Abstract. Designs for variety trials with very low replication. How do we allow for variation between the plots? Variety Testing

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1 Abstract Designs for variety trials with very low replication R A Bailey University of St Andrews QMUL (emerita) TU Dortmund, June 05 In the early stages of testing new varieties, it is common that there are only small quantities of seed of many new varieties In the UK (and some other countries with centuries of agriculture on the same land) variation within a field can be well represented by a division into blocks Even when that is not the case, subsequent phases (such as testing for milling quality, or evaluation in a laboratory) have natural blocks, such as days or runs of a machine I will discuss how to arrange the varieties in a block design when the average replication is less than two /50 /50 Variety Testing In breeding trials of new varieties, typically there is very little seed of each of the new varieties Traditionally, an experiment has one plot for each new variety and several plots for a well-established control : New Control Total for example, In the last years, Cullis and colleagues in Australia (and independently Bueno and Gilmour) have suggested replacing many occurrences of the the control by double replicates of a small number of new varieties: New New Control Total for example, This is an improvement if there are no blocks How do we allow for variation between the plots? on any given field agricultural operations, at least for centuries, have followed one of two directions, which are usually those of the rows and columns; consequently streaks of fertility, weed infestation, etc, do, in fact, occur predominantly in those two directions R A Fisher, letter to H Jeffreys, 30 May 938 (selected correspondence edited by J H Bennett) (This assumption is dubious for field trials in Australia) If field operations have been primarily in one direction for a long time, then it is reasonable to divide the fields into blocks whose length runs along that direction 3/50 4/50 Blocking in the second phase: an example The milling phase of a wheat variety trial has 4 varieties to be compared Only 0 can be milled in any one day The trial can take place over 8 days, so there are 8 blocks of size 0 There are only 80 4 = 56 experimental units spare for replication How should these be allocated? 8 blocks units 8 units controls varieties in every block 0 single replication Two possible designs for 4 varieties in 8 blocks of 0 8 blocks 8 blocks units 8 units controls varieties in every block 0 single replication 4 units 6 units 56 varieties 68 varieties all replicated twice all single replication 5/50 6/50

2 The problem Linear model, estimation and variance We measure the response Y on each unit in each block If that unit has variety i and block D, then we assume that We are given b blocks of size k We are given v varieties Assume that average replication = r = bk v How should we allocate varieties to blocks? What makes a block design good? Y = τ i + β D + random noise, where the random noise is independently normally distributed with zero mean and constant variance σ We want to estimate all the simple differences τ i τ j Put V ij σ = variance of the best linear unbiased estimator for τ i τ j We want all the V ij to be small 7/50 8/50 Optimality An example with 5n + 0 varieties in 5 blocks of size 4 + n Apart from the constant multiple σ, Put V ij = variance of the BLUE for τ i τ j v V T = v i= j=i+ V ij = sum of variances of variety differences Definition For given values of b (the number of blocks), k (the size of the blocks) and v (the number of varieties), a block design is A-optimal if it minimizes V T 3 4 A A n B B n C C n D D n E E n How do we calculate pairwise variances in a generic design? 9/50 0/50 Levi graph Levi graph: example 3 4 A A n B B n The Levi graph of the block design has C C n one vertex for each variety D D n one vertex for each block E E n one edge for each plot (aka experimental unit), so that the edge for plot ω joins the vertex for the variety on ω to the vertex for the block containing ω /50 B B n A n 5 C A 7 6 C n E n D 0 E D n /50

3 Electrical networks We can consider the Levi graph as an electrical network with a -ohm resistance in each edge Connect a -volt battery between vertices i and j Current flows in the network, according to these rules Ohm s Law: In every edge, voltage drop = current resistance = current Kirchhoff s Voltage Law: The total voltage drop from one vertex to any other vertex is the same whichever path we take from one to the other 3 Kirchhoff s Current Law: At every vertex which is not connected to the battery, the total current coming in is equal to the total current going out Find the total current I from i to j, then use Ohm s Law to define the effective resistance R ij between i and j as /I 3/50 Electrical networks: variance Reminder: V ij = variance of BLUE of τ i τ j for varieties i and j Theorem If R ij is the effective resistance between variety vertices i and j in the Levi graph then R ij = V ij Put: V CD = variance of BLUE of β C β D for blocks C and D, V ic = variance of BLUE of τ i + β C for variety i and block C Theorem If R CD and R ic are the effective resistances between vertices C and D, and between i and C respectively, in the Levi graph then R CD = V CD and R ic = V ic Effective resistances are easy to calculate without matrix inversion if the graph is sparse 4/50 Pairwise resistance (remove A,, E n to get core subdesign Γ) A n A Resistance(A, A ) = B B n 5 C 7 6 C n E n D 0 E D n Resistance(A, B ) = + Resistance(block A, block B) in Γ Resistance(A, 8) = + Resistance(block A, 8) in Γ Resistance(, 8) = Resistance(, 8) in Γ Silly names just for this talk Definition Call a variety a a drone if it has replication ; a queen-bee if it occurs in every block; a worker otherwise Is it better to put all the drones into one block (or a few blocks), or are they better distributed equally among all the blocks? 5/50 6/50 How should we distribute the drones? From now on, distribute drones as equally as possible n drones A rest of graph B If we move all the drones in block B into block A then we reduce nm variances from + R AB to m drones Then we have to remove m non-drones from block A, and this increases the resistance between A and the rest of the graph This increases the variances between these n + m drones and the remaining v n m varieties This more than compensates for the original reduction in variance b blocks k plots v varieties core subdesign Γ n plots bn drones all single replication whole design Whole design has v treatments in b blocks of size k = k + n; the core subdesign Γ has v core varieties in b blocks of size k, where v = v bn (The core varieties may include up to b extra drones) 7/50 8/50

4 Sum of the pairwise variances Sum of variances in whole design if Γ is equi-replicate Theorem (cf Herzberg and Jarrett, 007) If there are n drones in each block of, and the core subdesign Γ has v varieties in b blocks of size k then the sum of the variances of variety differences in where = V T ( ) = bn(bn + v ) + V T + nv BT + n V B, V T = the sum of the variances of variety differences in Γ V B = the sum of the variances of block differences in Γ V BT = the sum of the variances of sums of one treatment and one block in Γ 9/50 V T ( ) = bn(bn + v ) + V T + nv BT + n V B V T = the sum of the variances of variety differences in Γ V B = the sum of the variances of block differences in Γ V BT = the sum of the variances of sums of one treatment and one block in Γ If Γ is equi-replicate with replication r then k b V B b = r v V T v ; V BT = b v V T + v k (b v ), and so V B and V BT are both increasing functions of V T Consequence For a given choice of k, use the core subdesign Γ which minimizes V T 0/50 Sum of variances in whole design if there are many drones An example of this non-intuitive result V T ( ) = bn(bn + v ) + V T + nv BT + n V B V T = the sum of the variances of variety differences in Γ V B = the sum of the variances of block differences in Γ V BT = the sum of the variances of sums of one treatment and one block in Γ Consequence If n is large, we need to focus on reducing V B, so it may be best to increase the number of drones and decrease k (the size of blocks in the core subdesign Γ), so that average replication within Γ is more than If there are 4( + n) varieties in 4 blocks of size 4 + n, the design on the left is A-better than the design on the right if and only if n < n drones 5 6 n drones n drones n drones 3 n + drones 4 n + drones 3 4 n + drones 3 4 n + drones /50 /50 A definite result Theorem (Duals of BIBDs cannot be beaten) Suppose that we are given b blocks of size k, and v varieties For i =,, let design i have core subdesign Γ i with block size k i If Γ is the dual of a balanced incomplete block design and k > k then is worse than on the A criterion, no matter how big v is Design Core subdesign Γ Γ block size in subdesign k > k property of core subdesign dual of BIBD arbitrary then is A-better than An example of the good result If there are 7(3 + n) varieties in 7 blocks of size 6 + n, the design on the left is A-better than the design on the right, for all values of n n drones n drones n drones n drones n drones n drones n drones n + drones 5 7 n + drones 3 6 n + drones n + drones n + drones n + drones n + drones 3/50 4/50

5 Another example of the good result If there are 4n + 6 varieties in 4 blocks of size 3 + n, the design on the left is A-better than the design on the right, for all values of n 3 n drones 4 5 n drones 4 6 n drones n drones n + drones n + drones n + drones n + drones Strategy Given b, v and k, how do we find an A-optimal design for v varieties in b blocks of size k when bk Average replication v b(k ) +? Case b = or b = 3 (very small b) Maximum v for estimability Case v = b(k ) + or v = b(k ) (very large v) Case 3 k 0 b Case 4 < k 0 < b (small k 0 but not Case ) v bk k 0 = k = biggest space per block for non-drones b 5/50 6/50 Case Only blocks, of size k Morgan and Jin (007) showed that the A-optimal designs are those with n drones and q queen bees, where n = n 0 = v k and q = k = k 0 = k n 0 = k v Case continued 3 blocks of size k Using the result about drone-distribution and the nice theorem about duals of BIBDs, RAB has shown that the A-optimal designs are as follows when v is divisible by 3 (and presumably small changes deal with the other cases) There are 3w workers and 3n drones, where 3w = 3k v and n = n 0 = k w and k = k 0 = w 3 4 q A A A 3 A n 3 4 q B B B 3 B n queens drones 4 5 3w 3w A A A 3 A n w 3w B B B 3 B n w 3w C C C 3 C n w copies of design using all pairs from 3 drones 7/50 8/50 Case v = b(k ) + Case continued v = b(k ) This is the maximum number of varieties that can be tested in b blocks of size k with all comparisons estimable Mandal, Shah and Sinha (99), for k =, Dean and co-authors, and Bailey and Cameron (03), for general block size, showed that, no matter how many blocks there are, the A-optimal design has the following form The A-optimal designs were found for all cases by Krafft and Schaefer (997) small k and b increase k if b 5 then increase b A 3 B A A 3 B B A A B B 3 4 C 3 C C C C C 3 A A A 3 A k 4 5 D D D D 3 D D D 3 B B B 3 B k 5 6 E E E E 3 E E E 3 C C C 3 C k D D D 3 D k E E E 3 E k 6 F chain F F F 3 smaller chain F F F 3 G G G 3 queen queen v drones 9/50 Youden and Connor (953) had recommended chain designs 30/50

6 Case v = b(k ) revisited Theorem Consider a design with b blocks of size For s b, let Γ s be the design consisting of a chain of length s, one of whose varieties is in all blocks outside the chain, while all other varieties are drones Then Consequence V B (Γ s ) = 6 [ s3 + bs (6b 4)s + 6b 5b] If b = 3 then V B (Γ ) > V B (Γ 3 ) so there is no need for queens If b = 4 then V B (Γ ) = V B (Γ 4 ) < V B (Γ 3 ), but V T (Γ ) > V T (Γ 4 ) and V BT (Γ ) > V BT (Γ 4 ), so do not use Γ or Γ 3 (no need for queens) 3 If b 5 then V B (Γ ) < V B (Γ 3 ) < < V B (Γ b ), so we need to use smaller chains as v gets larger Case 3 k k 0 b For simplicity, assume that b divides v, so that Then n 0 = v bk b = minimum number of drones per block b(k b + ) v bk b(b ) Let Γ 0 be the design for b(b )/ varieties replicated twice in b blocks of size b in such a way that there is one variety in common to each pair of blocks This is A-optimal for these numbers 3/50 3/50 Case 3 continued k 0 b n 0 = minimal number of drones per block Construction Method put n 0 drones in each block; put in one copy of Γ 0 ; 3 put in as many further copies of Γ 0 as possible (if this uses up all the space, then the nice theorem shows that this is A-optimal); 4 in any remaining space, use a good design for workers with replication (so long as there is at least one copy of Γ 0, it probably doesn t make much difference which one is used) Case 3 Example: b = 8 and k = 5 (so 60 v 9) 60 varieties: all workers (n 0 = 0) one copy of Γ 0 another copy of Γ 0 33/50 34/50 Case 3 Example: b = 8 and k = 5 (so 60 v 9) 76 varieties: 44 workers, 3 drones (n 0 = 4) Case 3 Example: b = 8 and k = 5 (so 60 v 9) 9 varieties: 8 workers, 64 drones (n 0 = 8) A A A 3 A B B B 3 B C C C 3 C D D D 3 D E E E 3 E F F F 3 F G G G 3 G H H H 3 H 4 Γ 0 6 workers drones replication A A A 3 A 4 A 5 A 6 A 7 A B B B 3 B 4 B 5 B 6 B 7 B C C C 3 C 4 C 5 C 6 C 7 C D D D 3 D 4 D 5 D 6 D 7 D E E E 3 E 4 E 5 E 6 E 7 E F F F 3 F 4 F 5 F 6 F 7 F G G G 3 G 4 G 5 G 6 G 7 G H H H 3 H 4 H 5 H 6 H 7 H 8 Γ 0 drones 35/50 36/50

7 Case 3 A bad example: b = 4 and k 0 = 4 Case 4 < k 0 < b v = 4n + 8 and k = 4 + n My strategy gives which is worse than For various values of k i k 0, find the best core subdesign Γ i for v i varieties in b blocks of size k i (For equi-replicate core subdesigns, it is often easier to find the best dual design, which is obtained by interchanging the roles of blocks and varieties) 3 7 A A n B B n C C n D D n Γ 0 drones 3 A A n+ 4 B B n+ 3 4 C C n+ 3 4 D D n+ k = 3 drones rep 3 when n 50 37/50 V T (Γ i ) = the sum of the variances of variety differences in Γ i V B (Γ i ) = the sum of the variances of block differences in Γ i V BT (Γ i ) = the sum of the variances of sums of one treatment and one block in Γ i If there are n i drones in each block then, in the whole design, V T ( ) = bn i (bn i + v i ) + V T(Γ i ) + n i V BT (Γ i ) + n i V B(Γ i ) Use this formula to find the core subdesign which gives the smallest V T ( ) As the number of varieties increases, it becomes more important to choose Γ i with a small value of V B (Γ i ) 38/50 Case 4 continued k 0 = 4 < b, V B (Γ i ) b(b )/ Best design for b blocks known to RAB Γ Γ Γ 3 Γ 4 k i queens, queens, b workers b workers both boring workers (rep ) rep 3 rep b = b = b = b = 9 09 b = 0 b = b = 098 b = 3 07 b = 4 b = As v increases, Γ 3 becomes better than Γ 4 If b 4, then, as v increases, Γ and Γ become better than Γ 3 39/50 Case 4 continued < k 0 < b when b = 8: k 0 = 6 k = k 0 = 6, and 4 varieties, all workers, all replicated twice (One worker for each pair of blocks except for {A, B}, {C, D}, {E, F} and {G, H}) 40/50 Case 4 continued k = 5 and k = 6 when b = 8: k 0 = 5 Case 4 continued k = 5 and k = 6 when b = 8: k 0 = 4 k = 5 k = 6 0 varieties: 8 varieties: 0 workers, no drones 0 workers, 8 drones k = 5 k = 6 4 varieties: 3 varieties: 6 workers, 8 drones 8 workers, 4 drones A 3 4 A 4 A A A B B 3 5 B B B C 9 0 C C C C D D D D D E E E E E F F 6 7 F F F G G 7 8 G G G H 4/ H k = 4 rep = 8 3 H H H 3 k = 3 rep 3 4/50

8 Case 4 continued k = 5 and k = 6 when b = 8: k 0 = 3 Health Warning k = 5 k = 6 8 varieties: 36 varieties: workers, 6 drones workers, 4 drones 3 A A 4 5 B B C C D D 8 0 E E 3 A A A B B B C C C D D D E E E 3 The overall message is that there can be phase changes as the spare capacity for replication (bk v) decreases Therefore it is necessary to compare core subdesigns Γ i with different block size k i Although this overall message is correct, no one has checked the arithmetic in the examples presented, so individual cases may be wrong 5 0 F F 7 G G 3 9 H H 5 0 F F F 3 7 G G G H H H 3 43/50 44/50 Warning for specialists in optimality References: more equal replication The results given are for A-optimality A design is MV-optimal if it minimizes the value of the maximal V ij If v bk k then it is possible to put all the drones in a single block, and the MV-optimal design may have this form A design is D-optimal if it minimizes the volume of the confidence ellipsoid for the vector of fitted values (τ, τ, ) under the assumption of independent identically distributed normal errors The D-optimal designs have core subdesigns with the largest possible value of k (and so the smallest possible number of drones), no matter how large the number of varieties J S S Bueno Filho and S G Gilmour: Planning incomplete block experiments when treatments are genetically related Biometrics, 59, (003), B R Cullis, A B Smith and N E Coombes: On the design of early generation variety trials with correlated data Journal of Agricultural, Biological and Environmental Statistics,, (006), A B Smith, P Lim and B R Cullis: The design and analysis of multi-phase plant breeding experiments Journal of Agricultural Science, 44, (006), /50 46/50 References: Levi graph F W Levi: Finite Geometrical Systems, University of Calcutta, Calcutta, 94 iii + 5 pp N Gaffke: Optimale Versuchsplanung für linear Zwei-Faktor Modelle PhD thesis, Rheinisch-Westfälische Technische Hochschule, Aachen, N Gaffke: D-optimal block designs with at most six varieties, Journal of Statistical Planning and Inference, 6 (98), T Tjur: Block designs and electrical networks, Annals of Statistics, 9 (99), R A Bailey and P J Cameron: Using graphs to find the best block designs In Topics in Structural Graph Theory (eds L W Beineke and R J Wilson), Cambridge University Press, Cambridge, 03, pp 8 37 References: core subdesigns W J Youden and W S Connor: The chain block design Biometrics, 9, (953), 7 40 A M Herzberg and D F Andrews: The robustness of chain block designs and coat-of-mail designs Communications in Statistics Theory and Methods, 7 (978), A M Herzberg and R G Jarrett: A-optimal block designs with additional singly replicated treatments Journal of Applied Statistics, 34 (007), /50 48/50

9 References: few blocks References: (nearly) maximal number of varieties J P Morgan and B Jin: Optimal experimentation in two blocks Journal of Statistical Theory and Practice,, (007), N K Mandal, K R Shah and B K Sinha: Uncertain resources and optimal designs: problems and perspectives Calcutta Statistical Association Bulletin, 40, (99), 67 8 O Krafft and M Schaefer: A-optimal connected block designs with nearly minimal number of observations Journal of Statistical Planning and Inference, 65, (997), /50 50/50