Donald A. Preece

Some Row and Column Designs for Two Sets of Treatments

This paper provides some new, simple designs for the simultaneous estimation of the effects of two non-interacting sets of treatments, with two-way elimination of heterogeneity. The designs are suitable for use when a new set of treatments is to be applied to experimental material which may still be affected by an eailier set of treatments. One type of design, obtained by omitting a row or rows from a GraecoLatin square of order t, has each set arranged in a t X k Youden square design, t > k; this type is 0: OT: OTT, in the notation of Pearce [1963], and includes series of designs for which t is a prime of the form (4n + 3) and k = l(t 1 1). Closely related X L(t zL 1) 0: OP: OPP designs, for which t is a prime of the form (4n + 1), are also described and discussed. A further type of design, first discussed by Clarke [1963], is for one set of t treatments arranged in a t X k Youden square design, and one set of k treatments; 7 X 3 and 11 X 5 designs of this 0: OT: TOO type are given for the first time.

Round-dance neighbour designs from terraces

In a round-dance neighbour design, an odd number v of objects is arranged successively in (v-1)/2 rings (circular blocks) such that any two of the objects are adjacent to one another in exactly one ring. A round-dance neighbour design is also a Hamiltonian decomposition of the complete graph on v vertices. We show how such designs can be constructed from terraces, which are building blocks for row-complete Latin squares. A round-dance neighbour design is equivalent to a Tuscan square in which the reverse of each row is also a row. Terraces for the cyclic group of order n are used to construct elegantly patterned round-dance neighbour designs for n2^m+1 objects for any positive integer m.

Nested balanced incomplete block designs

Abstract If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v,b 1 ,r,k 1 ) are each partitioned into sub-blocks of size k 2 , and the b 2 =b 1 k 1 /k 2 sub-blocks themselves constitute a BIBD with parameters (v,b 2 ,r,k 2 ) , then the system of blocks, sub-blocks and treatments is, by definition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k 1 =2k 2 =4 . Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are defined and illustrated. A first-ever detailed table of NBIBDs with v⩽16 , r⩽30 is provided; this table contains many newly discovered NBIBDs.

Power-sequence terraces for Z n where n is an odd prime power

A power-sequence terrace for Zn is defined to be a terrace which can be partitioned into segments one of which contains merely the zero element of Zn, whilst each other segment is either (a) a sequence of successive powers of an element of Zn, or (b) such a sequence multiplied throughout by a constant. Many elegant families of such Zn terraces are constructed for values of n that are odd prime powers. The discovery of these families greatly increases the number of known constructions for terraces for Zn. Tables are provided to show clearly the constructions available for each prime power n satisfying n < 300.

Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

Abstract A power-sequence terrace for Z n is a Z n terrace that can be partitioned into segments one of which contains merely the zero element of Z n whilst each other segment is either (a) a sequence of successive powers of an element of Z n , or (b) such a sequence multiplied throughout by a constant. If n is odd, a Z n terrace (a1,a2,…,an) is a narcissistic half-and-half terrace if ai−ai−1=an+2−i−an+1−i for i=2,3,…,(n+1)/2. Constructions are provided for narcissistic half-and-half power-sequence terraces for Z n with n=pqt where p and q are distinct odd primes and t is a positive integer. All the constructions are for terraces with as few segments as possible. Attention is restricted to constructions covering values of n with n=pqt and n

On balanced incomplete-block designs with repeated blocks

Balanced incomplete-block designs (BIBDs) with repeated blocks are studied and constructed. We continue work initiated by van Lint and Ryser in 1972 and pursued by van Lint in 1973. We concentrate on constructing (v,b,r,k,@l)-BIBDs with repeated blocks, especially those with gcd(b,r,@l)=1 and r@?20. We obtain new bounds for the multiplicity of a block in terms of the parameters of a BIBD, and improvements to these bounds for a resolvable BIBD. This allows us to answer a question of van Lint about the sufficiency of certain conditions for the existence of a BIBD with repeated blocks.

Tight single-change covering designs with v = 12, k = 4

Abstract Standardised tight single-change covering designs with v = 12, k = 4 are enumerated and classified. There are 2554 of them, and these fall into 566 sets such that, within any set, the designs can be regarded as minor variants of one another. The sets pair off naturally, to give 283 classes of the designs. If any one design in a class is row-regular (or element-regular), then all the designs in the class are row-regular (or element-regular). Of the 283 classes, just 10 comprise row-regular designs; these 10 include the only one of the 283 classes that comprises element-regular designs. Representative members of the 10 row-regular classes are tabulated. Other properties of the designs are discussed. An indication is given of how each of the 10 representative row-regular designs can readily be converted into a row-regular tight single-change covering design with v = 13, k = 4.

Double Youden rectangles: an update with examples of size 5×11

Abstract The literature of double Youden rectangles (DYRs) is reviewed, to indicate what is known about their existence and construction. The discovery is reported of some 5 x 11 DYRs and of some similarly generated 6 x 11 DYRs.

Perfect Graeco-Latin balanced incomplete block designs (pergolas)

Abstract A PERfect GraecO-LAtin balanced incomplete block design (PERGOLA) is a block design for two sets of treatments, where (a) each set is arranged relative to the blocks in a balanced incomplete block design (BIBD), (b) each set is arranged relative to the other in a symmetric BIBD, and (c) the overall arrangement is such that there is adjusted orthogonality between the two sets. The currently small literature of pergolas is reviewed, and the topic is shown to be rich in combinatorial interest and unsolved problems. Isomorphism, automorphisms and duality are defined for pergolas, and matters of existence are discussed. A first-ever extensive table of pergolas with r ⩽ 20 is presented. For each of many of the 66 parameter-sets covered, the Table gives a selection of non-isomorphic pergolas, perhaps based on a selection of non-isomorphic BIBDs for that parameter-set.

Types of factor in experiments

Abstract Much statistical writing and practice is based on the notion that all factors in comparative experimentation are of one or other of two types, but there is conflict between the different dichotomies produced by different authors. The present paper argues that all such dichotomies (fixed versus random, treatment versus block, randomised versus unrandomised, etc.) are inadequate for categorising the factors that can be present in a single experiment. At the very least, a trichotomy is needed as a logical basis for designing and analysing individual experiments, the three basic types of factor being (A) treatment factors whose levels can be assigned at random to plots; (B) classification factors whose effects (and perhaps interactions with other factors) are to be studied; and (C) block factors. This trichotomy is a special case of one described by Cox (Int. Statist. Rev. 52 (1984) 1–31). Before a factor can be identified as belonging to a particular type, unaccustomed care may be needed in identifying and characterising the plot or experimental unit used in the experiment. A factor involving time may be of type (A) or (B), or may be of a further type that must be distinguished separately. The paper illustrates its arguments with examples from agricultural, industrial, medical and behavioural experimentation.