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
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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
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