A generalized general minimum lower order confounding criterion for nonregular designs

• We generalize the GMC theory into non regular designs.
• For any given design, the G2-AENP is unique up to isomorphism.
• Cores of criteria GMA and MMA are both functions on the G2-AENP.
• The G2-GMC criterion is more powerful than GMA and MMA in the design recognition.
• The G2-AENP is helpful for the arrangement of factors.

We extend the work of Zhang et al. [Statistica Sinica 18, 1689–1705] for nonregular designs and propose two new concepts, i.e., the generalized aliasing effect-number pattern (G2-AENP) and the generalized general minimum lower order confounding (G2-GMC). It proves that (i) isomorphic designs have the identical G2-AENP and (ii) the generalized minimum aberration (GMA) and minimum moment aberration (MMA) can both be treated as ones that optimize functions over the G2-AENP. That is, the G2-GMC criterion is more sensitive in the identification and classification of designs.