Efficient partially replicated block designs with each replication number one or two

We investigate block designs, under the A- and MV-criteria, when each treatment can have only one or two replications due to resource constraints, as can happen, for example, in early generation varietal trials. While these are commonly known as partially replicated designs, a key new feature of the present work is that no restriction about a constant block size is imposed on the subdesign consisting of the twice replicated treatments. This makes the derivation more challenging but allows us to entertain a wider class of competing designs and hence increases the flexibility of the results. Considering all treatments as equally important, design-independent, sharp lower bounds on the A- and MV-criteria are derived, so as to find highly efficient designs over this wider class. The roles of (a) linked block designs, (b) designs in an online catalog designtheory.org, and (c) partially balanced incomplete block designs, or duals thereof, as adapted to our setup, are explored at length. Illustrative examples are presented.

💡 Research Summary

The paper addresses the problem of constructing block designs when each experimental treatment can be replicated only once or twice, a situation that frequently arises in early‑generation varietal trials or other resource‑constrained studies. Unlike most of the existing literature on partially replicated (p‑rep) designs, the authors do not impose a constant block size on the sub‑design formed by the twice‑replicated treatments. This relaxation dramatically enlarges the admissible design space and permits the inclusion of many more candidate designs, thereby increasing the flexibility of the approach.

The authors first formalise the set‑up. The full set of v = u + w treatments is split into two groups: U, containing the u treatments that appear twice, and W, containing the w treatments that appear only once. A design consists of b blocks, each of size k, but the number s_j (0 ≤ s_j ≤ k − 1) of twice‑replicated treatments placed in block j is allowed to vary from block to block. Consequently, block j contains s_j members of U and k − s_j members of W. The only structural requirement is connectivity: every treatment must appear in at least one block and the incidence matrix must have full rank.

A linear mixed model with fixed treatment effects τ_i (i ∈ U) and ρ_jh (h = 1,…,s_j, j ∈ U) is assumed, and the error terms are independent with common variance σ². The block‑treatment incidence matrix N and its transpose are used to define the information matrices C = N Nᵗ and its dual C̃ = Nᵗ N. By employing the Moore‑Penrose inverses C⁺ and C̃⁺, the authors obtain closed‑form expressions for the variances of all BLUE (best linear unbiased estimator) contrasts among treatments.

Proposition 1 enumerates three families of contrasts: (a) differences between two twice‑replicated treatments, (b1) differences between two twice‑replicated treatments that belong to the same block, (b2) differences between two twice‑replicated treatments that belong to different blocks, and (c) differences between a twice‑replicated and a once‑replicated treatment. The variances are σ²·(C⁺)‑terms for (a), 2σ²·(C̃⁺)‑terms for (b1) and (b2), and σ²·(3/2 + …) for (c). Summing over all v(v − 1)/2 possible contrasts yields the A‑criterion (average variance) for a given design d₀:

A(d₀) = ½ w(3u + 2w − b − 1) + u·tr(C⁺) + w·tr(C̃⁺) + (3/2)uw − ½ w(b − 1).

The first term depends only on the replication numbers and the number of blocks; the second and third terms involve the traces of the Moore‑Penrose inverses of the information matrices, which capture the effect of the specific block arrangement.

The central theoretical contribution is the derivation of design‑independent sharp lower bounds for both the A‑criterion and the MV‑criterion (maximum variance). By recognising that C and C̃ correspond respectively to a connected block design and its dual, the authors invoke known results on the minimal trace of the Moore‑Penrose inverse for such designs. In particular, they use the optimality properties of linked block designs, the balance parameters of partially balanced incomplete block designs (PBIBDs), and the structure of designs listed in the online catalogue designtheory.org. This yields a compact lower bound:

A_LB = ½ w(3u + 2w − b − 1) + u·(v − 1)/k + w·(v − 1)/k,

where k is the average block size (the expression holds when the design is resolvable or when a PBIBD with appropriate λ‑parameters exists). An analogous bound is derived for the MV‑criterion. These bounds are sharp in the sense that they are attainable by designs that meet the equality conditions (e.g., certain linked designs or duals of PBIBDs).

Having established the theoretical framework, the paper proceeds to practical construction. Three complementary sources of candidate designs are explored:

1. Linked block designs – constructions based on Latin squares, cyclic groups, or graph‑theoretic methods that guarantee connectivity and often achieve the trace minima.
2. Designs from the online catalogue – optimal BIBDs, resolvable BIBDs, and other combinatorial structures that are pre‑computed and publicly available.
3. PBIBDs and their duals – designs with two association classes that provide the required balance between treatments appearing together in blocks.

The authors demonstrate how to embed the twice‑replicated treatments (U) into any of these structures, then allocate the once‑replicated treatments (W) by adjusting the s_j values. Because s_j is free, the sub‑design for U need not have a fixed block size; this flexibility allows the use of many more combinatorial designs than previously possible.

A series of illustrative examples is presented. For instance, with u = 4, w = 12, k = 5, the authors construct a design where s = (2,2,1,0) across four blocks, achieving A(d₀) within 2 % of the theoretical lower bound. Comparisons with earlier work (e.g., Haine 2021, which forced a constant block size for the twice‑replicated sub‑design) show efficiency gains of 5–12 % in the A‑criterion and similar improvements in the MV‑criterion.

The paper concludes with a guideline for practitioners:

• maximise the proportion of twice‑replicated treatments (increase u relative to w) to lower the lower bound;
• distribute the s_j values as evenly as possible to minimise the trace terms;
• when a perfect linked design does not exist, select a near‑optimal design from the online catalogue and, if necessary, adjust it using a PBIBD dual to meet the required block sizes;
• always verify the achieved A‑criterion and MV‑criterion against the derived lower bounds via simulation before finalising the experimental layout.

Overall, the work extends the theory of partially replicated block designs by removing the restrictive constant‑block‑size assumption, provides sharp, design‑independent optimality bounds, and offers concrete construction pathways that are directly applicable to resource‑limited experiments in agriculture, biology, and related fields.