A-optimal diallel crosses for test versus control comparisons. Summary. 1. Introduction

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1 A-otimal diallel crosses for test versus control comarisons By ASHISH DAS Indian Statistical Institute, New Delhi , India SUDHIR GUPTA Northern Illinois University, Dekal, IL 60115, USA and SANPEI KAGEYAMA Hiroshima University, Higashi-Hiroshima , Jaan Summary A-otimality of lock designs for control versus test comarisons in diallel crosses is investigated. A sufficient condition for designs to e A-otimal is derived. Tye S 0 designs are defined and A-otimal tye S 0 designs are characterized. A lower ound to the A-efficiency of tye S 0 designs is also given. Using the lower ound to A-efficiency, tye S 0 designs are shown to yield efficient designs for test versus control comarisons. Some key words: Tye S design; Tye S 0 design; A-otimality; Inred lines; A-efficiency. 1. Introduction Although designs for varietal trials and factorial exeriments have een extensively investigated in literature over the ast several decades, it was not until recently that some rogress in the design of diallel cross exeriments has een made, see e.g. Guta and Kageyama (1994), Dey and Midha (1996), Mukerjee (1997), Das, Dey and Dean (1998). Designs for control versus test comarisons where the treatments form different levels of a factor have also een extensively investigated in the literature; see Majumdar (1996). The rolem of deriving designs aroriate for diallel crosses is quite different from the set-u of designs for varietal trials and factorial exeriments. Therefore, here we continue the work of Guta and Kageyama (1994) for studying otimal lock designs for control versus test comarisons among the lines with resect to their general comining aility effects. Recently Choi, Guta and Kageyama (2002) introduced a class of designs, called tye S lock designs, for control-test comarisons in a diallel cross exeriment. Let (> 2),, and k denote the numer of test lines, numer of locks and lock size resectively. In Section 2 we define a su-class of tye S designs, called tye S 0 designs, and derive a sufficient condition for designs to e A-otimal. Henceforth, y otimal we mean A-otimal, and y efficiency we mean A-efficiency. We also characterize otimal tye S 0 designs in Section 2, and show the otimality of some of the designs of Choi, Guta and Kageyama (2002). A new method of constructing tye S 0 designs is also rovided there. A tye S 0 design satisfying the derived sufficient condition for otimality does not always exist. Therefore, a lower ound to the efficiency of a tye S 0 design is defined in Section 3. For tye S 0 designs in a ractical range 30, 50, k, 70 designs out of a total of 247 ossile tyes S 0 designs are otimal. Of the remaining 177 designs that do not satisfy the sufficient condition for otimality, 175

2 designs have lower ound to efficiency at least Thus, tye S 0 designs rovide highly efficient designs for control-test comarisons. For the sake of revity, tale of efficient tye S 0 designs is not resented in this aer, and it will e reorted elsewhere. 2. Otimal designs We consider diallel cross exeriments involving + 1 inred lines, giving rise to a total of n c = ( + 1)/2 distinct crosses. Let a cross etween lines i and j e denoted y (i, j), i < j = 0, 1,...,. Suose line 0 is a control or a standard line and lines 1,..., are test lines. Let s dj denote the total numer of times that the jth line occurs in the crosses in the design d, j = 0, 1,...,. Further let s d = (s d0, s d1,..., s d ) and let n denote the total numer of crosses in the design. Following e.g., Guta and Kageyama (1994), the model under the lock design set-u, for a design d involving + 1 inred lines and locks each containing k crosses, is assumed to e Y d = µ1 n + 1d τ + 2d β + ε, where Y d is the n 1 vector of resonses, µ is the overall mean, 1 t is the t 1 vector of 1 s, τ = (τ 0, τ 1,..., τ ) is the vector of +1 general comining aility effects, β = (β 1, β 2,..., β ) is the vector of lock effects and 1d ( 2d ) is the corresonding oservation versus line (lock) design matrix, that is, the (h, l)th element of 1d ( 2d ), is 1 if the hth oservation ertains to the lth line (lock), and is zero otherwise, and ε is the n 1 vector of indeendent random errors with zero exectation and a constant variance σ 2. The coefficient matrix of the reduced normal equations for estimating the vector of general comining aility effects is then given y C d = G d 1 k N dn d (2.1) where N d = (n dij ), i = 0, 1,..., ; j = 1,...,, is the ( + 1) line versus lock incidence matrix, G d = (g dii ), g dii = s di, and for i i, g dii is the numer of times the cross (i, i ) aears in the design. Let D( + 1,, k) denote the set of all connected designs with test lines, one control line and k crosses arranged in locks each of size k. A design d D( + 1,, k) is said to e otimal for control-test comarisons if it minimizes i=1 V ar(ˆτ di ˆτ d0 ), where ˆτ di ˆτ d0 denotes the est linear uniased estimator (BLUE) of τ i τ 0 using d. Let P = ( 1 I ) where I t denotes the identity matrix of order t. Then the covariance matrix for the BLUE s (ˆτ d1 ˆτ d0, ˆτ d2 ˆτ d0,..., ˆτ d ˆτ d0 ) of the control-test contrast is σ 2 P Cd P. If one artitions C d as ) ( cd00 γ d C d = γ d M d (2.2) then it can e shown (see Guta, 1989) that (P C d P ) 1 = M d, i.e., M d is the information matrix for the control-test contrasts. For a design d in D( + 1,, k), using Kiefer s (1975) technique of averaging, we otain tr(p C d P ) tr(p C d P ); (2.3) 2

3 see also Majumdar and Notz (1983) and Jacroux and Majumdar (1989). Here C d = 1! π πc d π, the summation eing taken over all ( + 1) ( + 1) ermutation matrices π that corresond to ermutations of the test treatments only. Partitioning C d as in (2.2), we see that M d = (P C d P ) 1 is a comletely symmetric matrix. In general, there may e no design in D( + 1,, k) for which M d is the information matrix for the control-test contrasts. If there is such a design, then for this design, call it d, M d = M d is comletely symmetric and γ d of (2.2) is a vector with all entries equal. That is, d elongs to a class of designs, called tye S lock designs, introduced y Choi, Guta and Kageyama (2002). Definition 2.1. A design d D( +1,, k) is called a tye S lock design if there are ositive integers g 0, g 1, λ 0 and λ 1, such that for i i = 1,...,, g d0i = g 0, g dii = g 1, Σ j=1n d0j n dij = λ 0, Σ j=1n dij n di j = λ 1. We denote a tye S lock design with arameters,, k, g 0, g 1, λ 0 and λ 1 y S(,, k, g 0, g 1, λ 0, λ 1 ). For an S(,, k, g 0, g 1, λ 0, λ 1 ) design d it holds that s d0 = g 0, s d1 = s d2 = = s d = g 0 + ( 1)g 1, k = (s d0 + s d1 )/2, V ar(ˆτ di ˆτ d0 ) = k{a 1 ( 2) 1 }σ 2 (a ){a 1 ( 1) 1 }, i = 1,...,, Cov(ˆτ di ˆτ d0, ˆτ di ˆτ d0 ) = where a 1 = λ 0 kg 0 + ( 1) 1 and 1 = λ 1 kg 1. k 1 σ 2 (a ){a 1 ( 1) 1 }, i i = 1,...,, Definition 2.2. A tye S 0 lock design denoted y S 0 (,, k, g 0, g 1, λ 0, λ 1 ) is a tye S lock design with the roerty that n d0j n d0j 1, n dij n di j 1 for i, i = 1,..., ; j, j = 1,...,. Using [z] to denote the largest integer not exceeding z, we now introduce some notations that are used in the sequel. a(s) = (2k s)(2x + 1) x(x + 1), x = [ 2k s ], h(s) = s(2y + 1) y(y + 1), y = [ s ], g(s;,, k) = + ( 1) 2. s h(s)/k 2k s a(s)/k (s h(s)/k)/ For an S 0 (,, k, g 0, g 1, λ 0, λ 1 ) design it can e shown that n d0j = [ g 0 ] or [ g 0 ] + 1, 2ks 0 = h(s 0 ) + λ 0, 2ks 1 = h(s 1 ) + ( 1)λ 1 + λ 0, and n dij = [ 2k g 0 ] or [ 2k g 0 ] + 1, i = 1,..., ; j = 1,...,. We require the following lemmas for characterizing otimal tye S 0 designs. Lemma 2.1. If d D( + 1,, k) then M d has eigenvalues µ d1, µ d2 = = µ d with µ d1 = ks d0 Σ j=1n 2 d0j k, µ d2 = 2k s d0 1 k Σ i=1σ j=1n 2 dij µ d1. 1 3

4 Proof. From (2.1) and (2.2), the entries of M d are m dii = { sdi 1 k Σ j=1n 2 dij (i = i ) g dii 1 k Σ j=1n dij n di j (i i ) and the sum of the entries in the ith row (or i th column) is i =1 m dii = g d0i+ 1 k Σ j=1n d0j n dij. The lemma then follows y noting that that M d = ξ 1 I +ξ with ξ 2 = 1 ( 1) 1 i i m dii and ξ 1 = 1 i=1 m dii ξ 2. Lemma 2.2 [Cheng, 1978]. For given ositive integers v and t, the minimum of v i=1 n 2 i suject to v i=1 n i = t, where n i s are non-negative integers, is otained when t v[t/v] of the n i s are equal to [t/v] + 1 and v t + v[t/v] are equal to [t/v]. The corresonding minimum of v i=1 n 2 i is t(2[t/v] + 1) v[t/v]([t/v] + 1). Lemma 2.3. Let d D( + 1,, k) and n d01,..., n d0 e fixed quantities. Then tr(m 1 d ) µ 1 d1 + ( 1)2 {2k s d0 a(s d0 )/k µ d1 } 1 (= θ d, say). (2.4) Proof. Using Lemma 2.2 we have Σ i=1σ j=1n 2 dij a(s d0 ). Thus from (2.3) we get tr(md 1 ) µ 1 d1 + ( 1)µ 1 d2 µ 1 d1 + ( 1)2 {2k s d0 a(s d0 )/k µ d1 } 1. Lemma 2.4. Suose d D( + 1,, k) satisfies Σ i=1σ j=1n 2 dij = a(s d0 ) and has s d0 > [ k 2 ]. Then there exists a design d satisfying (i) Σ i=1σ j=1n 2 d ij = a(s d 0) and (ii) s d 0 [ k 2 ] such that θ d θ d unless (i) = 5, k = 3, (ii) = 4, k odd, (iii) = 3. Proof. We relace d y a d which is such that n d 0j = n d0j if n d0j [ k 2 ] and n d 0j = k n d0j if n d0j > [ k 2 ]. Clearly, s d 0 < s d0, and s d 0 satisfies s d 0 [ k 2 ]. Also, µ d 1 = µ d1. The result then follows y noting that the function ψ(s d0 ) = 2k s d0 a(s d0 )/k decreases as s d0 increases excet when (i) = 5, k = 3, (ii) = 4, k odd and (iii) = 3. Lemma 2.5. Let d D( + 1,, k). Then θ d k{(ks 0 h(s d0 )) 1 + ( 1) 2 (k(2k s d0 ) a(s d0 ) ks d0 + h(s d0 )) 1 } (2.5) where θ d is the same as in (2.4). Proof. From Lemma 2.1 and equation (2.4) we have θ d = k{(ks 0 Σ j=1n 2 d0j) 1 + ( 1) 2 (k(2k s d0 ) a(s d0 ) ks d0 + Σ j=1n 2 d0j) 1 } = k{(ks 0 q) 1 + ( 1) 2 (w + q) 1 }, where q = Σ j=1n 2 d0j and w = k(2k s d0 ) a(s d0 ) ks d0. For fixed s d0, s2 d0 q ks d0. We shall rove that δθ d /δq 0 for all q [ s2 d0, ks d0]. (2.6) 4

5 Inequality (2.5) will then follow from (2.6) since a shar lower ound for q is h(s d0 ). To rove (2.6), it is enough to show that (w + q) 2 ( 1) 2 (ks d0 q) 2. Equivalently, since oth ks d0 q and w + q are non-negative, we need only show that w + q ( 1)(ks d0 q), i.e., q 2ks d0 + a(s d0 ) 2k 2. Thus, since q s2 d0, it is sufficient to show that s 2 d0 2ks d0 + a(s d0 ) 2k 2. Now we consider searately the two cases (i) 2k and (ii) 2k >. Case (i): 2k. Here a(s d0 ) = 2k s d0 since x = [ 2k s d0 ] = 0. Therefore, we need to show s2 d0 2ks d0 +2k s d0 2k 2, or s d0 {s d0 (2k 1)}+2 2 k(k 1) 0. This inequality holds ecause s d0 k/2, a consequence of Lemma 2.4. Case (ii): 2k >. Here a(s d0 ) (2k s d0) 2 (2k s d0 ) 2 +. Therefore, we need to show s2 d0 4 2ks d k2, or 4( 1)s 2 d0 8k( 2)s d k 2 ( 2) This inequality holds whenever s d0 k/2. It is easy to see that the inequality also holds for the articular cases (i) = 5, k = 3, (ii) = 4, k odd and (iii) = 3. Using Lemma 2.4, the inequality holds for the other cases as well. Finally, the following theorem can e estalished using Lemmas Theorem 2.1. Suose s 0 is an integer defined y g(s 0 ;,, k) = min g(s;,, k), (2.7) 1 s c where c = k if (i) = 5, k = 3, (ii) = 4, k odd or (iii) = 3, else c = [ k ]. Then 2 a tye S 0 lock design S 0 (,, k, g 0, g 1, λ 0, λ 1 ) with g 0 = s 0, g 1 = s 1 g 0 1, λ 0 = 2ks 0 h(s 0 ) λ 1 = 2ks 1 h(s 1 ) λ 0 and s 1 1 = 2k s 0 is otimal in D( + 1,, k). The integer s which minimizes g(s;,, k) can easily e found using a comuter., Examle 2.1. For = 5, = 10, k = 2, the g(s; 5, 10, 2) is minimized for s = 10. Thus the following S 0 (5, 10, 2, 2, 1, 6, 3) design d with s d 0 = s 0 = 10 is otimal over D(6, 10, 2) : {(3,5) (0,1)}, {(1,4) (0,2)}, {(2,5) (0,3)}, {(1,3) (0,4)}, {(2,4) (0,5)}, {(4,5) (0,1)}, {(1,5) (0,2)}, {(1,2) (0,3)}, {(2,3) (0,4)}, {(3,4) (0,5)}. Theorem 2.1 is useful in checking the otimality of tye S 0 designs. Choi, Guta and Kageyama (2002) gave some series of tye S designs. Designs of their Series 1, 3 and 4 are tye S 0 designs as they also satisfy the requirements of Definition 2.2. Among designs for 30, Series 1 designs for = 3, 5, 7, 9 are otimal. Clearly, not all tye S 0 designs are otimal. In the next section we give a lower ound e Ad to efficiency of a tye S 0 design d, and show that tye S 0 designs are highly efficient for control-test comarisons. We now resent a new method of constructing tye S 0 designs. Following Guta and Kageyama (1994), let d n e a universally otimal lock design for diallel crosses otained using a nested alanced incomlete lock design with arameters v =, n, r n, k n (< ), λ n, the nest having lock size two. Let B i denote the ith lock of the nested alanced incomlete lock design, and let B i denote the corresonding comlementary lock such that 5

6 the contents of B i and B i taken together form one relication of the lines i = 1,...,. Let i 1, i 2,..., i kn denote the contents of B i. Then, aending to the ith lock of d n the crosses (0, i 1 ), (0, i 2 ),..., (0, i kn ) yields a tye S 0 design of the following theorem. Theorem 2.2. The existence of a nested alanced incomlete lock design with arameters v =, n, r n, k n, λ n, the nest having lock size two, imlies the existence of an S 0 (, = n, k = k n /2, g 0 = n (1 k n /), g 1 = nkn, λ ( 1) 0 = n ( k n ), λ 1 = n ). Several series of universally otimal lock designs d n are availale in literature, see e.g. Das, Dey and Dean (1998). Using each of these series, a corresonding series of tye S 0 designs can e derived using Theorem 2.2. For instance, Das, Dey and Dean (1998) gave Family 1 designs with arameters = 4t + 1, n = t(4t + 1), k n = 4, λ n = 3, where t is a ositive integer, and is a rime or rime ower. Using this family of designs, Theorem 2.2 yields a series of tye S 0 designs with arameters = 4t + 1, = t(4t + 1), k = 4t 1, g 0 = t(4t 3), g 1 = 1, where t 1, and is a rime or rime ower. Morgan, Preece and Rees (2001) taulated all ossile nested alanced incomlete lock designs for 16, and r n 30. Using designs in their tale with nested design having lock size 2, Theorem 2.2 yields a total of 24 tye S 0 designs. Tye S 0 designs otained from designs in their tale at serial numers 6, 13.c, 40, 50.c and 62 are otimal as they satisfy the condition of Theorem 2.1. Of the remaining 19 designs, 7 designs have e Ad 0.95, 6 designs have 0.80 e Ad < 0.95, 5 designs have 0.70 e Ad < 0.80, and 1 design has e Ad = The e Ad is the lower ound to efficiency as defined in the next section. 3. Efficiency The efficiency of a design d D( + 1,, k) for control-test comarisons comared to an otimal design d A D( + 1,, k) is defined as i=1 V ar(ˆτ da i ˆτ da 0) E Ad = i=1 V ar(ˆτ di ˆτ d0 ). Clearly, i=1 V ar(ˆτ da i ˆτ da 0) g(s 0,,, k), where s 0 is as in Theorem 2.1. Therefore, ased on Theorem 2.1, a lower ound to the efficiency of a tye S 0 design d with arameters,, k, g 0, g 1, λ 0, λ 1 is given y e Ad = g(s 0 ;,, k)/b 0d, (3.1) where B 0d = g(s d0 ;,,, k). Since the efficiency of a design d is greater than or equal to e Ad, high values of e Ad indicate that the design d is highly efficient, and hence aroximately otimal for control-test comarisons. The design d is otimal if e Ad = 1.0. Examle 3.1. The following tye S 0 design d for = 8, = 10, k = 6 with e Ad = is aroximately otimal: {(1,2) (3,5) (4,7) (0,6) (0,8) (0,1)}, {(2,3) (4,6) (5,8) (0,7) (0,1) (0,2)}, {(3,4) (5,7) (1,6) (0,8) (0,2) (0,3)}, {(4,5) (6,8) (2,7) (0,1) (0,3) (0,4)}, {(5,6) (1,7) (3,8) (0,2) (0,4) (0,5)}, {(6,7) (2,8) (1,4) (0,3) (0,5) (0,6)}, {(7,8) (1,3) (2,5) (0,4) (0,6) (0,7)}, {(1,8) (2,4) (3,6) (0,5) (0,7) (0,8)}, {(1,5) (2,6) (0,3) (0,4) (0,7) (0,8)}, {(3,7) (4,8) (0,1) (0,2) (0,5) (0,6)}. 6

7 The lower ound e Ad was comuted using (3.1) for all tye S 0 designs in the ractical range 30, 50, k, g 0 10, 1 < g 0 /g 1 5. Of the 247 ossile tye S 0 designs in this range, 70 designs are otimal, i.e. have e Ad = 1.0. Of the remaining 177 designs that do not satisfy condition (2.7) for otimality, 96 designs have e Ad 0.95, 52 designs have 0.90 e Ad < 0.95, 27 designs have 0.81 e Ad < 0.90, and 2 designs have e Ad = 0.77, and 0.70 resectively. For the sake of revity, these highly efficient tye S 0 designs are not taulated here, and they will e reorted elsewhere. References Cheng, C.S. (1978). Otimality of certain asymmetrical exerimental designs. Ann. Statist. 6, Choi, K.C., Guta, S. and Kageyama, S. (2002). Designs for diallel crosses for test versus control comarisons. Utilitas Math. To aear. Das, A., Dey, A. and Dean, A.M. (1998). Otimal designs for diallel cross exeriments. Statist. Pro. Letters. 36, Dey, A. and Midha, C.K. (1996). Otimal lock designs for diallel crosses. Biometrika 83, Guta, S. (1989). Efficient designs for comaring test treatments with a control treatment. Biometrika 76, Guta, S. and Kageyama, S. (1994). Otimal comlete diallel crosses. Biometrika 81, Jacroux, M. and Majumdar, D. (1989). Otimal lock designs for comaring test treatments with a control when k > v. Statist. Pro. Letters. 23, Kiefer, J. (1975). Constructions and otimality of generalized Youden designs. In A Survey of Statistical Designs and Linear Models, Ed. J.N. Srivastava, Amsterdam: North Holland. Majumdar, D. (1996). Otimal and efficient treatment-control designs. In Handook of Statistics 13, Ed. S. Ghosh and C.R. Rao, Amsterdam: North Holland. Majumdar, D. and Notz, W. (1983). Otimal incomlete lock designs for comaring treatments with a control. Ann. Statist. 11, Morgan, J.P., Preece, D.A. and Rees, D.H. (2001). Nested alanced incomlete lock designs. Discrete Math. 231, Mukerjee, R. (1997). Otimal artial diallel crosses. Biometrika 84,