>m</sub> are all greater than zero. In order to reduce the number of experimental points the use of 2<sup>k-1</sup> fractional factorials instead of 2<sup>k</sup> full factorials is being considered. Of particular interest and importance are separate considerations of fractions of resolutions III, IV, and V. The construction of Box-Behnken designs using such fractions is described and the properties of the designs concerning estimability of regression coefficients are investigated. Using designs obtained from resolution V factorials have the same properties as those using full factorials. Resolutions III and IV designs may lead to non-estimability of certain coefficients and to correlated estimators. The final topic is concerned with Box-Behnken designs in which treatments are applied to experimental units sequentially in time or space and in which there may exist a linear trend effect. For this situation, one wants to find appropriate run orders for obtaining a linear trend-free Box-Behnken design to remove a linear trend effect so that a simple technique, analysis of variance, instead of a more complicated technique, analysis of covariance, to remove a linear trend effect can be used. Construction methods for linear trend-free Box-Behnken designs are introduced for different values of block size (for the underlying PBIB design) k. For k= 2 or 3, it may not always be possible to find linear trend-free Box-Behnken designs. However, for k ≥ 4 linear trend-free Box-Behnken designs can always be constructed.en
dc.description.degreePh. D.en
dc.format.extentvii, 113 leavesen
dc.format.mediumBTDen
dc.format.mimetypeapplication/pdfen
dc.identifier.otheretd-02012006-141730en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-02012006-141730/en
dc.identifier.urihttp://hdl.handle.net/10919/37247en
dc.language.isoenen
dc.publisherVirginia Techen
dc.relation.haspartLD5655.V856_1992.J6.pdfen
dc.relation.isformatofOCLC# 27696952en
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.lccLD5655.V856 1992.J6en
dc.subject.lcshFactorial experiment designsen
dc.subject.lcshIncomplete box designsen
dc.subject.lcshResponse surfaces (Statistics)en
dc.titleConstruction and properties of Box-Behnken designsen
dc.typeDissertationen
dc.type.dcmitypeTexten
thesis.degree.disciplineStatisticsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en
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