Abstract: For quantitative factors, the minimum $\mathrm{\beta <!--\; beta\; -->}$-aberration criterion is commonly used for examining the geometric isomorphism and searching for optimal designs. In this paper, we investigate the connection between the minimum $\mathrm{\beta <!--\; beta\; -->}$-aberration criterion and the minimum contamination criterion. Results reveal that in ranking designs by the two criteria, the optimal designs selected by them can be different. We provide statistical justifications showing that the minimum contamination criterion controls the expected total mean square error of the estimation and demonstrate that it is more powerful than the minimum $\mathrm{\beta <!--\; beta\; -->}$-aberration criterion for identifying geometrically nonisomorphic designs.

Key words and phrases: Alias matrix, generalized minimum aberration, geometric isomorphism, indicator function, J-characteristics.