of lower-arity balanced designs, for which explicit formulas for replication numbers,

Overview

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Key numbers

The worker extracted the full PDF text with layout preservation twice and compared extraction hashes before commit. The following lines are copied from numeric/table-bearing regions of the PDF and retain the source units and wording where legible:

• License:   This work is licensed under a Creative Commons Attribution 4.0 International License.
• Sudesh K. Srivastav∗,1 Apurv Srivastav2
• 1 Department of Biostatistics and Data Science, Tulane University, New Orleans, USA
• 2 Department of Electrical and Computer Engineering, Center of Bioinformatics and
• We introduce a master (k + 1)-ary design with explicit replication and pairwise concur-
• AMS Subject Classification (2020): 62K10 · 05B05
• first considered by Tocher (1) and later formalized as balanced n-ary designs, in which each
• possible values for treatment multiplicities (0, 1, … , n − 1), not the number of treatments
• n = 2. Throughout, we use the convention that an n-ary design permits multiplicities in
• 0, 1, … , n − 1; thus a design allowing multiplicity up to k is referred to as (k + 1)-ary.
• in {0, 1, … , n − 1}, and (iii) for any two distinct elements i and j, the pairwise index
• definition is consistent with the multiset design framework of Shah and Dey (4) and subse-
• quent generalizations by Billington (5; 6), in which balance is defined via constant weighted
• stood to allow multiplicities in {0, 1, … , n − 1}; in particular, the master multiset design
• D(v : k) introduced in Section 2 is (k+1)-ary. Derived subdesigns enforce smaller admissi-
• ble multiplicity ranges as specified in Section 3. As in classical balanced incomplete block
• introduced by Bose (2), which revolutionized the field by providing algebraic techniques to
• Dey (3) and Shah and Dey (4) extended these concepts to n-ary structures, many traditional
• Recent developments include nested balanced n-ary designs (Dey and Midha (8); Sharma and
• are the multiset designs introduced by Assaf et al. (11), which allow repeated elements
• all distinct multisets of size k drawn from a v-set forms a balanced (k+1)-ary design with
• arities, satisfy the generalized Fisher inequality b ≥ v (Preece (12)), and provide analyti-
• Assaf et al. (11), who allow repeated blocks and variable block sizes, our construction uses
• The remainder of the paper is organized as follows. In Section 2, we describe the multiset
• (k+1)-ary design. Section 3 develops systematic deletion schemes that generate balanced
• 2 Construction of Balanced (k+1)-ary Designs
• 2.3 Structural design parameters
• using a standard binomial summation identity (see Appendix A and, for example, Gould (14);
• Riordan (15)). This expression is consistent with the symmetry argument that, among all
• as derived in detail in Appendix B, using repeated hockey-stick identities; see also Gould (14)
• 2.4.1 Case 1: v = 3, k = 2 (balanced 3-ary design)
• The blocks are categorized by their multiplicity patterns in Table 1.
• Table 1: Block configurations for v = 3, k = 2.
• Distinct Pairs (xi = 1, xj = 1) {1, 2}, {1, 3}, {2, 3} 3
• 2.4.2 Case 2: v = 3, k = 3 (balanced 4-ary design)
• The block structure becomes more complex, as shown in Table 2.
• Table 2: Compact block configurations for v = 3, k = 3.
• All distinct (x1 = x2 = x3 = 1) {1, 2, 3} 1
• 2.4.3 Case 3: v = 2, k = 3 (explicit λ calculation)
• smallest non-trivial case D(2 : 3). The master design contains b = 3
• The master pairwise index is λmaster = 41 = 4. Verifying manually via the sum of products:
• Applying the deletion rule for s = 2 (deleting blocks where xi = 3), we remove B1 and B2 .

Methods (brief)

• The current framework is combinatorially distinct from Efficiency Balanced Sample De-
• the Efficiency Balanced Sample Design (EBSD) framework of Srivastav and Srivastav (13).
• treatments ← sample(1:4, N, replace=TRUE)
• blocks_unbal ← replicate(b, sample(1:v, k, replace = TRUE), simplify = FALSE)
• Sample Simulation Output

Implications

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• Fish — marine, non-predatory (sardines, anchovies, salmon, cod)
• Lead
• Aluminum

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