Statistical efficiency of incomplete block designs over complete block designs in agriculture: A review

Introduction

An experiment is a planned inquiry designed to obtain new facts or confirm previous findings. The choice of treatments, methods of assigning treatments to experimental units, and the arrangement of experimental units into various patterns together constitute the Design of Experiment. The responses of experimental units are influenced not only by treatments but also by extraneous factors. Variation caused by these factors is termed experimental error, and minimizing it is the central objective of experimental design. Three fundamental principles govern experimental design: (i) Replication enabling estimation of error variance and valid comparison of treatments; (ii) Randomization ensuring unbiased estimation of means; and (iii) Local Control reducing experimental error through grouping of experimental units into homogeneous blocks. When blocks become too large to maintain homogeneity typically when the number of treatments exceeds 16 to 20 complete block designs like RCBD lose their efficiency advantage. A survey conducted in Gujarat state revealed that over 95% of field experiments were conducted in RCBD (Anon., 2016) [3] . Yet when the number of genotypes is large, RCBD fails to adequately control soil heterogeneity within blocks. Incomplete Block Designs (IBDs) address this limitation by subdividing replications into smaller, more homogeneous sub-blocks, thereby improving precision of treatment comparisons. This review synthesizes theoretical foundations and empirical evidence on the relative efficiency of IBDs versus CBDs across multiple crops and experimental settings.

Classification of Experimental Designs 2.1 Complete Block Designs (CBD)

In complete block designs, every block contains a complete set of all treatments. The major designs in this category are:

Completely Randomized Design (CRD): Treatments are assigned completely at random to experimental units with equal probability. CRD is suitable only when experimental units are completely homogeneous. The ANOVA structure is given in

Theoretical Framework of Key Incomplete Block Designs 3.1 Balanced Incomplete Block Design (BIBD)

A BIBD with parameters (v, b, r, k, λ) arranges v treatments in b blocks of k plots each (k < v) such that each treatment appears in exactly r blocks and each pair of treatments occurs together in exactly λ blocks. The fundamental parametric relations are: bk = vr and λ(v-1) = r(k-1). BIBD ensures equal precision for all treatment comparisons, but does not exist for all parameter combinations and often requires a large number of replications.

Partially Balanced Incomplete Block Design (PBIBD)

PBIBD relaxes the strict balance requirement of BIBD by allowing different pairs of treatments to have different concurrences (λ values). Treatments are arranged according to an association scheme with m classes where the number of i-th associates ni is constant for each treatment. PBIBDs reduce the number of replications required while maintaining reasonable precision for all comparisons, making them more practically feasible than BIBDs for many experimental settings.

Lattice Designs

Lattice designs were introduced by Yates (1936) as quasifactorial designs. The number of treatments must be a perfect square (s2) with block size s. Construction uses a standard s×s array where rows form the first replication and columns form the second. Additional replications are obtained by superimposing mutually orthogonal Latin squares. Lattice designs simple (2 reps), triple (3 reps), and balanced are classified by the number of replications. When the complete set of (s-1) mutually orthogonal Latin squares is used, the design becomes a balanced (s+1)-lattice equivalent to BIBD.

Alpha Lattice Design (ALD)

Proposed by Patterson and Williams (1976), Alpha Lattice Designs are cyclic resolvable incomplete block designs that exist for a much wider range of parameter combinations than traditional lattice or BIBD designs. For parameters r (replications), k (block size), and s (blocks per replication), the total number of treatments t = sk. Key features include existence for many (r,k,s) combinations where conventional lattices do not exist; efficient elimination of within-block heterogeneity; maximization of comparisons between genotypes in the same block; and suitability for large numbers of treatments in limited experimental areas. The ANOVA structure for ALD is presented in Table B below.

Variance Balanced Design (VBD)

VBD is specifically constructed for complete diallel cross systems. For p parents, the first block consists of all crosses involving the first parent with the remaining p-1 parents. The remaining p-1 blocks are obtained cyclically. Parameters: r=2 replications, k=p-1 block size. Sharma and Fanta (2010) proposed a VBD that is balanced and available for every number of parents. Ghosh and Ahuja (2017) formalized the relative efficiency formula comparing VBD with RCBD using the ratio of treatment variance estimators. The ANOVA structure for VBD is presented in Table C below.

Augmented Designs

Introduced by Federer (1956), augmented designs are used when testing material is limited and complete replication is not possible. They incorporate both test genotypes (unreplicated) and check varieties (replicated). Three forms exist: Augmented CRD (Design I), Augmented RCBD (Design II), and Augmented Latin Square (Design III). Error estimation is achieved through check variety replication within blocks, making these designs valuable for large germplasm screening programs.

Relative Efficiency: Concept and Measurement

Relative Efficiency (RE) quantifies how much more precise one experimental design is compared to another. It is expressed as the ratio of error variances or other precision measures. Values >100 (for EMS, S.Em, CV%) indicate the IBD is more efficient; for F-values, values <100 indicate IBD superiority. The gain in efficiency is calculated as RE% -100 (for EMS/S.Em/CV%) or 100 -RE% (for F-values).

Review of Literature

A systematic review of published studies demonstrates the consistent superiority of IBDs over CBDs across diverse crops, environments and experimental conditions. Nine key case studies are reviewed below.

Relative Precision of Experimental Designs

Singh (1995) studied the relative precision of various experimental designs (CRD, RBD, LSD, Simple Lattice, Triple Lattice, Balanced Lattice) across different block sizes. For block sizes of 5 and above, lattice designs consistently outperformed RCBD in terms of precision, especially for larger treatment numbers.

Alpha Lattice Design in Wheat

El-Shafi (2014) [6] evaluated 54 wheat genotypes in Alpha Lattice Design versus RCBD during two consecutive seasons (2011-12 and 2012-13). Alpha lattice design proved more efficient for most traits, with highest RE for number of kernels/spike (2.04) and spikes/m2 (1.51). The average RE across all traits was 1.14.

IBD, Alpha & Augmented Designs in Durum Wheat

Twenty-four durum wheat varieties were evaluated using RCBD, IBD, Alpha Lattice and Augmented Design. Alpha Lattice emerged as the most efficient (RE=147.1%), followed by IBD (118.7%) and Augmented (113.7%). Alpha design also achieved the highest adjusted R 2 (0.80) and lowest CV% (10.50%).

Alpha Lattice in Triticale

Twenty triticale genotypes were evaluated using Alpha Lattice and RCBD across two seasons (2017 and 2018) for eight agronomic characters. Alpha Lattice achieved higher relative efficiency for most traits, with average RE ranging from 91.08% (tillering general) to 135.50% (spike length). Grain yield showed an average RE of 126.69%. [15] evaluated 306 wheat genotypes using ALD and RCBD at CSKHPKV, Palampur, in Rabi 2021-22 under irrigated and rainfed conditions. Results showed marginal differences between ALD and RCBD for most traits in individual environments, with grain yield showing slight RE advantages (1.03 pooled). With very large treatment numbers, ALD's advantage is realized through better block homogeneity across the pooled analysis.

Discussion

The collective evidence from nine case studies spanning five crops (wheat, brinjal, oat, triticale, durum wheat [12, 15] reveal that efficiency gains accumulate across environments, making IBDs especially valuable in multienvironment trials. The reduction in error variance through smaller, more homogeneous blocks translates to smaller CV (%), lower SEm and higher F-values for treatment comparisons.

While RCBD remains popular (>95% usage in Gujarat surveys), its adequacy diminishes sharply when treatment numbers exceed 16. The practical advantages of ALD and VBD easy construction, flexible parameter requirements and straightforward analysis make them accessible alternatives. The comparable simplicity of analysis software for IBDs further removes barriers to their adoption.

Conclusion

This review conclusively establishes that Incomplete Block Designs (IBDs) are superior to Complete Block Designs for agricultural field experimentation involving large numbers of treatments or genotypes. VBD proved the most efficient design for diallel cross evaluation, achieving an average efficiency gain of 58.06% over RCBD across EMS, S.Em, CV% and F estimates in brinjal, while Alpha Lattice Design consistently outperformed RCBD across diverse crops and environments with efficiency gains ranging from 14% in wheat to 147% in durum wheat and Square Lattice designs showed notable advantages of 17-55% over RCBD for larger treatment sets. The consistent trend of error variance (RCBD > ALD > VBD) confirms the increasing effectiveness of tighter block control. Pooled analyses across environments further amplify these advantages, making IBDs indispensable for multi-environment trials with large germplasm sets. Accordingly, plant breeders and agricultural scientists are strongly advocated to replace RCBD with VBD and ALD when evaluating breeding

References

1. Abdelkawy RNF, Turbayev AZ, Soloviev AA. Evaluation of relative efficiency of alpha lattice design and cluster analysis of twenty spring triticale field experiments. Indian Journal of Agricultural Research. 2020;54(3):301. doi:10.18805/IJARe.A-466.
2. Anastasios K, Christos K. Evaluation of experimental designs in durum wheat trials. Biometrical Letters. 2015;52(2):105-114.
3. Campbell BT, Bauer PJ. Improving the precision of cotton performance trials conducted on highly variable soils of the southeastern USA coastal plain. Plant Breeding. 2007;126(6):622-627.
4. Dean A, Voss D, Draguljic D. Incomplete block designs. In: Design and analysis of experiments. Cham: Springer; 2017. doi:10.1007/978-3-319-52250-0_11.
5. El-Shafi MAA. Efficiency of classical complete and incomplete block design in yield trial on bread wheat genotypes. Research Journal of Agriculture and Biological Science. 2014;10(1):17-23.
6. Federer WT, Reynolds M, Crossa J. Combining results from augmented designs over sites. Agronomy Journal. 2001;93(2):389-395. doi:10.2134/agronj2001.932389x.
7. Ghosh DK, Ahuja S. On variance balanced designs. Computational Statistics and Applications. 2017;16(2):124-137.
8. Patterson HD, Williams ER. A new class of resolvable incomplete block designs. Biometrika. 1976;63(1):8392. doi:10.1093/biomet/63.1.83.
9. Popat R. Comparison of randomized complete block design and variance balanced design for diallel crosses [PhD thesis]. Anand: Anand Agricultural University; 2021.
10. Raza I, Masood MA. Efficiency of lattice design in relation to randomized complete block design in agricultural field experiments. Pakistan Journal of Agricultural Research. 2009;22(3-4):150-153.
11. Sanadya SK, Sood VK, Sawan K, Sharma G. Relative efficiency of alpha lattice design and randomized complete block design in oat breeding experiment. Journal of Cereal Research. 2022;14(2):27-32.
12. Sharma MK, Fanta S. Variance balanced designs for complete diallel cross. Journal of Agricultural Biotechnology and Sustainable Development. 2010;2(4):56-60.
13. Singh VP. Variability analysis in field experimentation: A case study on maize [PhD thesis]. Sardarkrushinagar: Gujarat Agricultural University; 1995.
14. Varma S, Chaudhary HK, Sood VK, Singh K, Dhillon KS, Kumar N. Comparative efficiency of randomized complete block design versus alpha lattice design in wheat breeding experiments. Himachal Journal of Agricultural Research. 2024;50(1):24-24.
15. Vekariya PR, Darji VB, Pandya MM. Relative efficiency of variance balanced design (VBD) and alpha lattice design (ALD) in diallel crosses of brinjal. International Journal of Research in Agronomy. 2025;8(1):10-14.
16. Yates F. A new method of arranging variety trials involving a large number of varieties. Journal of Agricultural Science. 1936;26(3):424-455.