Construction and properties of BoxBehnken designs
 Publisher: Virginia Tech

Subject: Incomplete box designs  Response surfaces (Statistics)  Factorial experiment designs  LD5655.V856 1992.J6
BoxBehnken designs are used to estimate parameters in a secondorder 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>k1</sup> fractional factorials.
<p>In this dissertation, a more general mathematical formulation of the BoxBehnken
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 BoxBehnken designs with respect to the estimability of all parameters in
a secondorder 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
Î»<sub>1</sub>, Î»<sub>2</sub>, ...,Î»<sub>m</sub> are all greater than zero.
<p>In order to reduce the number of experimental points the use of 2<sup>k1</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 BoxBehJken 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 nonestimability of certain coefficients
and to correlated estimators.
<p>The final topic is concerned with BoxBehnken 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 trendfree BoxBehnken 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 trendfree BoxBehnken 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 trendfree BoxBehnken designs. However, for k â ¥ 4 linear trendfree BoxBehnken
designs can always be constructed.
Ph. D.
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