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

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 $R = \frac 1l \sum_{\alpha \in \mathcal L} e_\alpha X^\alpha$, 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 $R = \frac1{2^k}(1 + e_1X^{\alpha_1}) ... (1 + e_kX^{\alpha_k})$, $e_j = \pm 1$, $j=1 ... k$, the inclusion condition in term of the $b_\alpha$'s is % \begin{equation}b_0 + e_1 b_{\alpha_1} + >... + e_1 ... e_k b_{\alpha_1 + ... + \alpha_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.