2-level fractional factorial designs which are the union of non trivial regular designs

Abstract

Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a condition for the inclusion of a regular fraction as follows. Let $F={\sum }_{\alpha }{b}_{\alpha }{X}^{\alpha }$ be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 149-172. Regular fractions are characterized by $R=\frac{1}{l}{\sum }_{\alpha \in \mathcal{L}}{e}_{\alpha }{X}^{\alpha }$, where $\alpha \mapsto e_{\alpha }$ is an group homeomorphism from $\mathcal{L}\subset {\mathbb{Z}}_2^d$ into $\{-1,+1\}$. The regular $R$ is a subset of the fraction $F$ if $FR=R$, which in turn is equivalent to ${\sum }_{t}F(t)R(t)={\sum }_{t}R(t)$. If $\mathcal{H}=\{{\alpha }_1>...{\alpha }_k\}$ is a generating set of $\mathcal{L}$, and $R=\frac{1}{2^k}(1+e_1{X}^{{\alpha }_1})...(1+e_k{X}^{{\alpha }_k})$, $e_j=\pm 1$, $j=\mathrm{1...}k$, the inclusion condition in term of the $b_{\alpha }$'s is % \begin{equation}b_0 + e_1 b_{α_1} + >... + e_1 ... e_k b_{α_1 + ... + α_k} = 1. \tag{*}\end{equation} % The last part of the paper will discuss some examples to investigate the practical applicability of the previous condition (*). This paper is an offspring of the Alcotra 158 EU research contract on the planning of sequential designs for sample surveys in tourism statistics.

Presented by R. Fontana at the DAE 2007 Conference, The University of Memphis, November 2, 2007

References in corpus (1)

• Two polynomial representations of experimental design