m20 lattice

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

Good experimental technique – be careful with land choice, preparation, and husbandry during the

experiment– choose seeds of uniform viability– use consistent methods for data collection

Use of controls– systematically or randomly placed controls can be used to identify

site variability and adjust yields of the entries

Incomplete Block Designs We group into blocks to

– Increase precision

– 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.)

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

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– statistical analysis more complex

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=t2. 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

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

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 (block size)– 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

Lattice Designs Square lattice designs

– number of treatments must be a perfect square (t = k2)

– 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

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

The Basic Plan for a Square Lattice

Block Rep I Rep II Rep III Rep IV 1 1 2 3 1 4 7 1 5 9 1 6 8 2 4 5 6 2 5 8 2 6 7 2 4 9 3 7 8 9 3 6 9 3 4 8 3 5 7

Balance - each treatment occurs together in the same block with every other treatment an equal number of times (λ = 1)

Example of randomization of a 3 x 3 balanced lattice (t = 9)

1 Assign r random numbers

Random Sequence Rank 372 1 2 217 2 1 963 3 4 404 4 3

2 From Basic Plan


3 Randomize order of replications


4 Randomize blocks within reps

Rep I II III IV 3 2 3 1 2 1 1 3 1 3 2 2

5 Resulting new plan


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

Analysis Details are the same for simple, triple and

quadruple

Nomenclature:– yij(l) represents the yield of the j-th treatment in the l-th

block of the i-th replication

Two error terms are computed– Eb - Error for block = SSB/r(k-1)

– Ee - Experimental error = SSE/((k-1)(rk-k-1))

Computing Sums of Squares SSTot = yij(l)

2 - (G2/rk2)

SSR = (1/k2)Rj2 - (G2/rk2)

SSB = (1/kr(r-1))Cil2 - (1/k2r(r-1)) Ci

2

– Cil = sum over all replications of yields of all treatments in the l-th block of the i-th replication minus rBil

– Bil = sum of yields of the k plots in the l-th block of the i-th replication

– Ci = sum of Cil

SST = (1/r)T2 - (G2/rk2)

SSE = SSTot - SSR - SSB - SST

Adjustment factor Compare Eb with Ee - If Eb < Ee

– then blocks have no effect– analyze as if it were RBD using replications as blocks

If Eb > Ee then compute adjustment factor A

– A = (Eb - Ee )/(k(r-1)Eb)

– used to compute adjusted yields in data table and table of totals

Compute the effective error mean square– Ee’ = (1+(rkA)/(k+1))Ee

– except for small designs (k=3,4), used in t tests and interval estimates

ANOVA

Source df SS MSTotal rk2-1 SSTOTRep r-1 SSRTreatments k2-1 SSTBlock(adj) r(k-1) SSB Eb

Intrablock error (k-1)(rk-k-1) SSE Ee

Testing differences

To test significance among adjusted treatment means, compute an adjusted mean square– SSBu = (1/k)Bil

2 - (G2/rk2) - SSR

– SSTadj = SST- A k (r-1) [ ((rSSBu)/(r-1)(1+kA))-SSB]

Finally, compute the F statistic for testing the differences among the adjusted treatment means– F = (SSTadj / (k2-1))/ Ee

– with k2 - 1 degrees of freedom (k-1)(rk-k-1) from MSE

Standard Errors SE of adjusted treatment mean

– = Ee’ / r

SE of difference between adjusted means in same block– = (2Ee/r)(1+(r-1)A)

SE of difference between adjusted means in different blocks– = (2Ee/r)(1+rA)

For larger lattices (k > 4) sufficient to use – = 2Ee

’ / r

Relative Precision

Compute the error mean square of a RBD– ERB = ((SSB+SSE)/(k2-1)(r-1))

Then the relative precision of the lattice is– RP = ERB/Ee

’

Numerical Example - Simple Lattice

Rep Blk Yield BilCil Adj

I 1 (19) (16) (18) (17) (20) 18.2 13.0 9.5 6.7 10.1 57.5 17.6 1.54 2 (12) (13) (15) (14) (11) 13.3 11.4 14.2 11.9 13.4 64.2 4.2 .37 3 (1) (2) (3) (4) (5) 15.0 12.4 17.3 20.5 13.0 78.2 5.3 .46 4 (22) (24) (21) (25) (23) 7.0 5.9 14.1 19.2 7.8 54.0 -.5 -.04 5 (9) (7) (10) (8) (6) 11.9 15.2 17.2 16.3 16.0 76.6 3.9 .34 Sum 330.5 30.5 2.67

Second Rep

RepBlk Yield Bil Cil Adj

II 1 (23) (18) (3) (8) (13) 7.7 15.2 19.1 15.5 14.7 72.2 -9.9 -.86 2 (5) (20) (10) (15) (25) 15.8 18.0 18.8 14.4 20.0 87.0 -13.3 -1.17 3 (22) (12) (2) (17) (7) 10.2 11.5 17.0 11.0 15.3 65.0 -10.4 -.91 4 (14) (24) (9) (4) (19) 10.9 4.7 10.9 16.6 9.8 52.9 15.5 1.35 5 (6) (16) (11) (21) (1) 20.0 21.1 16.9 10.9 15.0 83.9 -12.4 -1.08 Sum 361.0 -30.5 -2.67 SUM 691.5 0.0 0.00

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

SUM 691.5

ANOVA

Source df SS MS

Total 49 805.42

Replication 1 18.60

Selection (unadj) 24 621.82

Block in rep (adj) 8 77.59 9.70=Eb

Intrablock error 16 87.41 5.46=Ee

Eb is greater than Ee so we compute the adjustment factor, A

A = (Eb - Ee )/(k(r-1)Eb ) = (9.70 - 5.46)/((5)(1)(9.70)) = 0.0874Multiply A by the treatment/block sums (C) to get the adjusted totals

Computing Effective Error Once you have calculated the Adjustment factor

(A), calculate the effective mean square (Ee’ )

– Ee’ = (1+(rkA)/(k+1))Ee = (1+(2*5*0.0874)/6)*5.46 = 6.26

To test significance, compute SSBu

– 1/k S Bil2 - Correction factor - SSR

– for this example 9834.23-9563.44-18.60=252.18

Then SST(adj)=– SST – A*k(r-1){ [ r * SSBu/(r-1)(1+kA) ] - SSB}

– 621.82-(0.0874)(5)(1) { [ (2 x 252.18)/1 * (1+5*0.0874)]-77.59} = 502.35

Test Statistics

FT to test differences among the adjusted treatment means:– (SST(adj)/(k2-1))/Ee

’

– (502.35/24)/6.26 = 3.34

Standard Error of a selection mean– = Ee

’/r = 6.26/2 = 1.77

LSI can be computed since k > 4– t 2Ee

’/r = 1.746 (2x6.26)/2 = 4.37

Relative precision How does the precision of the Lattice compare

to that of a randomized block design?– First compute MSE for the RBD as:

ERB = (SSB+SSE)/(k2 - 1)(r -1) =

(77.59 + 87.41)/(24)(1) = 6.88

Then % relative precision = – (ERB / Ee

’ )100 = (6.88/6.26)*100 = 110.0%

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

Cyclic designs

BlockTreatment

Label

1 0, 1, 3

2 1, 2, 4

3 2, 3, 5

4 3, 4, 6

5 4, 5, 7

6 5, 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

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– r is the number of replicates– k is the block size– s is the number of blocks per replicate– number of treatments t = ks

Efficient α-designs exist for some combinations for which conventional lattices do not exist

Can accommodate unequal block sizes

Alpha designs Design Software

– The current version of Gendex can generate designs with up to 10,000 entries

– http://www.designcomputing.net/gendex/– Evaluation copy is free

– Cost for perpetual/personal license is $199

– Others: Alpha+, CycDesigN

Analysis can be done with SAS

m20 lattice

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