In this paper, we introduce nested partially balanced bipartite block (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\text {NPBBB}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>NPBBB</mtext> </math></EquationSource> </InlineEquation>) designs for efficient comparison of test treatments with multiple control treatments under a nested block structure. Such designs are of practical importance in plant breeding and animal experiments where two sources of heterogeneity are present, with one blocking factor nested within the other. We formally define the class of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\text {NPBBB}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>NPBBB</mtext> </math></EquationSource> </InlineEquation> designs, derive necessary parametric conditions, and establish an A-optimality criterion for evaluating their efficiency. Several systematic construction methods are developed using partially balanced incomplete block designs, partially balanced bipartite block designs, nested partially balanced incomplete block designs, and nested balanced incomplete block designs. Each construction method is accompanied by a formal proof of parameter derivation. In addition, comprehensive catalogues of efficient <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text {NPBBB}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>NPBBB</mtext> </math></EquationSource> </InlineEquation> designs are provided for the parameter range with the number of control treatments <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$v_2 = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and replications of test treatments <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r_1 \le 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>≤</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>. Many of the constructed designs achieve A-optimality for both block and sub-block structures.

Nested Partially Balanced Bipartite Block Designs for Comparing Test Treatments with Multiple Controls

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2026年7月29日

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Vinayaka,
Rajender Parsad,
B. N. Mandal,
L. N. Vinaykumar

摘要

In this paper, we introduce nested partially balanced bipartite block ( $\text {NPBBB}$ $NPBBB$ ) designs for efficient comparison of test treatments with multiple control treatments under a nested block structure. Such designs are of practical importance in plant breeding and animal experiments where two sources of heterogeneity are present, with one blocking factor nested within the other. We formally define the class of $\text {NPBBB}$ $NPBBB$ designs, derive necessary parametric conditions, and establish an A-optimality criterion for evaluating their efficiency. Several systematic construction methods are developed using partially balanced incomplete block designs, partially balanced bipartite block designs, nested partially balanced incomplete block designs, and nested balanced incomplete block designs. Each construction method is accompanied by a formal proof of parameter derivation. In addition, comprehensive catalogues of efficient $\text {NPBBB}$ $NPBBB$ designs are provided for the parameter range with the number of control treatments $v_2 = 2$ $v_{2} = 2$ and replications of test treatments $r_1 \le 10$ $r_{1} \leq 10$ . Many of the constructed designs achieve A-optimality for both block and sub-block structures.