John P. Morgan

Optimality and construction of nested row and column designs

Abstract Conditions and constructions are given for optimal nested row and column designs. The new designs are compared to a class of balanced incomplete block designs with nested rows and columns. Comparisons are drawn to Youden designs and the non-regular case discussed.

Some series constructions for two-dimensional neighbor designs

Abstract Terraces and sequencing are used to construct bordered, two-dimensional neighbor designs in which row, column, and diagonal neighbors are accounted for. Columns are generated from terraces, then juxtaposed (with appropriate borders) in such a way that row and diagonal neighbors are balanced or nearly balanced. For v treatments, v × 2v side-bordered designs are constructed with row, column, and near diagonal balance, where v ≢ 0 (mod 3). Full bordering completes the balance, which for even v is directional. Other series include v × v non-directional designs for v ≡ 2 or 4 (mod 6), and several series of (v + 1) 2 × 2v designs for v odd.

Optimality of designs with generalized group divisible structure

Abstract The generalized group divisible designs with s groups, or GGDD( s )s, are defined in terms of the elements of the information matrix, rather than in terms of the elements of the concurrence matrix as has been done previously. This definition extends the class of designs to include nonbinary members, and allows for broader optimality results. Several sufficient conditions are derived for the designs to be E- and MV-optimal. It is further shown how augmentation of additional blocks to certain GGDD( s )s produces infinite series of other nonbinary, unequally replicated E- and MV-optimal block designs. Where nonbinary designs are found, they can be preferable to binary designs in terms of interpretability as well as one or more formal optimality criteria.

Weighted Optimality in Designed Experimentation

An optimality framework is developed for designing experiments in which not all treatments are of equal interest, such as those including an established control. Differential interest in treatments is formalized by assignment of weights, incorporated into optimality measures through a weighted version of the information matrix. All conventional measures of design efficacy are shown to have weighted analogs. The properties of weighted measures are explored, some general theory is developed, and weighted optimal designs are determined for unblocked experimentation. This new approach includes “test treatments versus control” experiments as a special case. Supplementary materials for the article are available online.

The completely symmetric designs with blocksize three

Abstract The proper block design setting consists of v treatments to be assigned to b blocks, each block consisting of k experimental units. A design for this setting is said to be a completely symmetric design, or CSD, if its bottom stratum information matrix for estimation of treatment contrasts is constant on the diagonal and constant off the diagonal. Write bk=vr+p for some integer p with 0⩽p⩽v−1. If p=0, the CSDs of maximal conceivable trace are the balanced incomplete block designs. This paper constructs all maximal trace CSDs for k=3 and p=1,2,3. All of these designs are shown to be optimal under the E-criterion, and all but two under the MV-criterion.

A Class Of Neighbor Balanced Complete Block Designs and their Efficiencies for Spatially Correlated Errors

Two dimensional designs for blocksize (ν/2) × 2 and an even number ν of treatments, which are neighbor balanced and which have an additional property on the design formed by the corner plots, are studied. Efficiency lower bounds for these resolvable row-column designs are obtained under two different error covariance structures: the doubly geometric, for which the correlations decay rapidly, and the autonormal, which exhibits a slower rate of decay. Two infinite series of the designs are constructed.

On pairs of Youden designs

Abstract It is well known that generalized Youden designs, or GYDs, enjoy a variety of optimality properties. Not being maximum trace designs, b copies of a non-regular GYD will not be optimum for b sufficiently large, opening the question of whether such a set will be so for any b. This paper explores the E-behavior of b = 2 non-regular GYDs. A general E-efficiency bound is derived and the E-optimality of a particular series is proven. That pairs of non-regular GYDs are not always E-optimal is shown by a counterexample.

Polycross designs with complete neighbor balance

SummaryPolycross designs for n clones in n2 replicates, composed of n n×n squares, are presented, n being any positive integer. The method depends on whether n is odd or even, and for even n the squares are Latin. In either case, each clone has every other clone as a nearest neighbor exactly n times in each of the four primary directions (N, S, E, W) and n−2 times in each of the four intermediate directions (NE, SE, SW, NW). Also, each clone has itself as nearest neighbor n−1 times in each intermediate direction.

Orthogonal sets of balanced incomplete block designs with nested rows and column

Abstract Known series of balanced incomplete block designs with nested rows and columns are used to find orthogonal sets of these designs, producing main effects plans in nested rows and columns. Two infinite series are so constructed and shown to be universally optimum for the analysis with recovery of row and column information, a benefit produced by the additional higher strata orthogonality they enjoy. One of these series achieves orthogonality with just v − 1 replicates of v treatments, fewer than required by Latin squares.

Constructions for Generalized Group Divisible Designs in Settings Admitting Symmetry

SYNOPTIC ABSTRACT Several methods for constructing generalized group divisible designs with two classes and blocksize three are presented. Two constructions of completely symmetric designs are also given. All of the constructed designs achieve a lower bound on the diagonal elements of their information matrices required for E- and MV-optimality arguments; for most of the designs these optimalities are established. Difference techniques and the finite fields play key roles, as do idempotent and half-idempotent Latin squares.