Optimal Paired Choice Block Designs. Supplementary Material

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1 Saisica Sinica: Supplemen Opimal Paired Choice Block Designs Rakhi Singh 1, Ashish Das 2 and Feng-Shun Chai 3 1 IITB-Monash Research Academy, Mumbai, India 2 Indian Insiue of Technology Bombay, Mumbai, India 3 Academia Sinica, Taipei, Taiwan Supplemenary Maerial Design for Example 3.1. k = 3, v 1 = 2, v 2 = 3, v 3 = 4, b = 1, N = s = 72. (000,111) (000,222) (000,333) (100,211) (100,322) (100,033) (020,131) (020,313) (120,302) (001,112) (001,223) (001,330) (101,212) (101,323) (101,030) (021,132) (021,310) (121,303) (002,113) (002,220) (002,331) (102,213) (102,320) (102,031) (022,133) (022,311) (122,300) (003,110) (003,221) (003,332) (103,210) (103,321) (103,032) (023,130) (023,312) (123,301) (010,121) (010,232) (010,303) (110,221) (110,332) (110,003) (020,202) (120,231) (120,013) (011,122) (011,233) (011,300) (111,222) (111,333) (111,000) (021,203) (121,232) (121,010) (012,123) (012,230) (012,301) (112,223) (112,330) (112,001) (022,200) (122,233) (122,011) (013,120) (013,231) (013,302) (113,220) (113,331) (113,002) (023,201) (123,230) (123,012) Lemma 1. A necessary and sufficien condiion for C M = C M o hold is ha for each block and each aribue, he frequency disribuion of he levels of he aribue are same for he wo opions. Proof. Le P Mj = ((P j) 1 (P j) (P j) b) where (P j) represens P Mj for he h block. Then

2 Rakhi Singh, Ashish Das and Feng-Shun Chai he condiion W P M = 0 is equivalen o he condiion 1 (P 1) = 1 (P 2), = 1,..., b. Le (P j) = ((P j) 1 (P j) w (P j) k ) where (P j) w is of order s (v w 1) and represens (P j) for he wh aribue. Therefore for = 1,..., b, if 1 (P 1) = 1 (P 2), hen 1 (P 1) w = 1 (P 2) w for every w and. Now, since he ih column of (P j) w provides frequency of level i and level v w in he wh aribue of he jh opion in he h block, herefore, 1 (P 1) w = 1 (P 2) w implies ha he frequency of each of he levels of aribue w is same in he wo opions among he s choice pairs in block. The converse follows by noing ha if for each block and each aribue, he frequency disribuion of he levels of he aribue are same for he wo opions, hen 1 (P 1) = 1 (P 2) for every. Proof of Theorem 1. The proof follows as a special case of Lemma 1. Proof of Theorem 2. Under he linear paired comparison model, a design d opimally esimaes he main effecs if C M = diag(c (1),..., C (k) ) (see Großmann and Schwabe (2015)) where C (i) = z i(i vi 1 + J vi 1) wih z i = 2N/(v i 1), i = 1,..., k. This implies ha C M normalized by number of pairs would aain an opimal srucure if C (i) = z i(i vi 1 + J vi 1) wih z i = 2/(v i 1), i = 1,..., k. Since he OA + G mehod of consrucion enails adding generaors o he orhogonal array of srengh, ( 2), he off-diagonal elemens of P M P M corresponding o wo differen aribues is zero since under each level of he firs aribue, all he levels of he second aribue occur equally ofen. Also, since in an orhogonal array, under each column (aribue) he levels are equally replicaed, o esablish ha each C (i) aains an opimal srucure of he form z i(i vi 1 + J vi 1), i is enough o show ha normalized P M P M corresponding o a paired choice design wih one aribue, say a v levels, aains he srucure z(i v 1 + J v 1), where z = 2/(v 1).

3 Opimal Paired Choice Block Designs Wihou loss of generaliy, we consider only v choice pairs for a ypical aribue since under each column, he n rows of he orhogonal array involves v symbols each replicaed n/v imes. While using he generaor g j, le P 0 1, P j 2 be he v (v 1) effecs-coded marix for he main effecs for he firs and second opions, respecively, corresponding o any one aribue a v levels. When h > 1, noe ha P M is he collecion of differen marices generaed ou of he corresponding {P 0 1, P j 2 }, j = 1..., h of choice pairs. For noaional simpliciy, we denoe P 0 1 by P 0 and P j 2 by Pj, j = 1,..., v 1. Also, noe ha 1 P j = 0 and v 1 j=0 Pj = 0. Consider he informaion marix P M P M normalized for v even. v(v 1)P M P M = v 1 (P0 P j) (P 0 P j) = v 1 (P 0P 0 + P jp j P 0P j P jp 0) = v 1 {2(Iv 1 + Jv 1)} P 0( v 1 Pj) ( v 1 P j)p 0 = {2(v 1)(I v 1+J v 1)} P 0( P 0) ( P 0)P 0 = 2{(v 1)(I v 1+J v 1)}+2P 0P 0 = 2v(I v 1 + J v 1). Thus, for v even, h = v 1 generaors of he ype g j = 1,..., v 1 leads o he opimal srucure of normalized P M P M. For v odd, we noe ha, if say, mh row of P 0 corresponds o he level i, hen he mh row of P v j corresponds o he level i j (mod v). Similarly, if say, lh row of P j corresponds o he level i, hen he lh row of P 0 corresponds o he level i j (mod v). This makes he lh row of P j and P 0 same as he mh row of P 0 and P v j for every wo rows l m = 1,..., v. Therefore, for v odd, P jp 0 = P 0P v j. Now, v(v 1)/2P M P M = (v 1)/2 (P 0 P j) (P 0 P j) = (v 1)/2 (P 0P 0 + P jp j P 0P j P jp 0) = (v 1)/2 {2(I v 1 + J v 1)} (v 1)/2 (P 0P j + P jp 0) = (v 1)(I (v 1)/2 v 1+J v 1) (P 0P j+p 0P v j) = (v 1)(I v 1+J v 1) P 0 (v 1)/2 (P j+p v j) = (v 1)(I v 1 +J v 1) P 0 v 1 Pj = (v 1)(Iv 1 +Jv 1) P 0( P 0) = v(i v 1 +J v 1). Thus, for v odd, h = (v 1)/2 generaors of he ype g j = 1,..., (v 1)/2 leads o he opimal srucure of normalized P M P M. Proof of Theorem 3. For a given OA(n 1, k + 1, v 1 v k δ, 2), corresponding o he k aribues a levels v i, i = 1,..., k, le d 1 be he design consruced hrough OA + G mehod

4 Rakhi Singh, Ashish Das and Feng-Shun Chai using h = lcm(v 1,..., v k ) generaors. Then d 1 wih parameers k, v 1,..., v k, b = 1, s = hn 1 is an opimal paired choice design. From d 1, he choice pairs obained hrough each of he h generaors consiue a block of size n 1. This is rue since n 1 rows of a block form he orhogonal array in he firs opion and, wih labels re-coded hrough he generaor, in he second opion and hence he condiions in Theorem 1 are saisfied. Finally, we use he δ symbols of he (k + 1)h column of he orhogonal array for furher blocking giving a paired choice block design d 2 wih parameers k, v 1,..., v k, b = hδ, s = n 1/δ. This is rue since for every aribue in each of he blocks so formed, each of he v i levels occurs equally ofen under ih aribue and hence by Theorem 1, d 2 is opimal in D k,b,s. Proofs for Theorem 4 and Theorem 5 require a resul from Dey (2009) ha is given below. Lemma 2 (Dey (2009)). Consider v(v 1)/2 combinaions involving v levels aken wo a a ime. Then, for v odd, he combinaions can be grouped ino (v 1)/2 replicaes each comprising v combinaions. The groups are {(i, v 2 i), (i+1, v 1 i),..., (i+v 1, v 2 (i (v 1)))} and he levels are reduced modulo v; i = 0,..., (v 3)/2. Proof of Theorem 4. Theorem 3 of Graßhoff e al. (2004) saes ha from m( k) rows of a Hadamard marix H m of order m, an opimal paired choice design d 3 wih parameers k, v, b = 1, s = mv(v 1)/2 is consruced using he v(v 1)/2 combinaions of v levels aken wo a a ime. From every row of {H m, H m}, v(v 1)/2 choice pairs are obained by replacing 1 in he row by he firs column of he combinaions and 1 in he row by he second column of he combinaions. If v is odd, hen (v 1)/2 is an ineger and he v(v 1)/2 combinaions can be arranged in rows such ha each of he wo columns have every level appearing equally ofen. Such an arrangemen is always possible and follows from sysems of disinc represenaives. Therefore, corresponding o each of he rows of {H m, H m}, using v(v 1)/2 choice pairs as a block, a paired choice block design wih parameers k, v, b = m, s = v(v 1)/2 is obained

5 Opimal Paired Choice Block Designs which, following Theorem 1 is opimal. Now for v odd, from Dey (2009), v(v 1)/2 combinaions involving v levels aken wo a a ime can be grouped ino (v 1)/2 replicaes each comprising v combinaions. Therefore, he blocks generaed by each row of H m can be furher broken ino (v 1)/2 blocks each of size v, which gives us d 4. Proof of Theorem 5. Consrucion 3.2 of Demirkale, Donovan, and Sree (2013) uses an OA(n 2, k + 1, v k v k+1, 2) wih v k+1 = n 2/v and forms v k+1 parallel ses each having v rows. ( Then, an opimal paired choice design wih parameers k, v, b = 1, s = v v k+1 2) is consruced using he v(v 1)/2 combinaions of v numbers {1,..., v} aken wo a a ime. Le {i, j} be a ypical row. Then, for each such row of size wo, corresponding rows i and j from each of he v k+1 parallel ses are chosen o form he choice pairs of he opimal paired choice design d 6. Again as earlier, for v odd, he v(v 1)/2 combinaions can be arranged in rows such ha each of he wo columns have every number appearing equally ofen. Considering he v(v 1)/2 choice pairs, obained from a parallel se, as a block, we ge he paired choice block design wih parameers k, v, b = v k+1, s = v(v 1)/2 which is opimal in D k,b,s. Furher proof follows on he same lines as he proof of Theorem 4 by reaing he pairs generaed by each parallel se as blocks. Proof of Theorem 6. Theorem 4 of Graßhoff e al. (2004) uses an OA(n 3, k + 1, m 1 m k δ, 2) wih m i = v i(v i 1)/2 for some odd v i o consruc an opimal paired choice design d 7 wih parameers k, v i,..., v k, b = 1, s = n 3. This mehod involves a one-one mapping beween m i levels of orhogonal array o he v i(v i 1)/2 combinaions on v i symbols. For a combinaion {i, j} corresponding o a symbol of an orhogonal array, he firs opion in a pair is obained by replacing i in place of ha symbol and he second opion has j in he corresponding posiion. Then, similar o consrucion of Theorem 3, using he δ ( 1) symbols of he (k + 1)h column of he orhogonal array for blocking gives us an opimal paired choice block design d 8 wih

6 Rakhi Singh, Ashish Das and Feng-Shun Chai parameers k, v i,..., v k, b = δ, s = n 3/δ. Noe ha his mehod is applicable only for odd v i since for even v i, i is no possible o arrange v i(v i 1)/2 combinaions in a posiion-balanced manner. Proof of Theorem 7. From Theorem 1, for each of he h generaors, a paired choice design using he OA+G mehod of consrucion is opimal under he broader main effecs block model if P M P I = 0. For a given generaor, o show ha P M P I = 0, i suffices o show ha he inner produc of he columns of P M corresponding o he mh main effec and he columns of P I corresponding o he wo-facor ineracion effec of ih and jh aribue is zero. Using an OA(n 1, k, v 1 v k, 3) in he OA + G mehod of consrucion, we esablish he resul hrough he following wo cases. Case (i) m = i: In an orhogonal array of srengh 2, each of he v iv j combinaions occur equally ofen n 1/(v iv j) imes as rows. Therefore, since he paired choice design is based on he orhogonal array, for showing ha P M P I = 0, i suffices o show ha P M P I = 0 for one of he n 1/(v iv j) ses of v iv j rows of he ype (i, j); i = 0,..., v i 1; j = 0,..., v j 1. For such v iv j rows, noe ha P My, (y = 1, 2), corresponding o he jh aribue, can be pariioned ino v i ses P My(j) each of v j disinc rows. Then, 1 P My(j) = 0. Le P Iy corresponding o he ih aribue fixed a level i l (i l = 0,..., v i 1) and he jh aribue aking v j disinc levels be represened by P Iy(il j). Then, he columns of P Iy(il j) are muliples of eiher P My(j) or 0 v. Therefore, 1 P Iy(il j) = 0 for y = 1, 2. Le P M corresponding o he ih aribue a level i l be represened by X il. Then, X il = 1x i l where x i l is a row vecor of size v i 1. Therefore, P M P I = v i 1 i l =0 X i l (P I1(il j) P I2(il j)) = vi 1 i l =0 xi l (1 P I1(il j) 1 P I2(il j)) = 0. Case (ii) m i: In an orhogonal array of srengh 3, each of he v iv jv m combinaions

7 Opimal Paired Choice Block Designs occur equally ofen n 1/(v mv iv j) imes as rows. Therefore, as in Case (i), for showing ha P M P I = 0, i suffices o show ha P M P I = 0 for one of he n 1/(v mv iv j) ses of v mv iv j rows of he ype (m, i, j); m = 0,..., v m 1; i = 0,..., v i 1; j = 0,..., v j 1. For such v mv iv j rows, noe ha P Iy, (y = 1, 2), corresponding o he ih and jh aribue, can be pariioned ino v m ses P Iy(ij) each of v iv j disinc rows. Therefore, 1 P Iy(ij) = 0 for y = 1, 2, since from Case (i), 1 P Iy(il j) = 0 for he ih aribue a level i l. Finally, since for he mh aribue a level m l (m l = 0,..., v m 1), he v iv j combinaions under aribues i and j occur equally ofen, herefore P M P I = v m 1 m l =0 X m l (P I1(ij) P I2(ij) ) = vm 1 m l =0 xm l (1 P I1(ij) 1 P I2(ij) ) = 0. Proof of Theorem 10. From Lemma 1, W P M = 0 if and only if for each aribue under he choice pairs having foldover in he second opion of a choice pair, he level l (l = 0, 1) appears equally ofen in boh he opions in every block and hus, he frequency of he pair (1, 0) is same as he frequency of he pair (0, 1) under every aribue in each block. Le P I = (Y 1 Y Y b ) where Y is he s k(k 1)/2 marix corresponding o he h block. Wih (P Ij) represening P Ij for he h block, Y = (P I1) (P I2). Then, he condiion W P I = 0 is equivalen o he condiion 1 (P I1) = 1 (P I2) for every = 1,..., b. Consider (P Ij) = ((P Ij) 12 (P Ij) lm (P Ij) (k 1)k ) where (P Ij) lm is of order s 1 and represens (P Ij) for he wo-facor ineracion beween he lh and he mh aribue. Therefore, he necessary and sufficien condiion for 1 (P I1) = 1 (P I2) is ha 1 (P I1) lm = 1 (P I2) lm for every l and m. In he h block, for he choice pairs where eiher boh he aribues have a foldover in he second opion or boh do no have a foldover in he second opion, he corresponding rows in (P I2) lm are same as he corresponding rows in (P I1) lm. However, for he pairs in which one aribue has a foldover in he second opion and anoher does no have foldover in he second opion, he corresponding rows in (P I2) lm are

8 Rakhi Singh, Ashish Das and Feng-Shun Chai negaive of he corresponding rows in (P I1) lm. In such a case, 1 (P I1) lm = 1 (P I2) lm if and only if 1 (P I1) lm = 1 (P I2) lm = 0. Now, 1 (P I1) lm = 0 if and only if he frequency of he pairs from he se {(01, 00), (01, 11), (10, 00),(10, 11)} is same as he frequency of he pairs from he se {(00, 01), (00, 10), (11, 01), (11, 10)} under he lh and he mh aribue. Proof of Theorem 11. In seps (iii)-(iv), corresponding o an elemen f of F, make he firs se of 2 α 1 2 k α 2 = 2 k 3 blocks having choice pairs (ab, a b), (ab, a b ), (a c, ac), (a c, ac ). Similarly, following he seps (iii)-(iv), we make an addiional se of 2 k 3 blocks having choice pairs (ac, a c), (ac, a c ), (a b, ab), (a b, ab ). Noe ha each of he consruced blocks saisfy condiions (i) and (ii) of Theorem 10. This gives rise o a oal of 2 k 2 ses of blocks each of size 4. The way we have consruced he choice pairs in seps (iii)-(iv), i follows ha he collecion of firs opion in he 2 k choice pairs forms a complee facorial having 2 k combinaions. Furhermore, he addiional se of 2 k 3 blocks, in he consrucion, is idenical o he firs se of 2 k 3 blocks. Accordingly, we reain only he firs se of 2 k 3 blocks. This gives rise o a oal of 2 k 1 choice pairs divided ino 2 k 3 blocks each of size 4. Therefore, sep (v) gives an opimal paired choice block design d I 2 wih parameers k, v = 2, s = 4, b where b = 2 k 3( k q) for k odd and b = 2 k 3( k+1 q+1) for k even. References Demirkale, F., D. Donovan, and D. J. Sree (2013). Consrucing D-opimal symmeric saed preference discree choice experimens. J. Sais. Plann. Inference 143 (8), Dey, A. (2009). Orhogonally blocked hree-level second order designs. J. Sais. Plann. Inference 139 (10),

9 REFERENCES Graßhoff, U., H. Großmann, H. Holling, and R. Schwabe (2004). Opimal designs for main effecs in linear paired comparison models. J. Sais. Plann. Inference 126 (1), Großmann, H. and R. Schwabe (2015). Design for discree choice experimens. In A. Dean, M. Morris, J. Sufken, and D. Bingham (Eds.), Handbook of Design and Analysis of Experimens, pp Boca Raon, FL: Chapman and Hall. Rakhi Singh, IITB-Monash Research Academy, Mumbai, India Ashish Das, Indian Insiue of Technology Bombay, Mumbai, India Feng-Shun Chai, Academia Sinica, Taipei, Taiwan