Semifoldover plans for three-level orthogonal arrays with quantitative factors

Abstract

Although foldover designs can de-alias many effects, they involve at least twice the original number of runs. A semifoldover design, one kind of the partial foldover designs, is typically much more efficient since such a design adds only half of the new runs of a foldover design to the initial design. Semifoldover designs for two-level orthogonal arrays have been investigated in recent literatures. With the use of linear-quadratic system, this paper considers semifoldover designs for three-level orthogonal arrays with quantitative factors. We examine when the linear effects can be de-aliased from their aliased two-factor interactions for regular and nonregular designs, and obtain some good properties via semifolding over on partial factors or all factors. Theoretical properties and some examples are provided to illustrate the usefulness of the proposed designs.

Appendix: Proofs

Proof of Proposition 1

According to the condition in Proposition 1, we have $\mathcal {F}_{\gamma _5 } =\left( \mathcal {F}^T, \mathcal {F}(3)^T\right) ^T$, where $\mathcal {F}(3)=(A_1, A_2, A_3 ^{\gamma _5 })$. Then we have

$$\begin{aligned} |(\mathcal {F}_{\gamma _5})_{T_1 T_2 L} |=|{\mathcal {F}}_{T_1 T_2 L} | + |({\mathcal {F}(3)})_{T_1 T_2 L} |=|{\mathcal {F}}_{T_1 T_2 L} | + |{\mathcal {F}}_{T_1 T_2 Q} | \end{aligned}$$

and

$$\begin{aligned} |(\mathcal {F}_{\gamma _5})_{T_1 T_2 Q} |=|{\mathcal {F}}_{T_1 T_2 Q} | + |({\mathcal {F}(3)})_{T_1 T_2 Q} |=|{\mathcal {F}}_{T_1 T_2 Q} | + |{\mathcal {F}}_{T_1 T_2 L} |. \end{aligned}$$

Clearly, $|(\mathcal {F}_{\gamma _5})_{T_1 T_2 L} |=|(\mathcal {F}_{\gamma _5})_{T_1 T_2 Q} |$. From Lemma 1, this completes the proof. $\square$

Proof of Theorem 1

Suppose $\mathbf{w}$ and $\mathbf{w'}$ are any two generalized words of odd length and even length corresponding to $b_\mathbf{t} \ne 0$ and $b_\mathbf{t'} \ne 0$ in $\mathcal {F}$, respectively. Then $b_\mathbf{t}$ and $b_\mathbf{t'}$ in $\mathcal {F}$ become $-b_\mathbf{t}$ and $b_\mathbf{t'}$, respectively, in the full semifoldover plan $\mathcal {F}({ 1 \ldots k })$. Since the combined design is $\left( \mathcal {F}^T, \mathcal {F}({ 1 \ldots k })^T\right) ^T,$ we have Theorem 1 immediately. $\square$

Proof of Theorem 2

The proof of Theorem 2 is similar to that of Theorem 1, we omit it here for saving space. $\square$