## Construction and properties of Box-Behnken designs

##### Abstract

^{k}full or 2

^{k-1}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^{k}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»_{1}, Î»_{2}, ...,Î»_{m} are all greater than zero.

In order to reduce the number of experimental points the use of 2^{k-1} fractional
factorials instead of 2^{k} 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-BehJken 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 â ¥ 4 linear trend-free Box-Behnken designs can always be constructed.

##### Collections

- Doctoral Dissertations [11232]