Covariate Allocation in Completely Randomized Designs under Capsule–Based Cost Constraints

Abstract

The incorporation of covariates into Completely Randomized Designs (CRDs) enhances inferential precision by accounting for systematic variation 
among experimental units, yet in practical contexts, structural and economic constraints often limit the freedom of covariate assignment. This paper 
presents a unified framework for optimal covariate allocation in CRDs under a capsule–based cost structure, where a capsule represents a subset of 
plots constrained to share identical covariate values. Such a formulation transforms the classical design problem into a discrete optimization setting 
that explicitly captures the trade–off between information gain and economic expenditure. For the single–covariate case, it is established that balanced 
two–capsule partitions achieve the theoretical upper bound on information while minimizing cost, thereby yielding a greedy allocation strategy that 
is provably D–optimal under the proposed cost model. For the multiple–covariate case, the same two–capsule allocation principle is adopted as far 
as possible, with higher capsule splits introduced only when necessary to maintain orthogonality or budget feasibility. If the available budget permits 
each treatment to employ its optimal capsule configuration, the resulting design achieves D–optimality for the covariate parameters; otherwise, 
partial implementation continues to yield near–optimal efficiency. The framework assumes capsule cost depends solely on its size, ensuring analytical 
tractability while realistically modelling cost escalation. The proposed methodology unifies theoretical optimality with practical feasibility and can be 
naturally extended to Randomized Block, Latin Square, and Balanced Incomplete Block Designs under similar capsule–based constraints.

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