[Paper Review] Bounds for orthogonal arrays with repeated rows

Douglas R. Stinson|arXiv (Cornell University)|Dec 12, 2018

Optimal Experimental Design Methods15 references4 citations

TL;DR

This paper strengthens the Plackett-Burman bound for orthogonal arrays of strength two by a factor of $m$ when a row is repeated $m$ times, using a variance-based proof technique. It generalizes this result to higher-strength orthogonal arrays via a theorem by Mukerjee, Qian, and Wu, providing tighter existence bounds for arrays with repeated rows, with applications in experimental design and combinatorial construction.

ABSTRACT

In this expository paper, we mainly study orthogonal arrays (OAs) of strength two having a row that is repeated $m$ times. It turns out that the Plackett-Burman bound (\cite{PB}) can be strengthened by a factor of $m$ for orthogonal arrays of strength two that contain a row that is repeated $m$ times. This is a consequence of a more general result due to Mukerjee, Qian and Wu \cite{Muk} that applies to orthogonal arrays of arbitrary strength $t$. We examine several proofs of the Plackett-Burman bound and discuss which of these proofs can be strengthened to yield the aforementioned bound for OAs of strength two with repeated rows. We also briefly discuss related bounds for $t$-designs, and OAs of strength $t$, when $t > 2$.

Motivation & Objective

• To establish tighter existence bounds for orthogonal arrays of strength two that contain a row repeated $m$ times.
• To generalize the strengthened Plackett-Burman bound to orthogonal arrays of arbitrary strength $t \geq 2$.
• To examine which standard proofs of the Plackett-Burman bound can be adapted to incorporate repeated rows.
• To draw analogies between bounds for orthogonal arrays and those for balanced incomplete block designs (BIBDs) and $t$-designs.
• To provide theoretical justification for constructing orthogonal arrays with high multiplicity rows, motivated by practical applications in experimental design.

Proposed method

• Adapts the classical variance method proof of the Plackett-Burman bound to account for $m$ repeated rows in an $\mathrm{OA}_\lambda(k,n)$.
• Derives a modified inequality by reweighting row sums and variances over the non-repeated rows, leading to a bound scaled by $m$.
• Applies the generalized result of Mukerjee, Qian, and Wu (2003) on nested orthogonal arrays to derive bounds for strength $t \geq 2$.
• Uses the Rao bound for orthogonal arrays as a foundation and scales it by $m$ when a row is repeated $m$ times.
• Compares proof techniques from Fisher’s inequality, Mann’s inequality, and Wilson’s cone condition to assess adaptability to repeated rows.
• Establishes analogies between bounds for orthogonal arrays and those for $t$-designs, particularly in the context of repeated blocks.

Experimental results

Research questions

• RQ1 Can the Plackett-Burman bound for orthogonal arrays of strength two be improved when a row is repeated $m$ times?
• RQ2 Which standard proofs of the Plackett-Burman bound can be modified to yield the strengthened bound under repeated rows?
• RQ3 Is there a generalization of the strengthened bound to orthogonal arrays of strength $t > 2$?
• RQ4 How do bounds for orthogonal arrays with repeated rows compare to known bounds for BIBDs and $t$-designs with repeated blocks?
• RQ5 What is the theoretical justification for constructing orthogonal arrays with high-multiplicity rows in practical experimental design?

Key findings

• For an $\mathrm{OA}_\lambda(k,n)$ with a row repeated $m$ times, the bound is strengthened to $\lambda \geq \frac{m(k(n-1) + 1)}{n^2}$, improving the original Plackett-Burman bound by a factor of $m$.
• The strengthened bound for strength two is derived via a variance-based proof technique that accounts for the reduced number of non-repeated rows.
• The result generalizes to higher-strength orthogonal arrays via the Mukerjee-Qian-Wu bound, which scales the Rao bound by $m$ when a row is repeated $m$ times.
• For $t$-designs with a repeated block of multiplicity $m$, Wilson’s inequality implies $b \geq m\binom{v}{s}$, generalizing Mann’s inequality for BIBDs.
• The proof techniques of Fisher, Mann, and Wilson are shown to be adaptable to repeated row/block cases, with Wilson’s method being the most general but complex.
• The bound $m \leq \frac{\lambda n^2}{k(n-1) + 1}$ is derived as a direct consequence, giving a maximum possible repetition count for a given array.