#### Date of Award

Summer 1996

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

John P. Morgan

#### Committee Member

Narasinga R. Chaganty

#### Committee Member

Ram Dahiya

#### Committee Member

Edward Markowski

#### Committee Member

Dayanand N. Naik

#### Abstract

This thesis is an investigation of the optimality and construction problems attendant to the assignment of v treatments to experimental units in b blocks of size k, paying special attention to settings for which equal replication of the treatments is not possible. The model is that of one way elimination of heterogeneity, in which the expectation of an observation on treatment *i* in block *j* is *T _{i}* + βj (treatment effect + block effect), where

*T*and βj are unknown constants, 1

_{i}*≤ i ≤ v*and 1

*≤ j ≤ b*. All observations are assumed to be uncorrelated with same variance.

The generalized group divisible design with s groups, or GGDD(s), is defined in terms of the elements of the information matrix, instead of in terms of the elements of the concurrence matrix as done by Adhikary (1965) and extended by Jacroux (1982). This definition extends the class of designs to include non-binary members, and allows for broader optimality results. Some sufficient conditions are derived for GGDD(s) to be E- and MV-optimal. It is also shown how augmentation of addition blocks to certain GGDD(s)s produces other nonbinary, unequally replicated E- and MV-optimal block designs. Where nonbinary designs are found, they are generally preferable to binary designs in terms of interpretability, and often in terms of one or more formal optimality criteria as well.

The class of generalized nearly balanced incomplete block designs with maximum concurrence range *l*, or NBBD(l), is defined. This class extends the nearly balance incomplete block designs as defined by Cheng & Wu (1981), and the semi-regular graph designs as defined by Jacroux (1985), to cases where off-diagonal entries of the concurrence matrix differ by at most the positive integer *l*. Sufficient conditions are derived for a NBBD(2) to be optimal under a given type-I criterion. The conditions are used to establish the A- and D-optimality of an infinite series of NBBD(2)s having unequal numbers of replicates. Also, a result from Jacroux (1985) is used to establish the A-optimality of a new series of NBBD(1)s.

Several methods of construction of GGDD(s)s are developed from which many infinite series of designs are derived. Generally these designs satisfy the obtained sufficient conditions for E- and MV-optimality.

Finally, in the nested row-column setting, the necessary conditions for existence of 2 x 2 balanced incomplete block designs with nested rows and columns (BIBRCs) are found to be sufficient. It is also shown that, sufficient for a BIBRC with p=q to generally balanced, is that the row and column classifications together form a balanced incomplete block design, as does the block classification. All of the 2 x 2 BIBRCs are constructed to have this property.

#### DOI

10.25777/jwxa-c533

#### ISBN

9780591048698

#### Recommended Citation

Srivastav, Sudesh K..
"Optimality and Construction of Designs with Generalized Group Divisible Structure"
(1996). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/jwxa-c533

https://digitalcommons.odu.edu/mathstat_etds/66