1. Handling Large Numbers of Entries Stratification of materials –group entries on the basis of certain traits - maturity, market class, color, etc. –then analyze in separate trials –if entries are random effects, you can conduct a number of smaller trials and pool results to get better estimates of variances Homogenization of experimental area –work to improve your experimental technique - take care with land choice, preparation, and husbandry during the experiment –choose seeds of uniform viability Use of controls –systematically or randomly placed controls can be used to identify site variability and adjust yields of the entries

2. Incomplete Block Designs We group into blocks to –Increase precision –Be able to make comparisons under more uniform conditions But the problem is –As blocks get larger, the conditions become more heterogeneous - precision decreases –So small blocks are preferred, but in a breeding program the number of new selections may be quite large –In other situations, natural groupings of experimental units into blocks may result in fewer units per block than required by the number of treatments (limited number of runs per growth chamber, treatments per animal, etc.)

3. Incomplete Block Designs Plots are grouped into blocks that are not large enough to contain all treatments (selections) Good references: –Kuehl – Chapters 9 and 10 –Cochran and Cox (1957) Experimental Designs

4. Types of incomplete block designs Balanced Incomplete Block Designs –each treatment occurs together in the same block with every other treatment an equal number of times - usually once –all pairs are compared with the same precision even though differences between blocks may be large –can balance any number of treatments and any size of block but...treatments and block size fix the number of replications required for balance –often the minimum number of replications required for balance is too large to be practical Partially balanced incomplete block designs –different treatment pairs occur in the same blocks an unequal number of times or some treatment pairs never occur together in the same block –mean comparisons have differing levels of precision –more difficult to analyze statistically

5. Types of incomplete block designs May have a single blocking criterion –Randomized incomplete blocks Other designs have two blocking criteria and are based on Latin Squares –Latin Square is a complete block design that requires N=t 2. May be impractical for large numbers of treatments. –Row-Column Designs – either rows or columns or both are incomplete blocks –Youden Squares – two or more rows omitted from the Latin Square

6. Resolvable incomplete block designs Blocks are grouped so that each group of blocks constitute one complete replication of the treatment Trials can be managed in the field on a replication-by- replication basis Field operations can be conducted in stages (planting, weeding, data collection, harvest) Complete replicates can be lost without losing the whole experiment If you have two or more complete replications, you can analyze as a RBD if the blocking turns out to be ineffective Lattice designs are a well-known type of resolvable incomplete block design

7. Balanced Incomplete Block Designs t = s*k and b = r*s ≥ t + r - 1 –t = number of treatments –s = number of blocks per replicate –k = number of units per block –b = total number of blocks in the experiment –r = number of complete replicates take home message - often the minimum number of replications required for balance is too large to be practical

8. Lattice Designs Square lattice designs –number of treatments must be a perfect square (t = k 2 ) –blocks per replicate (s) and plots per block (k) are equal (s = k) and are the square root of the number of treatments (t) –for complete balance, number of replicates (r) = k+1 Rectangular lattice designs –t = s*(s-1) and k = (s-1) –example: 4 x 5 lattice has 4 plots per block, 5 blocks per replicate, and 20 treatments Alpha lattices –t = s*k –more flexibility in choice of s and k

9. Randomization Field Arrangement –blocks composed of plots that are as homogeneous as possible Randomization Using Basic Plan –randomize order of blocks within replications –randomize the order of treatments within blocks

10. The Basic Plan for a Square Lattice BlockRep IRep IIRep IIIRep IV 11 2 31 4 71 5 91 6 8 24 5 62 5 82 6 72 4 9 37 8 93 6 93 4 83 5 7 Balance - each treatment occurs together in the same block with every other treatment an equal number of times ( λ = 1)

11. Example of randomization of a 3x3 balanced lattice (9 treatments) 1 From random number table RandomSequenceRank 37212 21721 96334 40443 2 From Basic Plan BlockRep IRep IIRep IIIRep IV 11 2 31 4 71 5 91 6 8 24 5 62 5 82 6 72 4 9 37 8 93 6 93 4 83 5 7 3 Randomize order of replications BlockRep IRep IIRep IIIRep IV 11 4 71 2 31 6 81 5 9 22 5 84 5 62 4 92 6 7 33 6 97 8 93 5 73 4 8 4 Randomize blocks within reps RepIIIIIIIV 3231 2113 1322 5 Resulting new plan BlockRep IRep IIRep IIIRep IV 13 6 94 5 63 5 71 5 9 22 5 81 2 31 6 83 4 8 31 4 77 8 92 4 92 6 7

12. Partially Balanced Lattices Simple Lattices –Two replications - use first two from basic plan –3x3 and 4x4 are no more precise than RBD because error df is too small Triple Lattices –Three replications - use first three from basic plan –Possible for all squares from 3x3 to 13x13 Quadruple Lattices –Four replications - use all four –Do not exist for 6x6 and 10x10 - can repeat simple lattice, but analysis is different

13. Analysis Details are the same for simple, triple and quadruple Nomenclature: –y ij(l) represents the yield of the j-th treatment in the l-th block of the i-th replication Two error terms are computed –E b - Error for block = SSB/r(k-1) –E e - Experimental error = SSE/((k-1)(rk-k-1))

14. Computing Sums of Squares SSTot =  y ij(l) 2 - (G 2 /rk 2 ) SSR = (1/k 2 )  R j 2 - (G 2 /rk 2 ) SSB = (1/kr(r-1))  C il 2 - (1/k 2 r(r-1))  C i 2 –C il = sum over all replications of yields of all treatments in the l-th block of the i-th replication minus rB il –B il = sum of yields of the k plots in the l-th block of the i-th replication –C i = sum of C il SST = (1/r)  T 2 - (G 2 /rk 2 ) SSE = SSTot - SSR - SSB - SST

15. Adjustment factor Compare E b with E e - If E b < E e –then blocks have no effect –analyze as if it were RBD using replications as blocks If E b > E e then compute adjustment factor A –A = (E b - E e )/(k(r-1)E b ) –used to compute adjusted yields in data table and table of totals Compute the effective error mean square –E e ’ = (1+(rkA)/(k+1))E e –except for small designs (k=3,4), used in t tests and interval estimates

16. ANOVA SourcedfSSMS Totalrk 2 -1SSTOT Repr-1SSR Treatmentsk 2 -1SST Block(adj)r(k-1)SSBE b Intrablock error(k-1)(rk-k-1)SSEE e

17. Testing differences To test significance among adjusted treatment means, compute an adjusted mean square –SSB u = (1/k)  B il 2 - (G 2 /rk 2 ) - SSR –SST adj = SST- A k (r-1) [ ((rSSB u )/(r-1)(1+kA))-SSB] Finally, compute the F statistic for testing the differences among the adjusted treatment means –F = (SST adj / (k 2 -1))/ E e –with k 2 - 1 degrees of freedom (k-1)(rk-k-1) from MSE

18. Standard Errors SE of adjusted treatment mean – = E e ’ / r SE of difference between adjusted means in same block – = (2E e /r)(1+(r-1)A) SE of difference between adjusted means in different blocks –= (2E e /r)(1+rA) For larger lattices (k > 4) sufficient to use –= 2E e ’ / r

19. Relative Precision Compute the error mean square of a RBD –E RB = ((SSB+SSE)/(k 2 -1)(r-1)) Then the relative precision of the lattice is –RP = E RB /E e ’

20. Numerical Example - Simple Lattice RepBlkYieldB il C il Adj I1(19)(16)(18)(17)(20) 18.213.09.56.710.157.517.61.54 2(12)(13)(15)(14)(11) 13.311.414.211.913.464.24.2.37 3(1)(2)(3)(4)(5) 15.012.417.320.513.078.25.3.46 4(22)(24)(21)(25)(23) 7.05.914.119.27.854.0-.5-.04 5(9)(7)(10)(8)(6) 11.915.217.216.316.076.63.9.34 Sum330.530.52.67 Excel

21. Second Rep RepBlkYield B il  C il Adj II1(23)(18)(3)(8)(13) 7.715.219.115.514.772.2-9.9-.86 2(5)(20)(10)(15)(25) 15.818.018.814.420.087.0-13.3-1.17 3(22)(12)(2)(17)(7) 10.211.517.011.015.365.0-10.4-.91 4(14)(24)(9)(4)(19) 10.94.710.916.69.852.915.51.35 5(6)(16)(11)(21)(1) 20.021.116.910.915.083.9-12.4-1.08 Sum361.0-30.5-2.67 SUM691.50.00.00

22. Unadjusted Yield Totals (1)30.0(2)29.4(3)36.4(4)37.1(5)28.8 (6)36.0(7)30.5(8)31.8(9)22.8(10)36.0 (11)30.3(12)24.8(13)26.1(14)22.8(15)28.6 (16)34.1(17)17.7(18)24.7(19)28.0(20)28.1 (21)25.0(22)17.2(23)15.5(24)10.6(25)39.2 SUM691.5

23. ANOVA SourcedfSSMS Total49805.42 Replication118.60 Selection (unadj)24621.82 Block in rep (adj)877.599.70=E b Intrablock error1687.415.46=E e E b is greater than E e so we compute the adjustment factor, A A = (E b - E e )/(k(r-1)E b ) = (9.70 - 5.46)/((5)(1)(9.70)) = 0.0874 Multiply A by the treatment/block sums (C) to get the adjusted totals

24. Computing Effective Error Once you have calculated the Adjustment factor (A), calculate the effective mean square (E e ’ ) –E e ’ = (1+(rkA)/(k+1))E e = (1+(2*5*0.0874)/6)*5.46 = 6.26 To test significance, compute SSB u –1/k S B il 2 - Correction factor - SSR –for this example 9834.23-9563.44-18.60=252.18 Then SST (adj) = –SST – A*k(r-1){ [ r * SSB u /(r-1)(1+kA) ] - SSB} –621.82-(0.0874)(5)(1) { [ (2 x 252.18)/1 * (1+5*0.0874)]- 77.59} = 502.35

25. Test Statistics F T to test differences among the adjusted treatment means: –(SST (adj) /(k 2 -1))/E e ’ –(502.35/24)/6.26 = 3.34 Standard Error of a selection mean – = E e ’ /r = 6.26/2 = 1.77 LSI can be computed since k > 4 – t  2E e ’ /r = 1.746 (2x6.26)/2 = 4.37

26. Relative precision How does the precision of the Lattice compare to that of a randomized block design? –First compute MSE for the RBD as: E RB = (SSB+SSE)/(k 2 - 1)(r -1) = (77.59 + 87.41)/(24)(1) = 6.88 Then % relative precision = –(E RB / E e ’ )100 = (6.88/6.26)*100 = 110.0%

27. Report of Statistical Analysis Because of variation in the experimental site and because of economic considerations, a 5x5 simple lattice design was used LSI at the 5% level was 4.37 Five new selections outyielded the long term check (12.80kg/plot) One new selection (4) with a yield of 19.46 significantly outyielded the local check (1) None of the new selections outyielded the late release whose mean yield was 19.00 Use of the simple lattice resulted in a 10% increase in precision when compared to a RBD

28. Cyclic designs BlockTreatment Label 10, 1, 3 21, 2, 4 32, 3, 5 43, 4, 6 54, 5, 7 65, 6, 0 Incomplete Block Designs discussed so far require extensive tables of design plans. Must be careful not to make mistakes when assigning treatments to experimental units and during field operations Cyclic designs are a type of incomplete block design that are relatively easy to construct and implement

29. Alpha designs Patterson and Williams, 1976 –Described a way to construct incomplete block designs for any number of treatments (t) and block size (k), such that t is a multiple of k. Includes  (0,1)-lattice designs. –α-designs are available for many (r,k,s) combinations where r is the number of replicates, k is the block size and s is the number of blocks per replicate (the number of treatments t=ks). Efficient α- designs exist for some combinations for which conventional lattices do not exist. Can also accommodate unequal block sizes. Design Software –The current version of Gendex can generate designs with up to 10,000 entries –http://designcomputing.net/gendex/alpha/http://designcomputing.net/gendex/alpha/ –Cost for individual license is $495 –Others: Alpha+, CycDesigN Analysis can be done with SAS