OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION

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1 Statistica Sinica 0 (010), OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION J. P. Morgan Virginia Tech Abstract: Amongst resolable incomplete block designs, affine resolable designs are optimal in many conentional senses. Howeer, different affine resolable designs for the same numbers of treatments and replicates, and the same block size, can differ in how well they estimate elementary treatment contrasts. An aberration criterion is employed to distinguish the best of the affine resolable designs for this task. Methods for constructing the best designs are detailed and an extensie online catalog is compiled. Key words and phrases: Affine resolable block design, design optimality, orthogonal array, pairwise ariance aberration. 1. Introduction An oft-sought property of incomplete block designs is resolability: an incomplete block design for treatments in blocks of size k (< ) is resolable if the blocks can be partitioned into sets containing each treatment exactly once. The sets of this partition may be used to accommodate a second blocking factor, orthogonal to treatments, and containing the first. Quite naturally, the sets, or large blocks, are termed replicates, the number of which is denoted by r. Examples of the use of resolable designs are abundant and may be found in seeral of the papers cited forthwith. One special class of resolable designs has receied special attention, for good reason. Suppose any two blocks from distinct replicates of a resolable design intersect in the same number of treatments, call this number µ. Such a design is said to be affine resolable (Bose (19)). Using s to denote the number of small blocks per replicate in a resolable design, and b the total number of small blocks, then = ks and b = rs. For an affine resolable design, the number µ is necessarily µ = k/s, and so = µs and k = µs. That k is a multiple of s is the limitation imposed by affineness relatie to all resolable designs. The adantage gained is a host of ery nice statistical properties. Bailey, Monod and Morgan (1995) established Schur-optimality of affine resolable designs amongst all resolable designs with the same, r, and k. Thus an affine resolable design minimizes (i) the aerage ariance of any complete set

2 7 J. P. MORGAN of orthonormal treatment contrasts, which is proportional to the aerage ariance of the ( 1)/ pairwise treatment contrasts (i.e., is A-optimal); (ii) the largest ariance oer all normalized treatment contrasts (i.e., is E-optimal); (iii) the olume of the confidence ellipsoid for any 1 orthonormal treatment contrasts (i.e., is D-optimal). Moreoer, the treatment contrasts estimation space is especially simple for an affine resolable design, there being just two canonical efficiency factors 1 and (r 1)/r. Gien these excellent statistical properties, one might think that eery affine resolable design with the same, r, and k should be equally efficacious. Inspection of the ariances of the elementary treatment contrasts, howeer, shows this notion to be false, for the distribution of these ariances depends on the particular affine resolable design selected. The purpose of this paper is to identify the best of the affine resolable designs for estimation of elementary contrasts. Section formalizes the notion of best, proides two representations of affine resolable designs useful in finding best designs, and constructs a family of best designs based on orthogonal Latin squares. Section 3 proides a full solution in up to fie replicates haing two blocks per replicate. The known best designs for up to 00 treatments are compiled in an online catalog, as discussed in Section. Section 5 includes additional discussion and examples.. Minimum PV Aberration Though not generally partially balanced, affine resolable designs share an important property with the partially balanced incomplete block designs. This property, stated next as a lemma, is the key to ordering the designs in terms of how well they estimate elementary contrasts. For a gien affine resolable design, let λ ij be the number of small blocks containing both treatments i and j. For any resolable design write p ij for the pairwise ariance Var( τ i τ j ). Lemma 1. (Bailey, Monod and Morgan (1995)) The pairwise ariance p ij when using an affine resolable design is a linear function of λ ij. Specifically, where σ is the plot ariance. p ij = [r λ ij + k(r 1)] σ, (.1) kr(r 1) The best affine resolable design for estimating elementary treatment contrasts is one that, in some appropriate sense, makes the ( 1)/ quantities p ij in (.1) small. Lemma 1 says that, equialently, the quantities λ ij should be made large in a correspondingly appropriate sense. Since eery resolable design has i j>i λ ij = bk(k 1)/, the aerage pairwise ariance for eery affine

3 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 717 resolable design is the same (and, as mentioned in Section 1, is minimal oer all resolable designs). If the collection of p ij (or λ ij ) for an affine design is thought of as a uniform distribution on its points, then the problem is one of selecting among distributions with the same mean. The statistically meaningful route is to consider the tails of these distributions. Specifically, minimizing the number of poorly estimated elementary contrasts is achieed by selecting a design for which the left tail of its λ ij distribution is dominant in a natural sense. This motiates the following definition. Definition 1. For any affine resolable design d, let η du = (i, j) : i < j and λ ij = u and write η d = (η d0, η d1,..., η dr ). Design d 1 is said to hae smaller pairwise ariance aberration (shortly, PV-aberration) than design d if, for some t, η d1 t < η d t and η d1 u = η d u for u < t. If no affine design has smaller PVaberration than d, then d has minimum PV-aberration. If a design has minimum PV-aberration, then it minimizes the maximal p ij, that is, it is MV-optimal amongst all affine resolable competitors. Minimal PVaberration is generally stronger than MV-optimality, howeer, for it examines more than just the largest p ij. When there are seeral MV-optimal designs, minimizing PV-aberration selects among them according to the next largest pairwise ariance, then the next, and so on, sequentially on the ordered p ij. The task of determining the best affine resolable design for pairwise comparisons has been translated into a study of the η du, beginning with η d0. Needed now is a description of affine resolable designs that lends itself to ealuating these quantities. Two such descriptions will be gien here, beginning with the sometimes useful connection to orthogonal arrays. Bailey, Monod and Morgan (1995) constructed affine resolable designs based on orthogonal arrays of strength two as follows. Begin with an orthogonal array OA(, r, s) haing rows, r columns, and s symbols in each column, such that the rows of any two columns produce the s ordered pairs of symbols µ times each; necessarily = µs. Placing treatment i in block m of replicate q if and only if the m th symbol occurs in row i, column q, produces an affine resolable design (, r, k) for k = /s. This process is reersible, i.e., strength two orthogonal arrays and affine resolable designs are equialent combinatorial objects (see Shrikhande and Bhagwandas (1969) and Morgan (1996)). Now λ ij is the number of columns in the orthogonal array for which rows i and j share the same symbol. These numbers are the basis for the power moments of a fractional factorial design as defined by Xu (003). Good factorial designs are found by sequentially minimizing the power moments (e.g., Theorem of Xu (003)); in a broad sense this says to make the λ ij small. Thus the orthogonal arrays sought here are quite different from those pursued in the fractional factorial literature.

4 71 J. P. MORGAN Rep #1 Rep # S 1 S.. S s S s+1 S s+. S s... S s s+1 S s s+. S s S 1 S s+1. S s s+1 S S s+. S s s+... S s S s. S s Figure 1. First two replicates of an arbitrary affine resolable design. Rep #1 S 1 S S 3 S Rep # S 1 S 3 S S Rep #3 S 11 S 1 S 31 S 1 S 1 S S 3 S Rep # Rep #5 S 111, S 11 S 11, S 1 S 311, S 31 S 11, S 1 S 11, S 1 S 1, S S 31, S 3 S 1, S S 1111, S 111, S 111, S 11 S 111, S 11, S 11, S 1 S 3111, S 311, S 311, S 31 S 111, S 11, S 11, S 1 S 111, S 11, S 11, S 1 S 11, S 1, S 1, S S 311, S 31, S 31, S 3 S 11, S 1, S 1, S Figure. Affine resolable design for s = with 5 replicates. The second description, here called the standard description, takes direct adantage of the block intersection property. Fix any ordering of the replicates, then the first two replicates produce a partition of the treatments into sets S 1, S,..., S s, each set of size µ (see Figure 1). Replicate x > may then be described in terms of subsets of sets of treatments appearing in replicate x 1, as follows: for e = 1,..., s and x = 3,..., r, the set S e m1 m m x to be the subset of S e m1 m m x 3 that appears in block m x of replicate x. This is displayed for fie replicates with s = in Figure where, for example, S 11 is the subset of S 1 appearing in the first block of replicate three, S 11 is the subset of S 11 appearing in the second block of replicate four, and S 111 is the subset of S 11 in the first block of replicate fie. Denote the number of treatments in S e m1 m m x by e m1 m m x 0. The two descriptions proide two paths for attacking the PV-aberration problem. The standard description is employed for the remainder of this section and throughout Section 3, proiding a common framework for all of the results obtained. Some of these, though not all, can also be deried with roughly equialent effort working with the OA formulation and OA identities. The OA

5 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 719 description is taken up again in Section, allowing best designs to be plucked from existing OA enumerations. The standard description admits useful formulae for the η du. Write m = (m 1,..., m r ) where each m i {1,..., s}. Starting with s = as an example, if m and m differ in eery coordinate, then members of S 1m hae neer occurred with members of S m, and members of S m hae neer occurred with members of S 3m. Obsere that η d0 simply counts the treatment pairs formed by each member of S 1m with each member of S m, and each member of S m with each member of S 3m. That is, η d0 = ( 1m m + m 3m ), the sums being oer all m and m such that the Hamming distance between m and m is h(m, m ) = r. Extending this perspectie, for any subscript e, let B(e) be the collection of subscripts for the sets among S 1,..., S s that are contained in a small block with S e in either of the first two replicates (other than e itself). For instance, B(1) = {,..., s} {s + 1, s + 1, (s 1)s + 1} (see Figure 1). Let A(e) be all other subscripts (other than e itself). Following the same reasoning for general s demonstrated for s = aboe gies η d0 = 1 e e A(e) m,m : h(m,m )=r em e m, (.) which is a special case of this more general expression for u = 0, 1,..., r 1: η du = 1 [ em e m + em e m e e + e e A(e) m,m : h(m,m )=r u m,m : h(m,m )=r u em em e B(e) m,m : h(m,m )=r 1 u ]. (.3) m em The number of treatment pairs in a block in eery replicate is η dr = e ( em 1)/. With these expressions in hand, methods for constructing minimum PVaberration designs can be obtained. A general method based on sets of mutually orthogonal Latin squares (MOLS) is gien next. Let L 1,..., L r be a set of r MOLS of order s, and let L 0 be the s s array whose entries are the s sets of size µ exactly as displayed in the first replicate of Figure 1. The first two replicates of an affine resolable design are as displayed in Figure 1. For each y = 1,..., r, block m of replicate y+ contains the sets of L 0 that are in the same cells as the m th symbol of L y. It is easy to see that this produces an affine resolable design with r replicates, since any two blocks from different replicates intersect in exactly one of the sets S 1,..., S s.

6 70 J. P. MORGAN Not obious, but proen next, is that this design, call it d, has minimum PVaberration. Examples of d are gien in the first paragraph of Section 3 and by Example 1 of Section 5. Theorem 1. The affine resolable design d has minimum PV-aberration. Proof. It will be shown that only designs of the form d minimize η d0, and that η d is the same ector regardless of the choice of orthogonal Latin squares. As always the first two replicates of any affine resolable design are as displayed in Figure 1. To begin, consider r = 3. Then m in S em is a singleton and η d0 is η d0 = 1 em e m = 1 em ( e e m) e e A(e) m m m e e A(e) m = 1 em (µ e m) = s (s 1) µ 1 ( ) em e m e e A(e) m e m e A(e) = s (s 1) µ 1 ( em (s 1)µ 1 ) e m (by affineness of reps 1 and 3) (s 1)(s ) = µ + 1 e m m e e B(e) e B(e) em e m. (.) From (.), η d0 attains its minimum alue if and only if em e m = 0 for eery m {1,..., s}, e {1,..., s }, and e B(e). That is, η d0 is minimized if and only if any two treatments from distinct sets and in the same block in replicates one or two, are in different blocks in replicate three. Now if the treatments in any set S e are not all in the same block in replicate three, then the minimum cannot be attained, for there are then at most s blocks into which treatments from the other s 1 sets occurring with S e in replicate one can be placed. Thus each block in replicate three must consist of s of the sets S 1,..., S s, and e B(e) S e and S e are in different blocks in replicate three. This says precisely that the minimum for η d0 is uniquely attained when the blocks of replicate three are disjoint transersals of L 0, that is, when they are formed as in d. For a design with two replicates the number of treatment pairs that hae not occurred in a block is η d0 = (s 1) µ/, and for three replicates the minimum PV-aberration design has η d0 = (s 1)(s )µ/, a decrease of (s 1)µ/ due to the third replicate. It is easy to see that any choice of L 1 to build d gies the same ector η d, which is ((s 1)(s )µ/, 3(s 1)µ/, 0, s µ(µ 1)/). Thus for three replicates, only designs of form d hae minimum PV-aberration. Moreoer, it is now clear that for r 3, the absolute minimum of η d0 is achieed if and only if, relatie to the first two replicates, each of replicates

7 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 71 3,..., r independently decreases η d0 by the maximal amount of (s 1)µ/. This happens if and only if (i) for each of these replicates the blocks are disjoint transersals of L 0, and (ii) no two of the sets S 1,..., S s occur together in more than one small block. Property (i) says that the replicates correspond to r Latin squares as in the construction for d, and property (ii) says those Latin squares are orthogonal. Thus minimization of η d0 requires a design of form d. Regardless of the MOLS L 1,..., L r employed to build d, ( (s 1)(s r + 1) (s 1)r η d = µ, µ, 0,..., 0, 1 ). (µ 1) This is because any two treatments in the same set S e occur together in one small block in eery replicate, and any other two treatments, from sets S e and S e say, occur together in no or one small block as S e and S e do the same. This completes the proof. 3. Minimum PV-Aberration with Two Blocks per Replicate Affine resolable designs with two blocks per replicate are shown for up to fie replicates, in the standard representation, in Figure. The block intersection number is µ = /, and so must be a multiple of. This section will determine the minimum PV-aberration designs for s = and r 5. For r = the best (and only) affine design is the first two replicates in Figure. To this add the replicate Rep #3 S S 1 (3.1) S 3 S to get the unique minimum-pv aberration design in three replicates, haing η d0 = 0. In terms of the third replicate in Figure, this choice results from selecting set sizes 11 = 1 = 0 and 1 = 31 = µ. This is an example of design d of Theorem 1. Affine resolability places a number of restrictions on the set sizes em. These restrictions may be written as a collection of linear equations, any solution to which specifies an affine resolable design, so long as that solution consists entirely of nonnegatie integers. For instance, there are eight set sizes for the third replicate in Figure, but in fact only one is linearly independent: all designs are specified by all alues of 11 {0,..., µ/}. While the numbers of sets, and consequently the number of independent set sizes, grow with r, they are still manageable for r 5. Minimum PV-aberration designs can be determined for

8 7 J. P. MORGAN these cases by soling the equations with the additional restriction that η d0 be minimized. When there are multiple solutions, these are compared on η d1, then η d, and so on. Details of this approach are presented in the subsections below coering four and fie replicates Four replicates with two blocks per replicate For four replicates there are set sizes 11, 1,..., and 111, 11,..., (cf. Figure ). Linear restrictions on these quantities arise from the basic properties of any affine resolable design for s = : each replicate contains all treatments, each block contains / treatments, and any two blocks from different replicates intersect in / treatments. Let B fg be the g th block in replicate f. As preiously mentioned, 11 determines all of the em for the third replicate, since = B 11 B 31 = = 11 = B 1 B 31 = = 11 (3.) = B 31 = e ( ) ( e1 = ) = 11, and e1 + e = / for e = 1,...,. For the fourth replicate, in addition to em = em em1 for e = 1,..., and m = 1,, there are four independent restrictions specified by / = B 11 B 1 = B 1 B 1 = B 31 B 1 = B 3 B 1. These yield 1 = = = = (3.3) There are thus fie free ariables 11, 111, 11, 11, 311. The determined ariables aboe satisfy 0 1 = 11, = 11, = 11, and 0 1 = / 11. Thus the free ariables, in addition to being nonnegatie integers, must satisfy , the last line with no loss of generality, and likewise 11 /. (3.)

9 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 73 The initial problem is to satisfy the constraints (3.) while minimizing η d0 = The best designs will be among those haing η d0 = 0, wheneer that alue is achieable. The possibilities for η d0 = 0 are explored in the following two cases. Case 1: 11 = 0. This says that the third replicate is the one displayed in (3.1). The inequalities (3.) gie 111 = 0 and 11 = 11 = 311 = /. Thus the fourth replicate is Rep # S 11 S 11 S 311 S 1 S 11 S S 3 S 1 (3.5) with each set displayed in (3.5) being of size /. Case : 11 > 0. Then (3.) and 11 / say that all the sets S e1 and S e are nonempty. If η d0 = 0, then each of the pairs (S 11, S ), (S 1, S 3 ), (S, S 31 ), and (S 1, S 1 ) must occur in the same block of replicate four. For the first three of these pairs this says that ( 111, 1 ) = (0, 0) or ( 11, ), ( 11, 31 ) = (0, 0) or ( 1, 3 ), and ( 1, 311 ) = (0, 0) or (, 31 ). But 111 = 11 contradicts the last inequality in (3.), so 111 = 1 = 0. This leaes four combinations to explore, each of which leads to a contradiction (e.g., there is no design with 111 = 1 = 11 = 31 = 1 = 311 = 0), or to a design that is isomorphic to that found in Case 1. To illustrate the latter, if ( 11, 31 ) = (0, 0) and ( 1, 311 ) = (, 31 ) then replicate four must be Rep # S 1 S S 31 S 1 S 11 S 1 S 3 S (3.6) and / = B 1 B 1 = = (/ 11 ) 11 = /, and so all sets in (3.6) hae size /. Now make new sets S11 = S 1, S1 = S, S1 = S 31, S = S 1, S31 = S 3, S3 = S, S1 = S 11, and S = S 1. Then among the blocks found here are B 11 = (S 11, S 1, S 1, S ) = (S1, S 11, S, S 1 ) = (S1, S ), B 31 = (S 11, S 1, S 31, S 1 ) = (S1, S, S 1, S ) = (S, S ), and B 1 = (S 1, S, S 31, S 1 ) = (S11, S 1, S 1, S ) = (S 1, S ), showing that replicates one, three, and four found here are identical to the first three replicates in Case 1, as claimed.

10 7 J. P. MORGAN Table 1. Minimum PV-aberration designs for four replicates, two blocks per replicate. ( 11, 111, 11, 11, 311 ) η d = (η d0, η d1, η d, η d3, η d ) 0 (mod ) (mod ) 0, 0,,, 0, 3, 0, 1,, + 6, 3 Design ariables not shown are determined by (3.) and (3.3). ( 5), 3, 3, ( ), ( ) +3, +( 9), ( ) Cases 1 and show that η d0 = 0 is uniquely achieed in four replicates, but only for a multiple of. This soles the minimum PV-aberration problem for 0 (mod ), but (mod ) requires further work leaning more heaily on the relations (3.) (3.). Writing (.) in terms of the fie free ariables gies ( η d0 = 3 x 0 z( x 0 ) ( z)θ + θ 3 i=1 x i ) (3.7) where, for notational simplicity, z= 11, x 0 = 111, x 1 = 11, x = 11, x 3 = 311 and θ = x 1 + x + x 3. The problem is to minimize (3.7) subject to (3.). The details are left to Appendix A, where again η d0 is found to be uniquely (up to isomorphism) minimized. In terms of the free ariables, the best designs for four replicates are specified in Table 1, along with their η-distributions. 3.. Fie replicates with two blocks per replicate As seen in Figure, the fifth replicate is represented by 3 additional ariables. Gien the ariables from replicate four, all those in the second block are determined by those in the first block ia em = em em1 (in this subsection m = (m 1, m ) and each m i {1, }). The sixteen first-block ariables are subject to the fie linearly independent constraints / = B 11 B 51 = B 1 B 51 = B 31 B 51 = B 1 B 51 = B B 51. These resole to 1 = = = (3.) 11 = =

11 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 75 Thus there are 11 free ariables for replicate fie, making a total of free ariables in the standard representation of the fie replicate design. While an analytic solution in the fashion of Section 3.1 would appear to be unwieldy (to say the least), judicious use of software makes a full solution feasible, as will be seen. The key insight is that ery small alues of η d0 gie an abundance of information about the ariables in replicate fie. Write e = A(e), which here is a singleton (e.g., A(1) = ), and m i = 3 m i. Consider η d0 = 0. The expression for η d0 in (.) is the sum of eight products 1m1 m m 3 m 1 m and eight products m 3 m1 m m 3 3m 1 m, which collectiely contain each of the 3 ariables once. If m 3 η d0 = 0 then eery one of these products is zero, implying that at least of the replicate fie ariables are zero. If em1 (say) is zero, then either em = 0 (determining a replicate four parameter) or em = em (determining a replicate fie parameter in terms of a nonzero replicate four parameter). The former case also forces em = 0, while e m 1 and e m may take any alues subject only to e m 1 + e m = e m. The latter case forces e m = e m and consequently e m 1 = 0. Here then is a route for determining all designs haing η d0 = 0: 1. Specify a subset of the replicate four ariables to be set to zero, all others taken to be positie.. Gien the selection in step 1, specify the corresponding information about the replicate fie ariables (as gien in the preceding paragraph). For each pair ( em, e m ) neither of which is set to zero in step 1, this entails a selection of which of the two pairs ( em1, e m 1) and ( em, e m ) is set to (0, 0), and which is set to ( em, e m ). 3. Sole the system of equations comprised of the fie constraints (3.) and the specifications in steps 1 and. Discard solutions iolating nonnegatie integer requirements.. Repeat for each distinct selection of ariables in step 1 and for each selection of pairs set to (0, 0) in step. This algorithm assumes 0 < 11 < / so that all third replicate ariables are positie. For 11 = 0 (for which any design is isomorphic to a design with 11 = /), the third replicate is that shown in (3.1) and η d0 = 0 has been achieed if there is any fie replicate solution including (3.1). Section 3.1 established that there was exactly one four replicate design including (3.1), making this a simple case to handle separately. This, in fact, produces the fie replicate solution for 0 (mod ) shown in Table. While any one iteration of this algorithm would be a straightforward task with pen and paper, there are far too many iterations to complete manually.

12 76 J. P. MORGAN Table. Minimum PV-aberration designs for fie replicates, two blocks per replicate. 0 (mod ) 0 (mod 36) 0 (mod ) ( 11, 111, 11, 11, 311, 1111, 111, 111, 11, 111, 11, 11, 3111, 311, 311, 111 ) η d = (η d0, η d1, η d, η d3, η d, η d5 ) (0, 0,,,, 0, 0,, 0, 0,, 0, 0,, 0, 0) (0,,, ( ), 0, ) ( 1, 0, 6, 1, 1, 0, 0, 1, 0, 1, 36, 0, 1, 36, 0, 36 ) (0, , 61 3, 9, 1, ) ( 3, 0, 1, 1, 3, 0, 1,, 1, 1, 0, 0,, 0, 0, 1 ) (0, 75 7, 5, 55 39, 5 9, 7 7 ) 0 (mod 1) (mod ) ( + ( 1, 0, 6, 1, 1, 0, 0, 6, 0, 0, 1, 0, 0, 1, 0, 1 ), 1, (0, 5, 5 36, 5 7, 0, 1 ), 1 (1, + 11,, 1 + 1,, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1) +, 19, 3 + 9) Best design for gien is in first applicable row. For example, for = 36 use second row (not fourth or fifth), for = 7 use first row (not second or fourth). Design ariables not shown are determined by (3.), (3.3), and (3.). This is a task for a computer; symbolic software for linear algebra can perform the iterations quickly and in full generality. It is a simple matter to write code to generate the selections and feed each to a symbolic equations soler (this author used Maple). Een with machine processing, the task would be too lengthy were it not possible to significantly reduce the number of iterations relatie to all. Reductions are achieed by application of the basic, design-presering symmetries apparent in Figure : The blocks within replicate three can be reersed, as can those within replicate four and those within replicate fie. Sets S 1 and S can be interchanged, as can sets S and S 3. The pair of sets (S 1, S ) can be interchanged with the pair (S, S 3 ). These symmetries can be used in combination with other considerations to great effect. For example, at most eight of the sixteen ariables from replicate four can be set to zero, for with 11 > 0, at most one of em1, em can be zero. Of the four em with the same e, at most two can be zero, and there are only two ways in which two can be set to zero (otherwise the design becomes isomorphic to one haing 11 = 0). Applying the symmetries aboe, additional restrictions

13 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 77 are: (i) #{m : 1m = 0} + #{m : m = 0} #{m : m = 0} + #{m : 3m = 0} (otherwise interchange (S 1, S ) with (S, S 3 )), (ii) #{m : 1m = 0} #{m : m = 0} (otherwise interchange S 1 and S ), (iii) #{m : m = 0} #{m : 3m = 0} (otherwise interchange S and S 3 ), (i) 111 = 0 (otherwise reerse blocks within replicate three and/or replicate four). Implementing this algorithm shows that, unlike for four replicates in Section 3.1, there is not a unique solution to η d0 = 0. Here solutions for the same are found haing different η d ectors, from which the best is selected in accord with Definition 1. Results appear in Table. Like for four replicates in Section 3.1, η d0 cannot achiee zero for eery. Setting η d0 = 1, now exactly one of the products em1 e m is nonzero, which can be taken (using suitable symmetries) to be 1111 = 1. Since then = 0, this implies either 111 = 1 or = 1, so (again using suitable symmetries) take 111 = 1111 = = 1. With this change, the ideas aboe for soling η d0 = 0 are easily transferred, so the details will not be repeated. Results appear in Table, completing the minimum PV-aberration problem for fie replicates.. A Catalog of Minimum PV-Aberration Designs The methods of Sections and 3 proide best affine resolable designs for the ranges of r shown in Table 3. The alues of s displayed are sufficient to coer all numbers of treatments up to = 00. A further result based on deletion of replicates is easily added. Wheneer an affine resolable design is a BIBD, all of its λ ij are identical so it triially has minimum PV-aberration. Remoing any one replicate decreases some λ ij by one, leaing the rest unchanged; this too is a minimum PV-aberration design. Now an affine resolable design is a BIBD wheneer r = (µs 1)/(s 1) = r max. Recalling from Section that each replicate in an affine resolable design corresponds to a column of an orthogonal array, further deletions can be used with small s in accord with the following result. Theorem. (Shrikhande and Bhagwandas (1969), and Vijayan (1976)) OA(, r max w, s) can be extended to OA(, r max, s) if (i) s =, w, or (ii) s = 3, w. Thus deletion of any two replicates from an affine resolable BIBD with s = or s = 3 produces a minimum PV-aberration design. For s =, the same result holds for deletion of three replicates proided that the deleted replicates, when considered as an affine resolable design d, minimize aberration of the ector (η d3, η d, η d1, η d0 ). Likewise deletion of a four-replicate subdesign d requires minimizing aberration of (η d, η d3, η d, η d1, η d0 ) oer all -replicate affine resolable designs.

14 7 J. P. MORGAN Table 3. Solutions found for s 1. s r So that they may be easily accessed for application, all of these designs (for up to 00 treatments and 3 replicates) hae been compiled in an online catalog at designtheory.org, a website deoted to free storage and access to block designs and many of their properties. The designs are stored there as xml files, in external representation format (see Bailey, Cameron, Dobcsanyi, Morgan and Soicher (006)), along with lists of canonical efficiency factors, pairwise ariances, and much more. Mutually orthogonal Latin squares, needed for the construction of d in Theorem 1, can be found in Abel, Colbourn and Dinitz (007). Taking further adantage of the orthogonal array representation of an affine resolable design, best designs can be found wheneer all nonisomorphic orthogonal arrays OA(, r, s) for s symbols in rows and r columns hae been enumerated. OA enumeration has been pursued with some igor of late, with the following cases now completed: OA(1, r, ) for r 11, OA(, r, ) for r 15, and OA(0, r, ) for r 19, all in Sun, Li and Ye (00); OA(, r, ) for r 7, OA(, r, ) for r 6, and OA(3, r, ) for r 6, all in Angelopoulos, Eangelaras, Koukouinos and Lappas (007); and OA(1, r, 3) for r 7 in Eangelaras, Koukouinos and Lappas (007). All of these lists hae been searched to determine a minimum PV-aberration design, and these designs hae all been added to the online catalog. At this writing the catalog contains 5 designs, including all parameter combinations with 0 for which an affine resolable design can exist. Because all of these designs are affine, they are excellent resolable designs. Being additionally optimized for estimation of pairwise contrasts will make them the preferred choice in most applications. 5. Examples and Discussion Two detailed examples are shown next, coering Theorem 1 and the methods of Section 3. While both example designs can be downloaded from the web catalog, numbers outside the catalog s range require the techniques they illustrate. Those interested in the analysis of data from affine resolable designs are referred to the thorough coerage of this topic in Caliński, Czajka and Pilarczyk (009). That paper includes examples of, and data from, applications of these designs in agricultural trials. Example 1. A four replicate, affine resolable design with minimal PVaberration, for = 1 treatments in blocks of size k = 6, can be found with

15 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 79 Rep #1 Rep # Rep #3 Rep # Figure 3. A minimal PV-aberration design for 1 treatments in blocks of size 6. the construction underlying Theorem 1. First write = µs = (3). The 1 treatments are partitioned into s = 9 sets of µ = treatments each: S 1 = {1, }, S = {3, },..., S 9 = {17, 1}. Placing the S i s into blocks as displayed in general form in Figure 1 gies the first two replicates of the design (Figure 3). Finding the remaining two replicates requires = mutually orthogonal Latin squares of order s = 3. They are: Superimpose the first of these squares on the first 3 3 square in Figure 1; sets S i coincident with number j in this Latin square are placed in block j of the third replicate. Thus the first block of the third replicate contains S 1, S 5, S 9, the second contains S 3, S, S, and the third S, S 6, S 7. Superimpose the second Latin square on the first 3 3 square in Figure 1 to similarly get the fourth replicate in Figure 3. Example. A fie replicate, affine resolable design with minimal PVaberration, for = treatments in blocks of size k =, results from the calculations in Section 3. The design can be built from the information in Table ; the alues there tell the sizes of the subsets in the third through fifth replicates shown in general form in Figure. The sets comprising the first two replicates are S 1 = {1,, 3, }, S = {5, 6, 7, }, S 3 = {9, 10, 11, 1}, and S = {13, 1, 15, }. Reading from the first row of Table, the releant information for the third replicate is 11 = 0. This says no part of S 1 is in the first block of replicate three so that S 11 = and S 1 = S 1. The remainder of replicate three follows from affineness with respect to the first two replicates; see Figure. For the fourth replicate, Table says 11 = 11 = 311 =. So any two treatments from each of S 1 = S 1, S 1 = S, and S 31 = S 3 are placed in the first

16 730 J. P. MORGAN Rep #1 Rep # Rep #3 Rep # Rep # Figure. A minimal PV-aberration design for treatments in blocks of size. block of replicate four. It follows that 1 = and that block, and consequently the fourth replicate, is completed by any two treatments from S = S. With the fourth replicate in place, now use 111 = 11 = 311 = and, from (3.) and the zeros specified in Table, 11 =. So S 111 = S 11, S 11 = S 1, S 311 = S 31, and S 11 = S 1, determining the fifth replicate of Figure. The technique of Section 3, based on the standard representation introduced in Section, becomes rapidly more demanding as either r or s grows. This is to be expected, for the equialent problem of searching all orthogonal arrays, beyond the smallest cases, is notoriously difficult. Pushing on to r 6 for s =, or to r 5 for s = 3, is likely to require additional knowledge as to how the problem can be reduced, be it through exploiting additional symmetries, mathematical deriation of additional restrictions on the ariables em that are consonant with minimum aberration, or some combination of the two. Acknowledgement J. P. Morgan was supported by the United States National Science Foundation grant DMS Appendix: Minimizing η d0 for (, r, ) = ( (mod ),, ) Noting that x 1, x, and x 3 are interchangeable (this corresponds in Figure to permutations of blocks within replicates, and of sets within blocks of the first two replicates), there is no loss of generality in taking x 1 x x 3 / z (the last inequality from (3.)). Minimizing (3.7) first for fixed z, x 0, and θ, says to maximize 3 i=1 x i for fixed θ. A simple majorization argument shows this is accomplished by sequentially minimizing x 1, and then x. Since θ is fixed,

17 OPTIMAL RESOLVABLE DESIGNS WITH MINIMUM PV ABERRATION 731 minimizing x 1 is equialent to maximizing x + x 3, and by the third line of (3.), x + x 3 / x 0. Putting x 1 = θ + x 0 / and x + x 3 = / x 0 in (3.7) gies η d0 3 ( x 0 z( x 0 ) ( z) x 1 x 0 + ) [ ( + x 1 x 0 + ) ( x ) ] 1 x x 0 x = x 0 x + ( x 0 )x + (z x 0 )x 1 + x 0 (A.1) where, since x + x 3 = / x 0 and x 1 x x 3 / z, 0 x 1 x 1 ( x 0 and, from the first and third lines of (3.), ) (A.) z x 0 x x 1 x 0 z. (A.3) Importantly, the RHS of (A.) need not be an integer, though x must be; this is why different solutions are found depending on the (mod ) alue of. The minimization proceeds in two steps. Step 1 : minimize with x 0 = 0. From (A.1) the quantity to be minimized is H(z, x 1, x ) = x + x + zx 1 (A.) which, from (A.) and (A.3), is subject to the constraints / z x x 1 x for ( z)/ x ( )/. If z = 0 there are no feasible alues for x, so z 1. Also z / z ( )/. Since (A.) is linear in x 1 with positie slope, H(z, x 1, x ) H (z, ) z x, x = x z zx + (x + z) H(z, x ). (A.5) Now H(z, x ) is concae in x, so is minimized as a function of x at either x = ( )/ or x = ( z)/, the latter alue depending on the parity of z. Thus three ealuations of (A.5) are required, each of which produces a concae function of z, and which are then minimized by ealuating at the endpoints for z. This produces the minimum alue of / 3 for η d0 when x 0 = 0 at the alues (z, x 0, x 1, x, x 3 ) = ( 11, 111, 11, 11, 311 ) = (( )/, 0, 1, ( )/, (+)/)

18 73 J. P. MORGAN and (1, 0, ( )/, ( )/, ( +)/). These yield the same η d ectors; indeed, it may be shown that they are isomorphic solutions. Step : minimize with x 0 > 0. Now fix a positie integer alue for x 0. The same sequence of ealuations is carried out as in Step 1, with the appropriate endpoints as gien by (A.) and (A.3) (which are now a bit more complicated). For x 0 > 1 many of the ealuations do not require attending to the integer nature of the endpoints as done in Step 1; this subcase is easily eliminated as inferior to x 0 = 0. Not surprisingly, x 0 = 1 requires care in strictly adhering to the exact integer endpoints, but it, too, produces a minimum larger than / 3. Thus the unique minimum PV-aberration design was identified in Step 1, as listed in Table 1. References Abel, R. J. R., Colbourn, C. J. and Dinitz, J. H. (007). Mutually orthogonal Latin squares (MOLS). In The CRC Handbook of Combinatorial Designs. nd edition. (Edited by C. J. Colbourn and J. H. Dinitz), 0-193, CRC Press, Boca Raton. Angelopoulos, P., Eangelaras, H., Koukouinos, C. and Lappas, E. (007). An effectie stepdown algorithm for the construction and the identification of nonisomorphic orthogonal arrays. Metrika 66, Bailey, R. A., Cameron, P. J., Dobcsanyi, P., Morgan, J. P. and Soicher, L. H. (006). Designs on the web. Discrete Math. 306, Bailey, R. A., Monod, H. and Morgan, J. P. (1995). Construction and optimality of affineresolable designs. Biometrika, Bose, R. C. (19). A note on resolability of balanced incomplete designs. Sankhyā 6, Caliński, T., Czajka, S. and Pilarczyk, W. (009). On the application of affine resolable designs to ariety trials. J. Statist. Appl., to appear. Eangelaras, H., Koukouinos, C. and Lappas, E. (007). 1-run nonisomorphic three leel orthogonal arrays. Metrika 66, Morgan, J. P. (1996). Nested designs. In Design and Analysis of Experiments, Handbook of Statistics 13 (Edited by S. Ghosh and C. R. Rao), North Holland, Amsterdam. Shrikhande, S. S. and Bhagwandas. (1969). On embedding of orthogonal arrays of strength two. In Combinatorial Mathematics and its Applications; Proceedings of the Conference Held at the Uniersity of North Carolina at Chapel Hill, April 10 1, 1967, (Edited by R. C. Bose and T. A. Dowling), Uniersity of North Carolina Press, Chapel Hill. Sun, D., Li, W. and Ye, K. (00). An algorithm for sequentially constructing non-isomorphic orthogonal designs and its applications, Statist. Appl. 6, Vijayan, K. (1976). Hadamard matrices and submatrices. J. Austral. Math. Soc. Ser. A, Xu, H. (003). Minimum moment aberration for nonregular designs and supersaturated designs. Statist. Sinica 13, Department of Statistics, Virginia Tech, Blacksburg, Virginia, , U.S.A. jpmorgan@t.edu (Receied September 00; accepted February 009)