  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.
