Minimum Contamination and $\beta$-Aberration Criteria for Screening Quantitative Factors

TL;DR

This paper critically examines the minimum $eta$-aberration criterion for selecting screening designs, revealing its limitations and proposing the minimum contamination criterion as a more appropriate alternative for estimating effects.

Contribution

It establishes the mathematical relationship between $eta$-wordlength patterns and contamination, and clarifies when each criterion is optimal for design ranking.

Findings

Minimum $eta$-aberration minimizes contamination of nonnegligible effects only when factors equal design strength plus one.

The minimum contamination criterion better estimates the general mean effects in screening designs.

The two criteria are equivalent only under specific conditions related to the number of factors and design strength.

Abstract

Tang and Xu [Biometrika 101 (2014) 333-350] applied the minimum $\beta$-aberration criterion to selecting optimal designs for screening quantitative factors. They provided a statistical justification showing that minimum $\beta$-aberration criterion minimizes contamination of nonnegligible $k$th-order effects on the estimation of linear effects for $k=2,\cdots,r$, where $r$ is the strength of a design. Unfortunately, this result does not hold for $k>r$. In this paper, we provide a complete mathematical connection between $\beta$-wordlength patterns and contaminations (on the estimation of linear effects) and reveal that the minimum $\beta$-aberration criterion is not necessarily equivalent to the minimum contamination criterion for ranking designs. We prove that they are equivalent only when the number of factors of a design equals the strength plus one. We emphasize that the minimum $\beta$-aberration criterion, in fact, sequentially minimizes the contamination of nonnegligible $k$th-order effects on the estimation of the general mean, not on the estimation of linear effects. Therefore, the minimum contamination criterion should be more appropriate than the minimum $\beta$-aberration criterion for selecting optimal designs for screening quantitative factors.