Resolvable designs with large blocks J. P. Morgan

Brian H. Reck

Department of Statistics

Department of Human Genetics

Virginia Tech

University of Pittsburgh

Blacksburg, Virginia 24061, USA

Pittsburgh, PA 15261, USA

10 February 2005

Abstract: Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix, and consequently in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities, and a detailed study of E-optimality offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.

1

Introduction

Block designs arise in comparative experimentation as a fundamental device for improving efficiency when working with heterogeneous experimental units. The blocks are simply a partition of the units into b (say) sets exhibiting homogeneity within sets. Given the blocks, of sizes k1 , k2 , . . . , kb , a block design is an assignment of v treatments to the

Pb

j=1 kj

units. Optimality theory for block designs

attempts to determine which of the many possible assignments is in some sense best. In some applications there are restrictions on the collection of possible assignments. A block design is resolvable if the blocks can be partitioned into replicates, defined as sets of blocks with the property that each treatment is assigned to one unit in each set. The practical impact of, and motivation for, resolvability is to gain orthogonality between treatments and nuisance factors of 1

concern. For instance, resolvability in sequential experimentation, with replicates corresponding to time periods, is used to mitigate time effects. Resolvability can likewise be useful in multi-site experiments, and in experiments with multiple individuals handling experimental runs. Notably, the United Kingdom has for some time required the use of resolvable designs in agricultural field trials (see Patterson and Silvey, 1980). The combinatorial study of resolvability in block designs goes back at least as far as the wellknown Kirkman’s (1850) schoolgirl problem. The notion entered the statistical lexicon with Yates’ work on square lattice designs (1936, 1940), though the term “resolvable design” was introduced by Bose (1942). Yates’ lattice designs were extended to rectangular lattices by Harshbarger (1946, 1949); also see Bailey and Speed (1986). Williams (1975) and Patterson and Williams (1976) introduced a large family of resolvable designs they termed α-designs. Williams et al (1976) derived resolvable designs with two replicates from BIBDs. Bailey, Monod, and Morgan (1995) proved strong optimality for the affine resolvable designs introduced by Bose (1942). Resolvable BIBDs (balanced incomplete block designs) have received significant attention in both the combinatorial and statistical literature; for a summary see Morgan (1996), who surveys the major classes of resolvable designs with many references. With α-designs Patterson and Williams (1976) provided a flexible method for obtaining reasonably efficient resolvable designs for a wide range of values v, k, and r. They also adapted their method to obtain resolvable designs with two different block sizes k and k − 1, a first attempt at addressing the obvious restriction that resolvability with equal block sizes can be achieved only for v a multiple of k. John, Russell, Williams, and Whitaker (1999), in revisiting that idea, concluded that the α-technique for two block sizes could produce relatively inefficient designs for small v, and recommended an interchange algorithm for construction of designs with better efficiency. John et al (1999) also discussed the practical need for resolvable designs with unequal block sizes; for example, about half of 245 experiments examined by Patterson and Hunter (1983) had unequal block sizes. Also see Patterson and Silvey (1980). To the authors’ knowledge, the literature contains no systematic work on determining optimal resolvable block designs when block sizes need not be equal. This paper will undertake such work, for the special case of two blocks per replicate. Only with two blocks in each replicate must a block be large in the sense of containing at least half of the treatments. And with two blocks per replicate, 2

the block sizes must be unequal for any odd v (though this work is not restricted to odd v). Let D(v, r; k1, k2) denote the class of all resolvable block designs with r replicates of v treatments, each replicate consisting of two blocks of sizes k1 and k2 , k1 + k2 = v. Of special interest is k1 = k, k2 = k − 1 for odd v = 2k − 1, but here no restrictions are placed on the two block sizes for the general theory. Let k1 denote the larger block size, k1 ≥ k2 , and so with no loss of v 2

generality take

≤ k1 ≤ v − 2. In the most common applications of resolvable designs the number

of treatments is large relative to the number of replicates; here r ≤ v − 1 is required, allowing optimality problems to be more easily attacked through the dual of the information matrix. This defines the framework for the remainder of the paper: determine the best design d ∈ D(v, r; k1, k2). The general setup is displayed in figure 1. An example of a resolvable design in D(9, 4; 5, 4) is shown in table 1 with the blocks written as columns. Later this design will be proven optimal with respect to many useful criteria.

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

.. .. .. k1

.. . k2

.. .. ..

... ...

.. .

.. .. ..

k2

k1

.. . k2

k1

Figure 1: Experimental units of resolvable design with 2 blocks per replicate and k1 ≥ k2 For any resolvable design d ∈ D, ignoring the replicate grouping leaves an underlying simple block design for v treatments in 2r blocks. If the roles of blocks and treatments are reversed in this underlying design, another simple block design with 2r treatments in v blocks of size r is produced.

3

Table 1: A Resolvable Design In D(9, 4; 4, 5) 1 2 3 4 5

6 7 8 9

1 4 5 7 9

2 3 6 8

1 2 3 7 9

4 5 6 8

1 2 5 8 9

3 4 6 7

1 3 4 7 8

2 5 6 9

˜ This dual design is denoted by d.

2

Model, information, and optimality criteria

Let yhjl denote the yield from the lth experimental unit in block j of replicate h. Thus the triples (h, j, l) identify the experimental units, and the design d corresponds to a map d[h, j, l] from the units to the set of treatments. The standard linear model for the yields incorporates a mean effect µ, replicate effects ρh , block effects βhj , treatment effects τd[h,j,l] , and mean zero, uncorrelated, equivariable random error terms ehjl : yhjl = µ + ρh + βhj + τd[h,j,l] + ehjl h = 1, . . . , r; j = 1, 2; l = 1, . . . , kj . This model may be written in matrix terms as y = µ1 + M ρ + Lβ + Ad τ + e where with the yhjl lexicographically ordered in the vr × 1 yield vector y,the replicate  incidence  1k1 0k1  matrix is Mvr×r = Ir ⊗ 1v , the block incidence matrix is Lvr×2r = Ir ⊗  , and the 0k2 1k2 design matrix is the vr × v incidence matrix Ad , for which row (h, j, l) has a 1 in column i if and only if d[h, j, l] = i (that is, unit (h, j, l) receives treatment i), and all other entries are zero. Replicate effects, block effects, treatment effects, and error vectors are ρr×1 , β2r×1 , τv×1 , and evr×1 , respectively. Choice of design is equivalently choice of Ad . Linear models theory says that the information matrix for estimation of the block effects τ is Cd = A0d (I − L(L0 L)−1 L0 )Ad = rI − Nd Ds−1 Nd0 4

(1)

where Ds is the diagonal matrix of block sizes, Ds = L0 L = Diag(k1 , k2 , k1 , k2 , . . . , k1 , k2 ). The v×2r matrix Nd is the treament/block incidence matrix. The general off-diagonal element (Nd Nd0 )i,i0 of Nd Nd0 is the number of blocks to which both treatments i and i0 are assigned, called a treatment concurrence count. Notice that M plays no role in (1); the same form of information matrix is obtained for any simple block design. It follows immediately that the information matrix for the underlying dual d˜ is 1 Cdual = Ds − Nd0 Nd r

(2)

The off-diagonal elements of Nd0 Nd are block concurrence counts. All treatment contrasts are estimable with design d if and only if Cd has rank v − 1. Any such d is said to be connected ; only connected designs are considered here. Most (but not all) commonly employed optimality criteria, including those to be used here, are functions of the v − 1 nonzero eigenvalues of Cd . These will be ordered and labelled zd1 ≤ zd2 ≤ · · · ≤ zd,v−1 . Much of the optimality work below will focus on minimizing functions of the form φf (zd ) =

v−1 X

f (zdi )

(3)

i=1

where f is convex and zd is the vector of nonzero eigenvalues. If f in (3) is Schur-convex (Schurconvex functions include the convex functions, see Bhatia, 1997, section II.3), then (3) is said to be a Schur-criterion. A design optimal with respect to all (i.e., minimizing all) Schur-convex criteria is said to be Schur-optimal or S-optimal. If f in (3) satisfies (i) f is continuously differentiable on (0, maxd∈D tr(Cd )) with f 0 < 0, f 00 > 0, and f 000 < 0, and (ii) limx→0 f (x) = ∞, then (3) is said to be a type-1 criterion (see Cheng, 1978). A design optimal with respect to all type-1 criteria is said to be type-1-optimal. One popular criterion belonging to both families just defined is the A-criterion specified by f (x) = x1 . A criterion not of the form (3) (though it can be written as a limit of such criteria) is to minimize φE (zd ) =

1 zd1

(4)

called the E-criterion. Equivalently, a E-optimal design maximizes zd1 . For a broader discussion of optimality criteria and their statistical meanings, see Shah and Sinha (1989). 1/2

It will next be shown how the zdi can be advantageously approached through Cdual . Let Ds −1/2

be the diagonal matrix of square roots of block sizes, and write Bd = Nd Ds 5

. Multiplying Cd by

1 r,

−1/2

and right and left multiplying Cdual by Ds

, equations (1) and (2) become

1 1 1 Cd = I − Nd Ds−1/2 NdT = I − Bd Bd0 = Cd∗ r r r

(5)

and Ds−1/2 Cdual Ds−1/2 = I −

1 1 −1/2 T ∗ D Nd Nd Ds−1/2 = I − Bd0 Bd = Cdual . r s r

(6)

Since (1) and (5) differ only by a constant, and since the nonzero eigenvalues of Bd BdT and BdT Bd are identical, an eigenvalue-based optimality investigation of designs in D(v, b; k1 , k2 ) can be per∗ formed by restricting attention to the eigenvalues of Cdual in (6). Doing so requires focus on block

concurrences in the formation of NdT Nd . But there is further structure that can be exploited. −1/2

Use Ad to denote the symmetric matrix BdT Bd , that is, Ad = BdT Bd = Ds 1/2

−1/2

NdT Nd Ds

,

1/2

∗ so that Cdual = I − 1r Ad . Regardless of the design d, Ad Ds 1 = rDs 1; r is an eigenvalue

of Ad corresponding to the zero eigenvalue common to Cdual and Cd . One term of the spectral decomposition of Ad is then 

 1/2 1/2 (Ds 1)(Ds 1)T 1/2 1/2 (Ds 1)T (Ds 1)

=



√ k1 k2 

1   k1 J ⊗  √ k1 + k2 k1 k2

k2



(7)

where J is a r × r matrix of 1s. Subtracting (7) from Ad yields the new matrix 



A∗d = Ad −



1   k1 J ⊗  √ k1 + k2 k1 k2

√ k1 k2  k2



(8)

All the eigenvalue-based optimality information for any d ∈ D is carried by (8). Let φhh0 be the block concurrence for the first blocks (those of size k1 ) in replicates h and h0 . Then the (f, g) element of 



φhh0

k1 − φhh0

k1 − φhh0

k2 − k1 + φhh0



Φhh0 = 

 .

is the block concurrence for block f of replicate h with block g of replicate h0 (this includes the case h = h0 , for which φhh = k1 ). Then NdT Nd is the partitioned matrix NdT Nd = (Φhh0 ) so that Ad is the partitioned matrix Ad = (Φ∗hh ) where Φ∗hh = I2 , and  

Φ∗hh0 = 

φhh0 k1 k1 −φhh0 √ k1 k2

6

k1 −φhh0 √ k1 k2 k2 −k1 +φhh0 k2

  .

(9)

The subtraction in (8) for A∗d can now be easily done, producing         1  A∗d = k1 k2       

 k1 k2 v

φ∗12

φ∗12

k1 k2 v

φ∗23

···

k1 k2 v

φ∗1r

    ∗ φ2r    k2    φ∗3r  ⊗ √  − k1 k2 ..  .    



√ − k1 k2  k1

 

(10)

k1 k2 v

where φ∗hh0 = φhh0 −

k12 . k1 + k2

(11)

Since the eigenvalues of the 2 × 2 matrix in (10) are 0 and k1 + k2 , the b = 2r eigenvalues of A∗d are r copies of 0 and

v k1 k2

times the r eigenvalues of         Md =        

 k1 k2 v

φ∗12

φ∗13

k1 k2 v

φ∗23 k1 k2 v

···

φ∗1r

   ∗ φ2r     ∗ φ3r  .  ..  .    

(12)

k1 k2 v

Let the eigenvalue of Md be ed1 ≥ ed2 ≥ · · · ≥ edr . Tracking back through the above argument, the eigenvalues of Cd are 0, max{0, v − r − 1} copies of r, and (r −

ved1 k1 k2 , r

−

ved2 k1 k2 , . . . , r

−

vedr k1 k2 ).

Evidently, an eigenvalue-based optimality analysis of resolvable designs d ∈ D(v, r; k1 , k2 ) can be based on the eigenvalues ed = (ed1 , ed2 , . . . , edr ) of Md , and so one can work with the set of r(r − 1) block concurrence counts {φ12 , φ13 , φ23 , . . . , φr,r−1 }. Md will be called the optimality matrix for design d ∈ D. Restating (3), the goal is to minimize

r X

f (r −

i=1

vedi ) k1 k2

(13)

for f in one of the earlier-defined families. The E-criterion (4) is restated as minimize ed1

7

(14)

So long as r ≤ v − 1 (as earlier required in section 1), Cd had v − r + 1 eigenvalues fixed at r. Working with the dual not only makes this evident, but allows these structurally fixed eigenvalues to be easily set aside. When r > v − 1 some of the edi are structurally fixed and there is no advantage to the dual approach. With the optimality problem recast in terms of Md and its eigenvalues, a crucial concept for proofs of Scur-optimality is now defined. Definition 1 Let {xi }ni=1 and {yi }ni=1 be nonincreasing sequences of real numbers such that Pn

i=1 yi .

Pn

i=1 xi

If l X

xi ≤

l X

i=1

yi ,

for all 1 ≤ l ≤ n,

i=1

or, equivalently, n−l+1 X

xi ≥

i=n

n−l+1 X

yi ,

for all 1 ≤ l ≤ n

i=n

then {yi }ni=1 is said to majorize {xi }ni=1 . The importance of majorization is evident in this result (see, e.g., Bhatia, 1997, page 40): Theorem 2 Let {xi }ni=1 and {yi }ni=1 be nonincreasing sequences of real numbers such that Pn

i=1 yi .

Then

Pn

i=1 f (xi )

≤

Pn

i=1 f (yi )

Pn

i=1 xi

=

for all real-valued convex functions f if and only if {yi }ni=1

majorizes {xi }ni=1 . If the sequences {xi }ni=1 and {yi }ni=1 are written as the elements of vectors x and y, then y majorizes x is written as y  x or x ≺ y. Let ed∗ and ed be the vectors of eigenvalues for the optimality matrices for designs d and d∗ in D(v, r; k1 , k2 ). If ed  ed∗ , and the two vectors are not identical, then design d∗ is said to be Schur-better than design d, and d is said to be Schur-inferior to d∗ . Design d∗ is Schur-optimal if it is Schur-better than or equal to every design in D. The list of majorization facts in the following lemma will be used extensively (and often without comment) in subsequent sections. For majorization comparisons, the third of these allows work directly with edi in (13) rather than r −

vedi k1 k2 .

Lemma 3 Let a and b be real numbers, and let {ai }m i=1 be a sequence of real numbers. Then (i) If x1 ≥ x2 = x3 = · · · = xn and y1 ≥ x1 , then {yi }ni=1 majorizes {xi }ni=1 . 8

=

(ii) If x1 = x2 = · · · = xn−1 ≥ xn and xn ≥ yn , then {yi }ni=1 majorizes {xi }ni=1 . (iii) If {yi }ni=1 majorizes {xi }ni=1 then {a −

yi n b }i=1

majorizes {a −

xi n b }i=1 .

n m (iv) If {yi }ni=1 majorizes {xi }ni=1 then {{yi }ni=1 ∪ {ai }m i=1 } majorizes {{yi }i=1 ∪ {ai }i=1 }.

3

Designs based on BIBDs

Though not in the spirit of most of this paper, the result presented here is simple and powerful. Theorem 4 Corresponding to a BIBD for v treatments in b blocks of size k, there is a Schuroptimal resolvable design in D(v, b; k, v − k). Proof The two blocks in a replicate are one block of the BIBD and its complement. The v − 1 nonzero eigenvalues of the information matrix Cd of the resolvable design are all equal, and so are majorized by the eigenvalues of every d ∈ D.

2

See Mathon and Rosa (1996) for a fairly recent listing of known BIBDs up to r = 41. The limitation of this approach is excessive replication, as any BIBD satisfies Fisher’s inequality: b ≥ v, which translates here to r ≥ 2v. Interestingly, for large r this complementation method will not generally work well if the starting block design is other than a BIBD, for if not balanced in an equireplicate starting design, treatment concurrences become more unbalanced in the resolvable design when the complement is added.

4

Equal Concurrence Designs and Global Optimality

Among simple block designs the BIBDs are Schur-optimal, a result which follows from equal treatment concurrences inducing complete symmetry of the information matrix. The analogous notion for duals is equal block concurrences, which this section explores for utility with resolvable designs. A resolvable design d ∈ D(v, r; k1 , k2 ) having block concurrence counts φ12 = φ13 = φ23 = · · · = φr−1,r = θ for some k1 −k2 ≤ θ ≤ k1 is called an equal concurrence design with common concurrence θ, or ECD(θ). For an ECD(θ) the optimality matrix (12) is Md = {k1 k2 − [θ(k1 + k2 ) − k12 ]}I + [θ(k1 + k2 ) − k12 ]J 9

(15)

where I is the r × r identity matrix, and J is the r × r matrix of ones. Like the information matrix for a BIBD, it is completely symmetric, but unlike that matrix, Md for an ECD(θ) is nonsingular and thus has two distinct, relevant eigenvalues, rather than just one. The eigenvalues of (15) are k1 k2 + [k12 − θ(k1 + k2 )] and

k1 k2 − (r − 1)[k12 − θ(k1 + k2 )]

(16)

with frequencies r − 1 and 1, respectively. Theorem 5 Suppose D(v, r; k1 , k2 ) is a resolvable design setting for which (k1 + k2 ) | k12 , and define θ∗ =

k12 . k1 + k2

(17)

Then ECD(θ∗ )s in D are Schur-optimal whenever they exist. Proof Given the conditions on D, the inequalities k1 − k2 ≤

k12 k1 +k2

≤ k1 imply that θ = θ∗ is an

admissible value for the common block concurrence of an ECD(θ). By (16) the eigenvalues of the optimality matrix for ECD(θ∗ ) are k1 k2 − (r − 1)[k12 − θ∗ (k1 + k2 )] = k1 k2 + [k12 − θ∗ (k1 + k2 )] = k1 k2 , that is, are identical, so they are majorized by the eigenvalues of every competing design. Corollary 3.4 of Bailey, Monod, and Morgan (1995) established that affine-resolvable designs are Schur-optimal. Theorem 5 generalizes that result when there are two blocks per replicate. Here the optimality condition is that the blocks of size k1 have the same concurrence (17). When k1 = k2 and 2 | k1 , ECD(θ∗ )s are affine-resolvable designs. Example Consider the setting D(9, 4; 6, 3). Since (k1 + k2 ) | k12 , then θ∗ = 4, and if an ECD(4) exists it is Schur-optimal. In fact, an ECD(4) does exist and is shown in table 2. Table 2: A Schur-optimal ECD(4) in D(9, 4; 6, 3) 1 7 1 5 1 3 3 1 2 8 2 6 2 4 4 2 3 9 3 9 5 9 5 9 4 4 6 6 5 7 7 7 6 8 8 8

10

The settings for which k12 is a multiple of k1 + k2 are relatively sparse (a situation much like that of BIBDs relative to all simple block design settings). For the 1225 pairs 2 ≤ k2 < k1 ≤ 51, only 23 meet the divisibility requirement implied by (17). Theorem 5 is thus only a start, albeit an important one. Good designs are expected to be “close” to ECD(θ∗ )s, suggesting this question: Is some equal concurrence design Schur-optimal when (k1 + k2 ) /| k12 , and if not, what are the optimal classes of designs for the various criteria? The subsequent discussion will first focus on evaluating ECDs when (k1 + k2 ) /| k12 . Define the block concurrence parameter θ¯ by Ã

k12 θ¯ = int k1 + k2

!

.

(18)

Writing γ=

k12 − θ¯ k1 + k2

(19)

then 0 ≤ γ < 1 and a necessary condition for existence of ECD(θ∗ ) is γ = 0. Consequently, γ is called the block discordancy coefficient; it measures the departure of the block sizes from that required for equality of all eigenvalues. ¯ designs in D(v, r; k1 , k2 ) can be classified into four categories: (1) ECD(θ)s having In terms of θ, ¯ that is, ECD(θ)s; ¯ (2) ECD(θ)s having θ = θ¯ + 1, that is, ECD(θ¯ + 1)s; (3) designs having all θ = θ, ¯ θ¯ + 1} with both of the possible values occurring at least once; and (4) designs having φii0 ∈ {θ, either φii0 < θ¯ or φii0 > θ¯ + 1 for at least one 1 ≤ i 6= i0 ≤ r. Designs falling into the first category are ECD(θ∗ )s when (k1 + k2 ) | k12 . Designs falling into the third category will be referred to as nearly equal concurrence designs or NECDs, and those in the fourth category are called unequal concurrence designs, or UECDs. The investigation of which designs are best when γ > 0 begins ¯ and ECD(θ¯ + 1)s, then will expand to incorporate the other two with a comparison of ECD(θ)s categories. Define the block concurrence discrepancy matrix ∆d = (δdii0 ), where

δdii0 =

    φii0 − θ¯ if i 6= i0    0

if i = i0 .

For i 6= i0 , the off-diagonal elements δdii0 will be referred to as block concurrence discrepancies. Let the product of the block sizes be denoted by p = k1 k2 . The optimality matrix (12) can now be 11

written Md =

p I − γ(J − I) + ∆d . v

(20)

¯ have φii0 = θ¯ for each 1 ≤ i 6= i0 ≤ r and so ∆d = 0 and Md = (p + γ)I − γJ, for Now ECD(θ)s which the eigenvalues are ξ1 (γ) =

p +γ v

and ξ2 (γ) =

p − (r − 1)γ v

(21)

with frequencies r − 1 and 1, respectively. The two distinct eigenvalues satisfy ξ1 (γ) ≥ ξ2 (γ). For ECD(θ¯ + 1)s, φii0 = θ¯ + 1, ∆d = (J − I), and Md = [ vp − (1 − γ)]I + (1 − γ)J. For this optimality matrix the eigenvalues are ξ1 (γ − 1) =

p p − (1 − γ) and ξ2 (γ − 1) = + (r − 1)(1 − γ) v v

(22)

with frequencies r − 1 and 1, respectively, and ξ2 (γ − 1) ≥ ξ1 (γ − 1). The following theorem due to Cheng (1978, Theorem 2.3) will be used to establish φf -optimality ¯ depending on the discordancy coefficient γ. for certain ECD(θ)s Theorem 6 Let D0 be a class of block designs with fixed trace of the information matrix. If there exists d¯ ∈ D0 for which (i) Cd¯ has two distinct eigenvalues zd1 ≤ zd,v−1 , and ¯ = zd2 ¯ = . . . = zd,v−2 ¯ ¯ P 2 over D , (ii) d¯ minimizes v−1 0 ¯ i=1 zdi

then d¯ is type-1-optimal over D0 . ¯ minimize tr C 2 , uniquely so if γ < 1 , and consequently are Theorem 7 For 0 ≤ γ ≤ 12 , ECD(θ)s d 2 type-1-optimal. Proof The eigenvalues of the information matrix for any design d ∈ D(v, r; k1 , k2 ) are 0 < zd1 ≤ zd2 ≤ · · · ≤ zdr and v − r − 1 copies of r, and

Pr

i=1 zdi

= r(r − 1) is constant for all designs in D.

¯ the zdi follow the form of Theorem 6 as shown by (21). Let Md be the optimality For ECD(θ)s, matrix for d ∈ D(v, r; k1 , k2 ), with tr Md = tr Cd2 =

v−1 X

pr v

and (Md )ii0 = (δdii0 − γ). Then

2 zdi

i=1

12

= (v − r − 1)r2 +

r µ X

r−

i=1

vei p

¶2

2vr v2 tr Md + 2 tr Md2 p p 2 2v XX = (v − 3)r2 + r + 2 (δdii0 − γ)2 , p i
so that tr Cd2 is minimized by designs that minimize unique minimum of tr Cd2 on 0 ≤ γ <

1 2

PP

i
− γ)2 . Since δdii0 is integral, the

¯ For γ = 1 , any is at δdii0 ≡ 0, achieved only by ECD(θ). 2

values δdii0 ∈ {0, 1} minimize tr Cd2 . Now define the F-criterion as the value of the largest eigenvalue of Cd that is not constrained by the setting to equal r, that is, φF (Cd ) = zdr = r −

vedr . p

Minimizing φF (Cd ) over D is equivalent to maximizing edr . This criterion can be important in establishing Schur-optimality, as shown next. ¯ Schur-better than a competitor with a different set of eigenvalues if and Theorem 8 An ECD(θ)is ¯ are Schur-optimal only if it is F-equivalent or better than that competitor. Consequently, ECD(θ)s if and only if they are F-optimal. Proof Follows from (21) and lemma 3. A result of similar flavor holds for ECD(θ¯ + 1)s using the E-criterion. As pointed out by Kunert (1985, page 385), designs with eigenvalues zdi in the form of lemma 3(ii) are Schur-best whenever they are E-optimal. For the current problem this is stated as Theorem 9 An ECD(θ¯ + 1)is Schur-better than a competitor with a different set of eigenvalues if and only if it is E-equivalent or better than that competitor. Consequently, ECD(θ¯ + 1)s are Schur-optimal if and only if they are E-optimal. ¯ are Schur-better than ECD(θ¯+ 1)s if and only if γ ≤ 1 , and ECD(θ¯+ 1)s Corollary 10 ECD(θ)s r ¯ if and only if γ ≥ are Schur-better than ECD(θ)s

r−1 r .

13

¯ and ECD(θ¯ + 1)s are never identical. By (21) and Proof Note that the eigenvalues of ECD(θ)s ¯ are F-equivalent or better than ECD(θ¯ + 1)s if and only if ξ1 (γ − 1) ≤ ξ2 (γ) which (22), ECD(θ)s is equivalent to γ ≥

¯ if and only if ECD(θ¯ + 1)s are E-equivalent or better than ECD(θ)s

1 r.

ξ1 (γ) ≥ ξ2 (γ − 1) which is equivalent to γ ≤

r−1 r .

Corollary 10 illustrates that Schur-optimality is stronger than type-1 optimality. For any r ≥ 3, ¯ are type-1-optimal for any 0 ≤ γ ≤ ECD(θ)s 1 r

Schur-optimal for

1 2

but (given existence of a ECD(θ¯ + 1)) cannot be

≤ γ ≤ 21 .

¯ are E-better than ECD(θ¯ + 1)s if and only if γ < Corollary 11 ECD(θ)s

r−1 r .

¯ and Now the scope will be widened. Having examined the optimality ordering of ECD(θ) ECD(θ¯ + 1), NECDS and UECDs can be evaluated relative to these two. Lemma 12 Let d ∈ D(v, r; k1 , k2 ) have concurrence discrepancy matrix ∆d = (δdii0 ) and optimality matrix Md for which the maximum and minimum eigenvalues are ed1 and edr , respectively. (i)

δd12 ≤ 0

⇒

ed1 ≥

p v

+ γ − δd12

and

(ii)

δd12 ≤ 0

⇒

ed1 ≥

p v

− γ + δd12

and

edr ≤ edr ≤

p v p v

− γ + δd12 .

+ γ − δd12 .

Proof The leading 2×2 minor of Md , which is Md11 = ( vp +γ −δd12 )I −(γ −δd12 )J, has eigenvalues p v

+ γ − δd12 and

bounds.

p v

− γ + δd12 . A Sturmian Separation Theorem (Rao, 1973, page 64) provides the

2

Corollary 13 Suppose d ∈ D(v, b; k1 , k2 ) is a UECD with δdii0 ≤ −α for some 1 ≤ i 6= i0 ≤ r and some integer α ≥ 1. Then ¯ are Schur-better than d if γ ≤ (i) ECD(θ)s

α r−2

(ii) ECD(θ¯ + 1)s are Schur better than d if γ ≥

r−α−1 . r

Proof For the UECDd as described in the corollary, take δd12 ≤ −α. Then from lemma 12, p v

ed1 ≥

ξ2 (γ) ≥ ed1 ≥

p v

+ γ − α, and p v

p v

¯ is Schur-better than d if − γ + α ≥ edr . By Theorem 8, an ECD(θ)

− γ + α ≥ edr ⇐⇒ γ ≤

α r−2 .

+ γ − α ≥ ξ2 (γ − 1) ⇐⇒ γ ≥

By Theorem 9, an ECD(θ¯ + 1) is Schur-better than d if

r−α−1 . r

14

Corollary 14 When r ≤ 4, all UECDs with δdii0 ≤ −1 for some 1 ≤ i 6= i0 ≤ r are Schur-inferior to an ECD, and when r = 5 or 6, UECDs with δdii0 ≤ −2 for some 1 ≤ i 6= i0 ≤ r are Schur-inferior to an ECD. Corollary 15 Suppose d ∈ D(v, b; k1 , k2 ) is a UECD with δdii0 ≥ α for some 1 ≤ i 6= i0 ≤ r and some integer α ≥ 2. Then ¯ are Schur-better than d if γ ≤ α , (i) ECD(θ)s r (ii) ECD(θ¯ + 1)s are Schur better than d if γ ≥

r−α−1 r−2 .

Proof For the UECD d as described in the lemma, take δd12 ≥ α ≥ 2. Then from lemma 12, p v

ed1 ≥

ξ2 (γ) ≥ ed1 ≥

p v

− γ + α, and p v

p v

¯ is Schur-better than d if + γ − α ≥ edr . By Theorem 8, an ECD(θ)

+ γ − α ≥ edr ⇐⇒ γ ≤

α r.

− γ + α ≥ ξ2 (γ − 1) ⇐⇒ γ ≥

By Theorem 9, an ECD(θ¯ + 1) is Schur-better than d if

r−α−1 r−2 .

Corollary 16 When r ≤ 4, all UECDs with δdii0 ≥ 2 for some 1 ≤ i 6= i0 ≤ r are Schur-inferior to an ECD, and when r = 5 or 6, UECDs with δdii0 ≥ 3 for some 1 ≤ i 6= i0 ≤ r are Schur-inferior to an ECD. Corollaries 14 and 16 say that optimal (with respect to any convex criterion) designs in settings ¯ an ECD(θ¯ + 1), or an NECD, and optimal designs D(v, r; k1 , k2 ) with r ≤ 4 must be an ECD(θ), in settings with r = 5 or 6 must have block concurrence discrepancies δdii0 ∈ {−1, 0, 1, 2} for all 1 ≤ i 6= i0 ≤ r. It is unlikely that such global statements can be much improved. The importance of these results lies in the prevalence of small r in the application of resolvable designs.

5

E-optimality

Sufficient conditions, which are necessary given existence, will be developed for E-optimality of designs in D(v, r; k1 , k2 ). UECDs are immediately ruled out. ¯ are E-better than UECDs. Corollary 17 For all r ≥ 2 and 0 ≤ γ < 1, ECD(θ)s

15

Proof For the UECD d suppose δd12 ≤ −α for some integer α ≥ 1. Than by lemma 12, ed1 ≥ p v

¯ are E-better than d. If δd12 ≥ α for some integer α ≥ 2, then + γ − δd12 > ξ1 (γ), and ECD(θ)s

another application of the lemma gives ed1 ≥

p v

¯ are E-better − γ + δd12 > ξ1 (γ) and again ECD(θ)s

than d. ¯ are E-optimal, uniquely so when γ 6= 1 . Corollary 18 When 0 ≤ γ ≤ 12 , ECD(θ)s 2 ¯ are E-equivalent or better than ECD(θ¯ + 1)s when γ ≤ 1 , with Proof By corollary 11, ECD(θ)s 2 ¯ are always E-better than UECDs by corollary E-equivalence only when r = 2 and γ = 21 . ECD(θ)s ¯ in any resolvable design setting 17. The maximum eigenvalue of the optimality matrix for ECD(θ)s (v, r; k1 , k2 ) is ξ1 (γ) =

p v

+ γ, and with a proper labelling of the replications, the optimality matrix

of a NECD has δd12 = 1. Then, from lemma 12, zd1 ≥

p v

+ (1 − γ) ≥ ξ1 (γ) for 0 ≤ γ ≤ 12 , with

equality only when γ = 12 . The next lemma provides bounds for the maximum and minimum eigenvalues of the optimality matrix in terms of the eigenvalues of the corresponding centered discrepancy matrix. Lemma 19 For d ∈ D(v, r; k1 , k2 ) let ud1 and udr be the maximum and minimum eigenvalues of ∆d0 = P T ∆d P , where P = (I − 1r J). Then ed1 ≥

p + γ + ud1 v

edr ≤

p + γ + udr . v

provided ud1 > 0, and

Proof ed1 =

max xT Md x

xT x=1

p + γ)I − γJ + ∆d ]x v xT x=1 p ≥ max xT [( + γ)I − γJ + ∆d ]x T v x x=1 =

max xT [(

xT 1=0

=

p + γ + max xT ∆d x v xT x=1

=

p + γ + max xT PT ∆d Px v xT x=1

xT 1=0

xT 1=0

16

(23)

= =

p + γ + max xT PT ∆d Px v xT x=1 p + γ + ud1 . v

(24)

Equality (24) holds since ud1 > 0, 1T PT ∆d P1 = 0, and P T ∆d P 1 = 01 (that is, 1 is an eigenvector of P T ∆d P with eigenvalue 0). Likewise edr = =

min xT Md x

xT x=1

min xT [(

xT x=1

p + γ)I − γJ + ∆d ]x v

≤

p + γ + min xT PT ∆d Px v xT x=1

=

p + γ + udr v

xT 1=0

(25)

Equality (25) is true provided udr < 0, for similar reasons to above. If udr > 0, the bound still holds, since edr ≤

tr(Md ) p p = ≤ + γ + udr . 2 r v v

As a corollary, here is a result that is vital to deriving a characterization of E-optimal designs. A bit more is learned about the Schur comparisons studied in the previous section as well. Corollary 20 Let d ∈ D(v, r; k1 , k2 ) be a resolvable design with optimality matrix Md , whose ¯ or an ECD(θ¯ + 1). Let ud1 and udr be the eigenvalues are not identical to those of an ECD(θ) maximum and minimum eigenvalues, respectively, of ∆d0 = P T ∆d P , P = (I − 1r J). If γ<−

udr r

(26)

¯ are Schur-better than d. If ud1 > 0 and then ECD(θ)s µ

γ>

r − ud1 − 1 r

¶

(27)

then ECD(θ¯ + 1)s are Schur-better than d. Furthermore, if ud1 > 0 ¯ are E-better, but not necessarily Schur-better, than d. then ECD(θ)s Proof The result follows immediately from Theorems 8 and 9 and lemma 19. 17

(28)

¯ and for the E-criterion, UECDs are ruled out by corollary 17. For all Now, relative to ECD(θ)s ¯ θ¯ + 1} for all i 6= i0 and so ∆d is the adjacency matrix remaining competitors, φdii0 must be in {θ, for a simple, undirected graph on r vertices. Inequality (28) of corollary 20 further pares the class of competitors when combined with the next lemma. Lemma 21 Let A be the adjacency matrix for a simple undirected graph of r vertices. All the eigenvalues of P AP are nonpositive if and only if A − J may be written (possibly after vertex permutation) in block diagonal form 

  −Jt1     A−J =     

−Jt2 ..

. −Jtn

for some positive integers t1 ≤ t2 ≤ . . . ≤ tn and n with

Pn

i=1 ti

          

(29)

= r.

Proof Suppose A − J is of the suggested form. Write t(i) for t1 + t2 + . . . + ti and t(0) = 0. Then (since P J = 0) P AP = P (A − J)P , implying that max eigenvalue(P AP ) = max l0 P (A − J)P l 0 l l=1

= =

max

y 0 (A − J)y

max 0

−

y=P l,l0 l=1

y=P l,l l=1

t X

(

n

(i) X

yj )2

i=1 j=n(i−1) +1

≤ 0. Now suppose all of the eigenvalues of P AP are nonpositive. Exhaustive enumeration shows that A must have the form (29) for r = 3, 4, 5. Subscripting A by its dimension, suppose nonpositivity implies that Ar must have that form for a given r, and write r∗ = r + 1. Since every principal

18

minor of Ar∗ is an adjacency matrix for a graph with fewer vertices, 

  −Jt1       Ar∗ − J =        

     ar×1          

−Jt2 ..

. −Jtn

a0

−1

for some vector ar×1 of 0’s and -1’s. Indeed, by the induction hypothesis, every s × s principal minor of Ar∗ − J must have the form (29), so a may be partitioned as a0 = (a01 , a02 , . . . , a0n ) where ai is either −1ti or 0ti . If n = 1, the proof is done. If n = 2 and a2 = −1t2 , then 



 −Jt1   Ar∗ − J =   0  

a01

0 −Jt2 −10t2

a1

   −1t2  .  

−1





 −1   If also a1 = −1t1 , then Ar∗ − J contains a principal minor M3 =   0  

0

−1 

  −1 −1   which is not  

−1 −1 −1

of the form (29), contradicting the fact that the result holds for r = 3. Thus at most one of a1 , a2 is nonzero. It follows immediately that for n > 2 the same statement holds for any pair ai , ai0 . Permuting rows and columns as needed, it can be assumed that ai = 0ti for i < n and consequently Ar∗ − J has the form (29). ¯ exists, then an E-optimal design is either an ECD(θ), ¯ an ECD(θ¯ + 1), Lemma 22 If an ECD(θ) or an NECD with discrepancy matrix ∆d for which ∆d − J has the form (29). ¯ eliminates NAECDs from contention, and Proof As already mentioned, existence of an ECD(θ) for any other design ∆d is the adjacency matrix for a simple undirected graph. If P ∆d P has a positive eigenvalue, d is eliminated by corollary 20. Lemma 21 gives the result. ¯ n = 1 and t1 = r, Indeed, all of the designs of lemma 22 have ∆d − J of form (29). For ECD(θ), while for ECD(θ¯ + 1), n = r and t1 = · · · = tr = 1. Such designs are said to be group-affine: 19

concurrences φdii0 are constant within groups of sizes t1 , . . . , tn , and are constant across groups. A group affine design is said to be uniform if, for given n, the range of group sizes ti is at most one. For any group-affine design, Md = ( vp + γ)I + ∆d − γJ where for given t1 ≤ . . . ≤ tn 



γJt1 ,t1     (γ − 1)J  t2 ,t1 ∆d − γJ = (−1)   .  ..   

(γ − 1)Jtn ,t1

(γ − 1)Jt1 ,t2 γJt2 ,t2

· · · (γ − 1)Jt1 ,tn 

     ≡ (−1)Hd .     

· · · (γ − 1)Jt2 ,tn

.. .

..

(γ − 1)Jtn ,t2

···

.. .

.

γJtn ,tn

An E-optimal design will maximize the minimum eigenvalue of Hd . Clearly Hd has

Pn

(30)

i=1 (ti

− 1) =

r − n eigenvalues of zero (corresponding to eigenvectors which are orthogonal contrasts within groups of sizes t1 , . . . , tn ). So all of these designs have Hd with at least one eigenvalue of zero, except ECD(θ¯ + 1), for which the eigenvalues of Hd are 1 (frequency r − 1) and 1 + r(γ − 1). ECD(θ¯ + 1) is therefore E-optimal if 1 + r(γ − 1) ≥ 0, that is, if γ ≥

r−1 r

(in which case it is

Schur-optimal; see Theorem 9). Otherwise, all designs for which Hd is nonnegative definite are E-optimal. Needed now are the eigenvalues of Hd other than zero. Let Dt be the diagonal matrix with (t1 , . . . , tn ) diagonal elements. Lemma 23 The eigenvalues of Hd specified by (30) are 0 (with frequency r − n) and the eigenvalues 1/2

1/2

of Dt EDt , where E = In − (1 − γ)Jn . Proof See appendix. The best of the optimality matrices Md can now be characterized for the E-criterion. Theorem 24 Any group-affine design is E-optimal if n ≤

1 1−γ

and γ ≤

r−1 r .

exists, then all such designs form the class of all E-optimal designs. If γ ≥

r−1 r ,

If any such design then ECD(θ¯ + 1)s

are Schur-optimal. 1/2

1/2

Proof E-optimality follows if Hd is nonnegative definite, or equivalently (by lemma 23) if Dt EDt

is nonnegative definite. This is so if and only if E is nonnegative definite, that is, if and only if 1 − γ ≤ n1 .

20

Though not much discussed in the literature, the class of E-optimal designs in a given simple block design setting often contains a variety of designs with different information matrices (see, e.g., Morgan 2005 and Morgan and Uddin 1992). Theorem 24 reveals the same situation for resolvable designs settings. This allows other design criteria to be brought to bear: one should choose the best of the E-optimal designs. Let DE ≡ DE (v, r; k1 , k2 ) denote the class of all E-optimal designs as identified in Theorem 24. The design d∗ ∈ DE is said to be E-Schur optimal if it is Schur-optimal 1 amongst all members of DE . Let nγ = int( 1−γ ). For each d ∈ DE , let t(d) denote the vector of ti ’s

arranged in increasing order and with nγ elements, padding with zeros as necessary. For example, if r = 7, γ =

7 9

and d has ti values 1, 3 and 3, then nγ = 4 and t(d) = (0, 1, 3, 3).

Theorem 25 For d1 , d2 ∈ DE , if t(d1 ) ≺ t(d2 ) then d1 is Schur-better than d2 . Design d∗ is E-Schur optimal if it is uniform with n = nγ . Proof It is shown in the appendix that the eigenvalues of Hd , aside from r − nγ zeros, are the eigenvalues of E 1/2 Dt E 1/2 . Now t(d1 ) ≺ t(d2 ) is equivalent to t(d1 ) = St(d2 ) where S is a doubly stochastic matrix; S =

Pm

i=1 ai Pi

for permutation matrices Pi and positive numbers ai summing to

1 (Bhatia, 1997, pages 33 and 37). Thus X

E 1/2 Dt(d1 ) E 1/2 = E 1/2 DSt(d2 ) E 1/2 = E 1/2 [ the last step because E 1/2 = (I − n1 J) +

q

1 n

i

ai Pi Dt(d2 ) Pi ]E 1/2 =

X i

ai Pi [E 1/2 Dt(d2 ) E 1/2 ]Pi

− (1 − γ)J commutes with any permutation matrix.

This shows that E 1/2 Dt(d1 ) E 1/2 is a symmetrization of E 1/2 Dt(d2 ) E 1/2 and thus the eigenvalues of Hd1 are majorized by those of Hd2 (this follows from Bhatia, page 69). Since t(d∗ ) ≺ t(d) for every d ∈ DE , d∗ is E-Schur optimal. Amongst all conceivable optimality matrices Md , Theorem 24 characterizes the E-best. In this potentially large class of E-best structures, Theorem 25 identifies the best with respect to all Schur-convex criteria.

6

Special Cases: (k1 − k2 ) ≤ 2

In this section the three important special cases of k1 and k2 being equal or nearly so, k2 ∈ {k1 , k1 − 1, k1 − 2}, are investigated in light of the results in sections 4 and 5. Writing k2 = k1 − m, 21

then (k1 − k2 ) ≤ 2 says that m = 0, 1, or 2, and for any m k12 k12 k1 m m2 = = + + . k1 + k2 2k1 − m 2 4 4(2k1 − m) Recall that θ¯ is the integer part of (31). The values for γ =

k12 k1 +k2

(31)

− θ¯ in the corollaries below are

easily found using (31). Corollary 26 For k1 = k2 , (i) if 2 | k1 then γ = 0, (k1 + k2 ) | k12 , and ECD(θ∗ )s are Schur-optimal. 1 2

(ii) if 2 /| k1 then γ =

¯ are E-Schur and type-1-optimal. and ECD(θ)s

When k1 = k2 and 2 | k1 , Schur-optimality also follows from Bailey, Monod, and Morgan (corollary 3.4, 1995). For 2 /| k1 , the result is from Theorems 7 and 25. Corollary 27 For k1 − k2 = 1, (i) if 2 | k1 , then γ =

v+1 4v ,

(ii) if 2 /| k1 , then γ =

¯ are E-Schur and type-1-optimal. and ECD(θ)s

3v+1 4v

and uniform group-affine designs with four groups are E-Schur-

optimal if r ≥ 5. For r ≤ 4, ECD(θ¯ + 1)s are Schur-optimal. As an example, the design in Table 1 is E-Schur-optimal for 9 treatments in 5 replicates with block sizes 4 and 5. The design consisiting of the first four replicates is Schur-optimal. Corollary 28 For k1 − k2 = 2, (i) if 2 | k1 then γ =

v+2 2v ,

and uniform group-affine designs are E-Schur-optimal if the number of

groups is 3 for v = 6; and 2 for v ≥ 10. For v = 6 and r = 2, ECD(θ¯ + 1)s are Schur-optimal. ¯ are Schur-optimal. (ii) if 2 /| k1 , then γ = v1 , and ECD(θ)s The Schur-optimality in part (ii) of corollary 28 is from Theorem 8 and lemma 12.

22

7

ECDs, balanced arrays, and design construction

Balanced arrays were introduced by Chakravarti (1956, 1961) as a useful device for fractional factorial designs, and have since been investigated by a plethora of authors including of late Kuriki (1993), Fuji-Hara and Miyamoto (2000), Sinha et al (2002), and Ghosh and Teschmacher (2002). Here only strength two arrays on two symbols will be needed. A balanced array of strength 2, BA(N, m, 2), on the symbols 0 and 1, is a N × m array with the property that for any selection of two columns, the N pairs formed by the rows are (0, 0), (0, 1), (1, 0), and (1, 1) with frequencies µ0 , µ1 , µ1 , and µ2 , respectively. Two-symbol orthogonal arrays of strength two are the special case µ0 = µ1 = µ2 . When the number of 0’s in each column is specified (as is the case below), the µi ’s are all determined by θ = µ0 and the array will be denoted BA(N, m, 2; θ). Bailey, Monod, and Morgan (1995) demonstrate the combinatorial equivalence between affine resolvable designs and orthogonal arrays. Their method can be used to express any resolvable design as a combinatorial array, as follows. Given a resolvable design for v treatments in r replicates, each consisting of s blocks, arbitrarily label the blocks within a replicate 0, . . . , s − 1. Construct a v × r array by identifying rows of the array with treatments of the resolvable design, and columns with replicates: symbol l ∈ {0, 1, . . . , s − 1} is placed in row i, column j if and only if treatment i is in block l of replicate j. Evidently, Theorem 29 Each ECD(θ) in D(v, r; k1 , k2 ) is equivalent to a BA(v, r, 2; θ). A group affine design with n groups of sizes t1 , . . . , tn is a juxtaposition (BA1 ,BA2 ,. . .,BAn ) of balanced arrays BAi =BA(v, ti , 2; θ) so that any two columns from different BAi ’s form a BA(v, 2, 2; θ+ 1). Call such a juxtaposition a grouped balanced array, denoted by GBA(v, (t1 , . . . , tn ), 2; θ). Constructions for the designs in corollaries 26-28 of section 6 are now listed. These are stated in terms of Hadamard matrices: orthogonal matrices for which every element is 1 or −1. The Hadamard conjecture says that a Hadamard matrix exists for every order a multiple of four. Existence is know for all such orders up to 428, and for infinitely many other values (see Craigen, 1996) A Hadamard matrix is said to be standardized if the first row and column are all ones; this can always be done. Theorem 30 If k1 = k2 is even, then a Schur-optimal ECD(θ∗ ) corresponds to an OA(v, r, 2; v2 ). Existence of a Hadamard matrix of order v implies existence of the OA for every r ≤ v − 1. 23

¯ corresponds to a BA(v, r, 2; v−2 ). Theorem 31 If k1 = k2 is odd, then a type-1-optimal ECD(θ) 4 Existence of a Hadamard matrix of order v + 2 implies existence of the BA for every r ≤ v2 . 2 Proof The value of θ¯ is int( (v/2) v ) =

v−2 4 .

Given the standardized Hadamard matrix, permute

columns (except the first) so that the second row has 1 in the first first two rows and the first

v+2 2

v+2 2

columns. Now delete the

columns, then replace −1 by 0 throughout. Clearly the result is

BA(v, v2 , 2; v−2 4 ). ¯ corresponds to a BA(v, r, 2; v+1 ). Theorem 32 If k1 = k2 +1 is even, then a type-1-optimal ECD(θ) 4 Existence of a Hadamard matrix of order v + 1 implies existence of the BA for every r ≤ v. 2 Proof The value of θ¯ is int( ((v+1)/2) )= v

v+1 4 .

Given the standardized Hadamard matrix, delete

the first row and column, then replace −1 by 0 throughout. The result is BA(v, v, 2; v+1 4 ). Theorem 33 If k1 = k2 + 1 is odd, then a type-1-optimal ECD(θ¯ + 1) with r ≤ 4 corresponds to a BA(v, r, 2; v+3 4 ), which always exists. For r ≥ 5 and v ≥ 9, an E-Schur-optimal group affine design corresponds to a GBA(v, (t1 , t2 , t3 , t4 ), 2; v−1 4 ) with t4 − t1 ≤ 1. Existence of a Hadamard matrix of order v + 3 with a 4 × r submatrix of the form            



1t1

1t2

1t3

1t1

1t2

−1t3

1t1

−1t2

1t3

−1t1

1t2

1t3

−1t4    1t4     1t4    

1t4

implies existence of the GBA. 2 Proof The value of θ¯ is int( ((v+1)/2) )= v

v−1 4 .

For r ≤ 4 take OA(v − 1, r, 2) on {0, 1} and add one

row of 0’s to get BA(v, r, 2; v+3 4 ). Given the assumed Hadamard matrix, delete v + 3 − r columns not containing the submatrix, then delete the submatrix and add a row of 1’s. Replacing −1 by 0 throughout gives the GBA. The maximum r admitted for given v by the Hadamard construction in Theorem 33 depends on the particular Hadamard matrix chosen. All nonisomorphic Hadamard matrices are known up through 24

order 28, and for these, a complete search has produced (v, r) = (9, 5), (13, 5), (17, 9), (21, 9), (25, 13). A search of a few known Hadamard matrices of orders up to 48 (compiled by N. J. A. Sloane in the online catalog http://www.research.att.com/ ∼njas/hadamard/ ) has further produced (v, r) = (29, 13), (33, 16), (37, 18), (41, 24), (45, 19). The GBAs appear in Appendix B. Theorem 34 If k1 = k2 +2 is even, for v ≥ 10, an E-Schur-optimal group affine design corresponds to a GBA(v, (t1 , t2 ), 2; v+2 4 ) with t2 − t1 ≤ 1. Existence of a Hadamard matrix of order v + 2 implies existence of the GBA for every t1 + t2 = r ≤ 2 Proof The value of θ¯ is int( ((v+2)/2) )= v

v 2

v+2 4 .

+ 1. Given the standardized Hadamard matrix of order

v + 2 = 4h, permute columns (except the first) so that the first three rows are 



 1h     1h  

1h

1h

−1h

1h

1h −1h −1h

1h 

  −1h    

1h

Now delete the first 2h columns, delete the first three rows, then add one row of −1’s, and fiv+2 v+2 nally replace −1 by 0 throughout. The result is GBA(v, ( v+2 4 , 4 ), 2; 4 ) with two juxtaposed v+2 r BA(v, v+2 4 , 2; 4 ) being the first h and last h columns. For smaller r, delete int(h − 2 ) columns

from the first component BA, and int(h −

r+1 2 )

from the second.

¯ corresponds to a BA(v, r, 2; v+4 ). Theorem 35 If k1 = k2 + 2 is odd, a Schur-optimal ECD(θ) 4 Existence of a Hadamard matrix of order v implies existence of the BA for every r ≤ v − 1. 2 Proof The value of θ¯ is int( ((v+2)/2) )= v

v+4 4 .

Given the standardized Hadamard matrix, delete

the first row and column, add a row of −1’s, then replace −1 by 0 throughout. The result is BA(v, v − 1, 2; v+4 4 ).

8

Comments

By exploiting properties of the dual, optimality theory for resolvable designs with two blocks per replicate has been developed. Section 4 establishes conditions for Schur-optimality of equal concurrence designs depending on the block discordancy coefficient γ. For γ where Schur-optimality 25

is not established, the class of competitors has been significantly narrowed via Schur-ordering for small r. Section 5 gives a complete solution to the E-optimality problem whenever any grouped affine design exists, and further determines the best of the E-optimal designs. Sections 6 and 7 apply these results for the important cases k1 − k2 ≤ 2, including explicit design constructions. It is evident from the constructions that many other designs, eliminated by the Schur-domination argument in Theorem 25, do exist. The optimality route is clear for γ ≤ 12 , but for larger γ there are still unanswered questions. For instance, A- and E-optimality need not coincide, and the problem of determining an A-best design remains open. For r ≤ 4 the authors have solved the A-optimality problem in its entirety, including construction. These and related results will be reported elsewhere.

ACKNOWLEDGEMENT This research is partially based on work in Reck’s doctoral dissertation (Reck, 2002), for which Morgan was director. Morgan completed his Ph.D. under the direction of I. M. Chakravarti. We dedicate this paper to the memory of Professor Chakravarti, and in doing so humbly acknowledge the ongoing impact of his work on our own, and on that of the statistical and mathematical communities at large. J. P. Morgan was supported by National Science Foundation grant DMS0104195. Brian Reck was supported by the NIMH training grant 5T32MH020053-05

A

Eigenvalue equations for Hd

Any vector of the form (c01 , c02 , . . . c0n ), where ci ∈ Rti is either a contrast vector or the zero vector, is an eigenvector of Hd with eigenvalue 0. Consequently all other eigenvectors are of the form e0 = (x1 10t1 , x2 10t2 , . . . , xn 10tn ) for some scalars x1 , . . . , xn . The first equation in the system Hd e = λe is γt1 x1 − (1 − γ)t2 x2 − · · · − (1 − γ)tn xn = λx1 ; the other equations may be written similarly to see that Hd e = λe are equivalent to [Dt −(1−γ)1n t0 ]x = λx where t = (t1 , . . . , tn )0 and x = (x1 , . . . , xn )0 . Thus the remaining eigenvalues of Hd are the right eigenvalues of Dt −(1−γ)1n t0 . Now Dt is positive definite and E is nonnegative definite for γ ≥

n−1 n ,

so both have symmetric square root matrices

and Dt is invertible. Thus |Dt − (1 − γ)1n t0 − λI| = 0

⇐⇒

|Dt − (1 − γ)1n t0 − λI||Dt−1 | = 0 26

⇐⇒

|E − λDt−1 | = 0

⇐⇒

|Dt EDt

⇐⇒

|E 1/2 Dt E 1/2 − λI| = 0 (needed in Theorem 25)

1/2

1/2

− λI| = 0 (proving lemma 23)

1/2

the last step because GG0 and G0 G have the same eigenvalues for any square G (here G = Dt E 1/2 ). Now in the proof of Theorem 25 some elements of t are allowed to be zero (without loss of generality, t1 = t2 = · · · = tz = 0 for some integer z ≥ 1), and then Dt is not invertible. In this case there are z additional zero eigenvalues plus a reduced system of n − z equations in tz+1 , . . . , tn . It is easy to see 1/2

1/2

that |Dt EDt

˜ 1/2 E ˜D ˜ 1/2 | where E ˜ and D ˜ t are the lower right submatrices − λI| = |0z,z − λIz ||D t t 1/2

1/2

of E and Dt of order n − z. Thus the n eigenvalues sought are still those of Dt EDt

and thus

of E 1/2 Dt E 1/2 .

B

GBAs with four groups

GBA(9, (1, 1, 1, 2), 2; 2)

1 0 0 0 0 1 0 1 1

1 0 1 1 0 0 0 1 0

1 0 0 0 1 0 1 1 0

1 0 0 1 1 1 0 0 0

1 0 1 0 0 0 1 0 1

GBA(13, (1, 1, 1, 2), 2; 3)

1 1 0 0 1 0 1 0 1 0 0 0 1

1 1 1 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 1 1 1 0 0 0

1 1 0 1 1 0 0 1 0 0 1 0 0

1 1 0 0 0 1 1 0 0 1 0 1 0

GBA(17, (2, 2, 2, 3), 2; 4)

1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 0

1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1

27

1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1

1 0 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0

1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 0

1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 0

1 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 1

1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 0

GBA(21, (2, 2, 2, 3), 2; 5)

1 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 1

1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0

1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0

1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0

1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 1 1

1 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1

1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0

1 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0

1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 1

1 1 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1

GBA(25, (3, 3, 3, 4), 2; 6) 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1

1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 1 0 1 1 1

1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0

1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0

1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0

1 1 0 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0

1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0

1 1 0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0

1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 1

1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1

1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1

GBA(29, (3, 3, 3, 4), 2; 7) 1 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0

1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1

1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0

28

1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1

1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 0 0

1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0

1 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0

1 0 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 1

1 0 1 0 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0

1 0 0 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0

1 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1

1 0 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0

1 0 0 0 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 0 1 0 1

1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0 1 0 1 0

1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1

GBA(33, (4, 4, 4, 4), 2; 8) 1 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1

1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1

1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0

1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0

1 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0

1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0

1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0

1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1

1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0

1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1

1 1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0

1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1

GBA(37, (4, 4, 5, 5), 2; 9) 1 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1

1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1

1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0

1 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0

1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 1

29

1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1

1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1

1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0

1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0

1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1

1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0

1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0

1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0

1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 1

1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1

1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0

1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1

1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 0

1 0 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1

1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0

1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1

1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 0

GBA(41, (6, 6, 6, 6), 2; 10) 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 1

1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1

1 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 1

1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0

1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1

1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1

1 0 0 1 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1

1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1

1 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0

1 0 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0

1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1

1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 1 0

1 1 1 0 1 1 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0

30

1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0

1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1

1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 1

1 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0

1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 1 0 1

1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0

1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0

1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1

1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0

1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0

1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1

GBA(45, (4, 5, 5, 5), 2; 11) 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0

1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0

1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1

1 0 1 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 0

1 0 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0

1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1

1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1

1 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1

1 0 0 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0

1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0

31

1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1

1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0

1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 0 1 1

1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1

1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0

1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0

1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 0

1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 1

1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1

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