Covariate Allocation in Completely Randomized Designs under Capsule–Based Cost Constraints - Bikas K. Sinha; Manisha Pal; Ganesh Dutta (Journal of the Indian Society of Agricultural Statistics, 2026)
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.Dutta, G., P. Das, and N.K. Mandal. (2009). “Optimum Covariate Designs in Partially Balanced Incomplete Block (PBIB) Design Set-Ups.” Journal of Statistical Planning and Inference, 139, 2823-35.Dutta, G., P. Das, and N.K. Mandal. (2010). “Optimum Covariate Designs in Binary Proper Equi-Replicate Block Design Set-up.” Discrete Mathematics, 310, 1037-49.Dutta, G., and Bikas Kumar Sinha. 2017a. “Nearly Optimal Covariate Designs—Part i.” Communications in Statistics - Theory and Methods, 46(19), 9691-702. https://doi.org/10.1080/03610926.2016.1217020.Dutta, G., and Bikas Kumar Sinha. 2017b. “Nearly Optimal Covariate Designs—Part II.” Communications in Statistics - Theory and Methods, 46(19), 9703-25. https://doi.org/10.1080/03610926.2016.1217021.Harville, David A. (1974). “Nearly Optimal Allocation Procedures.” Journal of the American Statistical Association, 69(346), 501-7.Hore, Samrat, Anup Dewanji, and Aditya Chatterjee. (2014). “Design Issues Related to Allocation of Experimental Units with Known Covariates into Two Treatment Groups.” Journal of Statistical Planning and Inference, 155, 117-26.Rao, P.S.S.N.V.P., S.B. Rao, G.M. Saha, and B.K. Sinha. (2003). “Optimal Designs for Covariates’ Models and Mixed Orthogonal Arrays.” Electronic Notes in Discrete Mathematics, 15, 155-58.Saha Ray, R., and G. Dutta. (2019). “Optimal and Efficient Treatment Control Design in the Heteroscedastic One-Way Layout with Covariates.” Statistics, 53(2), 459-69. https://doi.org/10.1080/02331888.2019.1573821.Sapam, S., G. Dutta, and B.K. Sinha. (2024). “Optimal Covariate Allocation in Asymmetrical Factorial Designs: A Comparative Analysis of Design Choices.” International Journal of Statistical Sciences, 24(2 Special), 49-66.Sapam, S., G. Dutta, and B.K. Sinha. (2025). “Optimal Covariate Allocation in Symmetric 32 Factorial Designs: A Comparative Study of Efficiency and Treatment Contrast Estimation.” Unpublished manuscript.Sarkar, B., A. Majumder, and G. Dutta. (2025). “Optimizing Covariate Allocation in Factorial Designs: Advancing d-Optimality with Multiple Covariates.” Unpublished manuscript.Sinha, B.K., and P.S.S.N.V.P. Rao. (2019). “Some Aspects of Optimal Covariate Designs in Factorial Experiments.” Statistics and Applications, 17, 77-86.Sinha, B.K., P.S.S.N.V.P. Rao, T. Mathew, and S.B. Rao. (2014). “A New Class of Optimal Designs in the Presence of a Quantitative Covariate.” International Journal of Statistical Sciences, 14, 1-15.Troya, L.J. 1982a. “Cyclic Designs for a Covariate Model.” Journal of Statistical Planning and Inference, 7, 49-75.Troya, L.J. 1982b. “Optimal Designs for Covariate Models.” Journal of Statistical Planning and Inference, 6, 373-419.