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

Fontana, Roberto; Pistone, Giovanni

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

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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 $F = \sum_αb_αX^α$ be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 149-172. Regular fractions are characterized by $R = \frac 1l \sum_{α\in \mathcal L} e_αX^α$, where $α\mapsto e_α$ 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 $\mathcal H = \{α_1 >... α_k\}$ is a generating set of $\mathcal L$, and $R = \frac1{2^k}(1 + e_1X^{α_1}) ... (1 + e_kX^{α_k})$, $e_j = \pm 1$, $j=1 ... k$, the inclusion condition in term of the $b_α$'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