Construction and a-Efficiency of Nested Partially Balanced Bipartite Block Designs.
NPBBBdesigns
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.
What the package does
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:
- the reduced information matrix
C = R - N K^{-1} N'of either classification (info_matrix); - the A-value of the design, that is, the sum of the variances of the
v1 * v2elementary test-versus-control contrasts (a_value); - the A-value of the A-optimal completely symmetric reference design (
a_value_optimal); - the block and sub-block A-efficiencies as the ratio of the two (
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.
Installation
# from a local source tarball
install.packages("NPBBBdesigns_1.0.0.tar.gz", repos = NULL, type = "source")
Quick start
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
The six construction methods
| 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, ....
References
Hedayat AS, Majumdar D (1984) A-optimal incomplete block designs for control-test treatment comparisons. Technometrics26, 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 Inference19, 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).
License
GPL-3.