a_value_optimal

Computes the smallest attainable A-value (sum of variances of the
 \(v_1 v_2\) test-versus-control elementary contrasts) within the class of
 connected (sub-)block designs that are completely symmetric in the test and in
 the control treatments, for a control replication \(r_0\). This is the
 benchmark against which the A-efficiency is measured. The expression is the
 nested-design analogue of the Hedayat-Majumdar / Stufken optimal A-value; see
 Vinayaka et al. (2026).

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) <doi:10.1080/03610929608831743> and Vinayaka et al. (2024) <doi:10.1080/03610926.2023.2251623>. These designs are particularly suited to agricultural, animal husbandry, industrial, and clinical trials involving multiple standard checks under nested experimental conditions, such as multi-environment trials where field heterogeneity (blocks) and within-field variation (sub-blocks) must be controlled simultaneously.

Vinayaka 

NPBBBdesigns

Construction and a-Efficiency of Nested Partially Balanced
Bipartite Block Designs

Rajender Parsad

B.N. Mandal

L.N. Vinaykumar

Gopalareddy Krishnappa

a_value_optimal function

<dl><dt>v1</dt>
<dd>Number of test treatments.</dd>
<dt>v2</dt>
<dd>Number of control treatments.</dd>
<dt>b</dt>
<dd>Number of blocks (or sub-blocks) in the classification.</dd>
<dt>k</dt>
<dd>Block (or sub-block) size.</dd>
<dt>r0</dt>
<dd>Replication of each control treatment in the classification.</dd></dl>

Arguments

A-Value of the A-Optimal Completely Symmetric Reference Design — a_value_optimal

<dl>

<dt>v1</dt>
<dd>Number of test treatments.</dd>


<dt>v2</dt>
<dd>Number of control treatments.</dd>


<dt>b</dt>
<dd>Number of blocks (or sub-blocks) in the classification.</dd>


<dt>k</dt>
<dd>Block (or sub-block) size.</dd>


<dt>r0</dt>
<dd>Replication of each control treatment in the classification.</dd>

</dl>

a_value_optimal: A-Value of the A-Optimal Completely Symmetric Reference Design

Description

Usage

Value

Arguments

References

Examples