Confounding analysis of s-level designs with multi-block variables

In practical experiments, block variables often arise from multiple sources of heterogeneity. To address the confounding problem, this paper proposes a blocked aliased component-number pattern (B${}^2$-ACNP) to analyze the confounding properties of s-level designs with multi-block variables. We calculate the values of (B${}^2$-ACNP) via a blocked wordlength distribution matrix. The classification patterns of existing criteria can be expressed as functions of specific elements within the B${}^2$-ACNP, thereby stablishing connections within a unified framework. Further, we provide confounding algorithms and visualization methods of the B${}^2$-ACNP. Finally, case analysis clarifies the significant role of the B${}^2$-ACNP. The Python code is available in the Appendix.