Construction and evaluation of nested partially balanced bipartite block (NPBBB) designs for comparing a set of test treatments with a set of control treatments under a nested (blocks within blocks) structure. Six systematic construction methods are provided: composing partially balanced bipartite block designs with nested balanced incomplete block designs; augmenting nested partially balanced incomplete block designs with controls; merging rows of group-divisible nested designs; direct construction from group-divisible schemes; and expansion of partially balanced incomplete block designs (Vinayaka et al. 2026: In press). The A-efficiencies of the block and sub-block classifications are computed against the A-optimal completely symmetric reference design, following the test-versus-control optimality framework of Gupta and Parsad (1996)
Construction and A-efficiency of nested partially balanced bipartite block (NPBBB) designs for comparing a set of test treatments with a set of control treatments under a nested (blocks within blocks) structure.
The package accompanies the manuscript Nested partially balanced bipartite block designs for comparing test treatments with multiple controls by Vinayaka, R. Parsad, B. N. Mandal and L. N. Vinaykumar (2026). It provides the six construction methods described there, together with an exact evaluation of the A-efficiency of both the block and the sub-block classifications.
An NPBBB design has two nested systems of grouping: blocks, and sub-blocks within blocks. The test treatments follow a partially balanced (two-associate) association scheme, while the control treatments are completely symmetric and appear so that every test-versus-control comparison is estimated with high precision. For such a design the package computes:
C = R - N K^{-1} N' of either classification
(info_matrix);v1 * v2
elementary test-versus-control contrasts (a_value);a_value_optimal);npbbb_efficiency).The reference A-value is the nested-design analogue of the optimal A-value for test-versus-control designs studied by Hedayat and Majumdar (1984) and Stufken (1988). An efficiency of 1 means the design is A-optimal for that classification.
# from a local source tarball
install.packages("NPBBBdesigns_1.0.0.tar.gz", repos = NULL, type = "source")
library(NPBBBdesigns)
# Method 4: direct construction from a 4 x 3 group-divisible scheme,
# two controls. A-optimal for both classifications.
d <- construct_method4(m = 4, n = 3, v2 = 2)
d
n_units(d) # total number of experimental units
# Evaluate the efficiency of any design supplied as block / sub-block matrices
eff <- npbbb_efficiency(d$block_design, d$subblock_design, v1 = d$v1, v2 = d$v2)
eff
| Function | Method | Parent design | Notes |
|---|---|---|---|
construct_method1 |
Compose a PBBB design with an NBIB design | PBBB + NBIB | general v2; high replication, infinite families |
construct_method2 |
Augment an NPBIB design with controls | NPBIB | general v2; needs block size raised by v2 |
construct_method3 |
Merge rows of a group-divisible NPBIB design | GD-NPBIB | general v2; may create degenerate sub-blocks (Note 3.1) |
construct_method4 |
Direct construction from a group-divisible scheme | GD scheme | general v2; A-optimal, no redundancy |
construct_method5 |
Expand size-4 blocks of a PBIB design | PBIB, k = 4 |
v2 = 2 only |
construct_method6 |
Expand size-2 blocks of a PBIB design | PBIB, k = 2 |
v2 = 2 only |
Each constructor returns an object of class "npbbb" whose print method shows
the design parameters, the two A-efficiencies and the full physical layout of
the blocks and sub-blocks, with the controls displayed as 0_1, 0_2, ....
Hedayat AS, Majumdar D (1984) A-optimal incomplete block designs for control-test treatment comparisons. Technometrics 26, 363-370.
Stufken J (1988) On bounds for the efficiency of block designs for comparing test treatments with a control. Journal of Statistical Planning and Inference 19, 361-372.
Vinayaka, Parsad R, Mandal BN, LN Vinaykumar (2026) Nested partially balanced bipartite block designs for comparing test treatments with multiple controls. Journal of Statistical Theory and Practice. (In press).
GPL-3.