Incomplete block designs for plant breeding experiments

Full text

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*Corresponding author : pritpal@pau.edu
Date of receipt : 14.04.2017, Date of acceptance : 17.10.2017
Agric Res J 54 (4) : 607-611, December 2017
DOI No. 10.5958/2395-146X.2017.00119.3
Crop improvement is a continuous process and
development of new cultivars better than the existing
ones is the ultimate objective of plant breeding. In order
to meet this objective, plant breeders regularly generate
experimental materials and evaluate them in an experimental
design with elite cultivars termed as checks. The choice of
the experimental design depends critically on the number
of entries being evaluated and availability of seeds for
replications. In early stages of the experiment, plant breeder
has a large number of treatments (usually > 20) with sucient
seeds to be evaluated with respect to check cultivars. At
these stages, the randomized complete block design (RCBD)
has always been preferred due to its ease in the layout and
data analysis. On the other hand, a plant breeder develops
dierent mapping populations including recombinant inbred
lines (RILs), backcross inbred lines (BILs), etc. depending
on the purpose of study. The size of such populations is
also large enough for precise evaluation in RCBD. The
evaluation of more than 20-25 entries in RCBD increases soil
heterogeneity within large size blocks, thereby, increasing
the experimental error (Cochran and Cox, 1957). As a result,
the error sum of squares is large as compared to the sum of
squares attributable to the model and hence, small treatment
dierences may not be detected as signicant. The analysis
of data also reveals that the coecient of variation (CV) in
many of such experiments is high and as a consequence, the
precision of treatment comparisons is low. Thus, there is an
urgent need to shift the plant breeding eld experiments from
RCBD to such experimental designs that would increase the
eciency of evaluation of large number of treatments by
reducing the soil heterogeneity and increasing the precision
of treatment comparisons. The incomplete block designs
(IBD) are the type of designs that could better replace the
RCBD. As compared to RCBD, IBD gets its name from the
incomplete set of treatments in each block. In this design,
plots are grouped into blocks that are not large enough to
contain all the treatments, thereby, a large number of entries
can be evaluated in a number of incomplete and more
uniform blocks. These designs were introduced by Yates
(1936) in order to eliminate heterogeneity to a greater extent
as compared to RCBD and Latin squares, when the number
of treatments is large.
The precision of the estimate of a treatment eect
depends on the number of replications of the treatment; large
is the number of replications, more is the precision. Similar
is the case for the precision of estimate of the dierence
between two treatment eects. If a pair of treatments occurs
together more number of times in the design, the dierence
between these two treatment eects can be estimated with
more precision. To ensure equal precision of comparisons
of dierent pair of treatment eects, the treatments are so
allocated to the experimental units in dierent blocks of
equal sizes such that each treatment occurs at most once in
a block and it has an equal number of replications and each
pair of treatments has the same number of replications.
When the number of replications of all pairs of treatments
in a design is same, then the class of IBD is called Balanced
Incomplete Block (BIB) designs and when there are unequal
number of replications for dierent pair of treatments, then
the designs are called as Partially Balanced Incomplete Block
(PBIB) designs. In plant breeding, the minimum number
of replications required for balance is often too large to be
practical due to large number of treatments. Another class of
IBD is the resolvable incomplete block designs also called
as lattice designs. The resolvable IBD or Lattice designs
are the type of IBD, where the blocks must be capable of
arrangement in complete replications of all the treatments
and plant breeder has more exibility choosing the number
of replications depending upon the availability of resources
(Patterson and Williams, 1976). In the article, dierent types
of Lattice designs, their method of construction and analysis
using SAS (SAS Institute Inc.) software are discussed.
Lattice designs
Lattice designs are the most useful incomplete block
designs for the plant breeders. In lattice design, blocks
are grouped such that each group of block constitutes one
complete replication of treatments. We can call a group of
incomplete blocks that contain one complete replication
of treatments as superblock. These designs provide more
exibility in managing an experimental trial on replication
basis. Based on the number of replications, Lattice design
could be simple lattice if the design has two replications of
the treatments; triple-lattice if it has 3 replications and so on.
In general, if the number of replications is more, it is called
m-ple lattice (Hinkelmann and Kempthrone, 2005). Lattice
designs can also be analyzed as RCBD, if the blocking turns
out to be ineective. Based on the number of treatments,
lattice designs can be square lattice, rectangular lattice and
α-designs.
INCOMPLETE BLOCK DESIGNS FOR PLANT BREEDING EXPERIMENTS
Pritpal Singh* and Dharminder Bhatia
Department of Plant Breeding and Genetics
Punjab Agricultural University, Ludhiana – 141 004, Punjab
Opinion Paper

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Square lattice designs
The characteristic features of these designs are that the
number of treatments “t” should be equal to square of the
number of blocks “k” so that t=k2. A balanced square lattice
with k2 treatments has r = k+1 replications and are arranged
in k (k+1) blocks. The numbers of replication of partially
balanced square lattices are less than required for balanced
square lattices. So, each replication has k blocks and contains
every treatment. In this design, every pair of treatments
occurs together once in the same incomplete block. Let
“λ” be an integer number indicating how many times each
treatment occurs together in same block, and the relationship
among the number of treatments t, block size k, and number
of replications r. Numerically, it is dened as λ = r(k -1)/(t-1)
(Montgomery, 2005).
Design construction
Assume that there are t treatments labelled as 1, 2… k2
with treatment numbers arranged in a k x k square referred as
standard array. For example, in a 3 x 3 square, there are nine
treatment numbers arranged in a specic order such that each
row of the square array is considered as a block containing
three treatments. Standard array forms the Replication I of
square lattice (Table 2). To construct Replication II, each
column of the array for Replication I is taken to form the
three blocks in Replication II (Table 2). Replications III
and IV are formed using two mutually orthogonal Latin
squares (MOLS). Two Latin squares are said to be mutually
orthogonal, when superimposed, all ordered pairs are distinct
(Table 1). From the standard array, upon superimposing, the
treatment numbers that fall on the same letter in a Latin square
are taken to form a block. For example, from latin square 1
(Table 1), treatment numbers 1, 6 and 8 fall on the letter A, so
treatments 1, 6 and 8 are in the same block in Replication III
(Table 1). Similarly, B falls on 2, 4, 9 and C falls on 3, 5, and
7 that form other two blocks. Analogous to Replication III,
Replication IV is constructed from the second latin square.
Another method of constructing lattice square designs can be
found in Federer and Wright (1988) who proposed a simple
method for constructing lattice square designs with greater
than 3 numbers of blocks.
Randomization
Randomization is important for giving equal chance
to assignment of treatment to the experimental block.
Randomization of square lattice design should be carried out
with following steps: (i) Allot the treatments to treatment
numbers at random; (ii) Randomize the replications; (iii)
Randomize the blocks separately and independently within
replication; (iv) Randomize the treatments separately and
independently within each block.
Steps (ii) and (iii) give each treatment an equal chance
of being allotted to any experimental unit. If dierences
among blocks are large, the error variance per plot for the
mean of a group of treatments that lie in the same block
may be considerably higher than the average error variance.
This additional randomization ensures that the average error
variance may be used, in nearly all cases, for comparisons
among groups of treatments.
Statistical analysis using SAS
The LATTICE procedure in SAS is very handy
programme to analyse data from balanced square lattices,
partially balanced square lattices, and some rectangular
lattices. The LATTICE procedure determines the type of
lattice design from the data set and gives the message if
data is not valid. The data must consist of variables named
“Group”, “Block”, “Treatment”, and “Rep”. The values of
“Group” are 1, 2…r, where r is the number of replicates in
a balanced design. The variable “Rep” is needed when there
are more than 1 repetition of the entire basic plan. The values
of “Rep” are 1, 2…..p, where p is the number of replications
of the entire basic plan. Hence, the experiment has a total of r
x p replications. The variable “Treatment” should be written
as shown. Details can be found in SAS/STAT 9.3 User’s
Guide (SAS Institute Inc, 2011: http://support.sas.com/
documentation/cdl/en/statug/63033/HTML/default/viewer.
htm#lattice_toc.htm).
Table 2. 3 X 3 balanced lattice (t= 9, k = 3, r = 4, λ = 1)
Block Replication I Block Replication II Block Replication III Block Replication IV
B11, 2, 3 B41, 4, 7 B71, 6, 8 B10 1, 5, 9
B24, 5, 6 B52, 5, 8 B82, 4, 9 B11 2, 6, 7
B37, 8, 9 B63, 6, 9 B93, 5, 7 B12 3, 4, 8
Table 1. Standard array and two orthogonal Latin squares (MOLS)
Standard array Latin square I Latin square II Superimposed
1 2 3 A B C A B C (A,A) (B,B) (C,C)
4 5 6 B C A C A B (B,C) (C,A) (A,B)
7 8 9 C A B B C A (C,B) (A,C) (B,A)

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The SAS script of square lattice design is:
Proc lattice data;
Var [variables];
Run;
The output of the Lattice procedure using Proc lattice
displays the Analysis of variance (ANOVA) table and
related statistics. ANOVA includes Replications, Blocks
within Replications (adjusted for treatments), Treatments
(unadjusted), Intra-block Error, Randomized Complete
Block Error as the sources of variation. The Blocks within
Replications sum of squares is divided into “Component
“A” and Component “B”. Both the components may show
dierence, if there is repetition of the basic plan. The
variation due to Randomized Complete Block Error is the
sum of variation due to Blocks within Replications and Intra-
block Error. However, this SAS procedure does not give the
adjusted treatment sum of squares. In addition, it also gives
the eciency of lattice design as compared to RCBD and
adjusted means due to block eects. The related statistics
include the LSD values and variance of mean dierence in
the same and dierent blocks.
Rectangular lattice designs
To avoid the restriction of using perfect square number
of entries in square lattice design, Harshbarger (1946)
introduced rectangular lattice design, with details and
extensions provided in further series of papers by Harshbarger
(1947, 1949, 1951) and Harshbarger and Davis (1952). The
design was a useful addition to the square lattice design. A
rectangular lattice design is a resolvable incomplete block
design for t treatments in r replicates of k blocks of size k - 1,
where t = k(k - 1) and 2 ≤ r ≤ k, for some integer k. The design
has the property that any pair of treatments occurs together in
at the most one block. The design is constructed from a set of
r - 2 mutually orthogonal n x n Latin squares.
Method of construction
There are several methods of constructing rectangular
lattice designs. Here we describe the simple method
suggested by G.S. Watson (Robinson and Watson, 1949).
This method uses a Latin square with k number of rows and
columns, in which every letter in the diagonal is dierent.
For constructing rectangular lattice with t=12, k=4, r=3, rst
write down a Latin square with dierent letters assigned to
diagonal elements and give a number to all letters except
those in the leading diagonal, as illustrated below for a 4 x 4
Latin square (Table 3).
Table 3. 4 X 4 Latin square with assigned numbers
A B1C2D3
D4C B5A6
B7A8D C9
C10 D11 A12 B
The numbers in the same row of the Latin square will
form blocks of Replication I, whereas the numbers in the
same column will form the Replication II. The Replication
III will be formed by all numbers that have the same latin
letter (Table 4).
By using the rst 2 replications from any plan we
obtain a rectangular lattice in 2 replications, which by means
of repetitions can be used for an experiment in 4,6,8, etc.,
replications, by using all 3 replications of the plan we have
designs for 3,6,9 etc., replications.
Statistical analysis using SAS
The statistical analysis using SAS can be done with
Lattice procedure described above.
Alpha (α) lattice designs
Square and Rectangular lattices deigns are more
restrictive to incorporate a statutory number of treatments.
The requirements to evaluate non-statutory number of
treatments led to the development of generalized lattice
designs (Patterson and Williams, 1976). Alpha lattice design
is a class of resolvable block design that is more exible
for number of treatments to be evaluated in the experiment.
The original catalogue of designs provides designs suitable
for any number of treatments up to 100, grown in 2, 3 or 4
replications. The experimenter is able to specify the size of the
block and hence to x the number of blocks per replicate. In
addition, if the total number of treatments does not factorise
exactly, the design will permit some blocks to contain more
or fewer treatments than all the other blocks. For example, an
alpha lattice design of 46 treatments can be constructed with
8 blocks in each replication, of which 6 blocks will contain
6 treatments and rest 2 blocks will have 5 treatments. Alpha
designs are not balanced. Lack of balance, in this sense,
means that not all possible pairs of varieties occur together
within an incomplete block.
Method of construction
Alpha designs are described as 0, 1 designs if pairs of
varieties concur once or never. They are described as 0, 1, 2
designs if, in addition, some pairs of varieties concur twice
and so on. The steps below can generate alpha design with
number of treatments t, for which t = sk where s is the block
size (number of treatments per block) and k is the number of
blocks per replicate. For example, a design for 200 varieties
can be produced in k=20 blocks per replicate, each with s=10
treatments per block.
i) α-array: It is a k x r array of elements consisting of
residue modulo class of s, i.e., the elements can be 0, 1,
2, --------, s-1, i.e. if s = 3, the k x r array will have (upto
s-1) 0, 1, 2 as its elements (Table 5). Under a suitable
relabeling or permutations convert the α-array to reduced
array where the rst row and rst column consist of
zero as elements. We shall construct the α-designs with

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the help of the reduced array. We can determine the
concurrences of the dierent pairs by the α-array.
ii) Intermediate array (α*): Construction of the
intermediate array is the intermediate or second step to
develop α-designs that helps to generate the replications.
One can have (s-1) new columns from a column of
reduced array by developing the column cyclically
modulo s. By this way we can construct r(s-1) new
columns by cyclically developing each column of the
α-array modulo s (Table 5). This arrangement k x sr of
array is called intermediate array.
iii) α-designs: Now, take the intermediate array and add
{1+(j-1)s} to the elements of jth row of that array (j=1,2,-
---k), this gives us an array of size (k x sr) (Table 6).
For example, take the rst row of the intermediate array
(Table 5) written as (0,1,2), add {1+(j-1)s} =1 (j = 1 for
rst row and s = 3), this will give 1,2,3 as the rst row
of replication 1 (Table 6). Similarly, taking one to one
correspondence between the treatments and the elements
of the nal array, we get an α-designs with parameters t
= ks, b = rs, r, k.
Given the generating array α, the intermediate array
α* and hence designs are easily constructed as per the steps
given above.
Concurrences
The number of concurrence of two treatments is the
number of blocks containing pair of treatments. In the
above example pair of treatments 1 and 7 concur once,
while treatments 1 and 3 do not concur at all. A design with
concurrence g1,g2-----gn will referred as α (g1,g2,------gn)
Table 4. 4 x 3 rectangular lattice (t= 12, k = 4, r = 3)
Block Replication I Block Replication II Block Replication III
B11, 2, 3 B54, 7,10 B96, 8, 12
B24, 5, 6 B61, 8, 11 B10 1, 5, 7
B37, 8, 9 B72, 5, 12 B11 2,9,10
B410, 11, 12 B83, 6, 9 B12 3,4,11
Table 5. Construction of reduced α array and Intermediate array α* for generating α-design with t=24, k=4, r=3 s=6, b=18
Reduced array α
000
034
015
043
Intermediate array α*
012345 012345 012345
012345 345012 450123
012345 123450 501234
012345 450123 345012
design; the above example is therefore an α (0,1) design.
Softwares for generating alpha lattice designs
The software packages can be a great help to construct alpha
lattice designs. A catalogue of designs for number of varieties
up to 100 has been published, and can be accessed from within
GenStat (http://www.edgarweb.org.uk/). For larger number
of varieties, the packages Gendex (http://designcomputing.
net/gendex/alpha/), CropStat7.2 (https://drive.google.com/
le/d/0Bw-PTBz1SHmsWWlRMXdybWJWT0k/view) or
CycDesignN (http://www.vsni.co.uk/downloads/ cycdesign)
and will create the designs. The R package “Agricolae”
(https://cran.rproject.org/web/packages/agricolae/index.
html) and SAS procedure “Proc Plan” (http://www.stat.ncsu.
edu/people/dickey/courses/st711/Demos/) can also be used
for constructing optimal or near optimal alpha lattice designs.
Statistical analysis using SAS
The statistical analysis of alpha lattice design is discussed
here using an example. The data used for analysis consists of
evaluation of 15 rice varieties in alpha lattice design for tiller
number (hypothetical example). The linear model for SAS
lattice design can be written as
Y ijk = μ + τ i + γ j + ρ k(j) + ε ijk ,
where Yijk is the phenotype of ith treatment in jth replication
and k block; τ i = treatment eect i=1, 2…….k; γ j = replicate
eect j= 1, 2,……r; ρ k(j) = block within replicate eect k=
1,2,….s; ε ijk = random error. The SAS procedures Proc GLM
can be used if all the eects in the above model are xed
eects, whereas, Proc Mixed can be used if some eects are
random in nature.

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SAS code for xed eects
Proc glm; /* procedure generalized linear model*/
Class rep blk trt;
Model y = rep blk(rep) trt ; /* ‘y’ is any trait, ‘blk(rep)’ shows
blocks within replications*/
Lsmeans trt/PDIFF=all lines adjust=tukey; /* ‘PDIFF=all’
will generate pairwise
Run; dierences of all lsmeans, Tukey test is used for multiple
comparisons of means, lines is used to show the alphabets
showing signicant dierences*/
SAS codes for mixed eects including both random
and xed eects
Proc mixed; /* computes REML and ML estimates of variance
parameters, generally preferred over ANOVA*/
Class rep blk trt;
Model y = trt;
Random rep blk(rep)/solution; /* replications and blocks
within replications are taken as
ods output solution=y; random eects, ‘solution’ will produce
parameter estimates
Store y; ods output makes dataset ‘y’ of estimates, and stores
the results in ‘y’*/
Run;
Proc plm source= y; /* uses the dataset ‘y’ and do the post
tting statistical analysis*/
Lsmeans trt /pdi=all lines adjust=tukey;
Run;
Advantages of incomplete block designs
Incomplete block designs (IBD) such as alpha lattice
designs have several advantages.
Small blocks are more homogeneous than large blocks,
hence experimental error is lower. IBD can be used when
there is variability within larger blocks and also to increase
the precision of the experiment.
Lattice designs, a class of resolvable incomplete block
designs, are more precise than RCBD, where the numbers
of treatments are more than 20, which not only reduces the
error sum of square, but also helps to detect small treatment
dierences. Out of three lattice designs discussed, square and
rectangular lattice designs can incorporate a xed number of
treatments, whereas, alpha lattice designs are more exible
for number of treatments to be evaluated and the number
of blocks per replicate. Data generated from dierent eld
experiments in lattice designs can be analyzed more eciently
by using Statistical Analysis System (SAS) software.
LITERATURE CITED
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Harshbarger B 1946. Preliminary report on the rectangular lattices.
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Sci 2: 13-27.
Harshbarger B and Davis LL 1952. Latinized rectangular lattices.
Biometrics 8: 73-84.
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Experiments -Volume 2 Advanced Experiment Designs. Wiley,
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Montgomery D C 2005. Design and Analysis of Experiments (6th
ed.), Wiley, New York, 146p.
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incomplete block designs. Biometrika 63: 83-92.
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rectangular lattice designs. North Carolina Agr Exp Sta Tech
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SAS Institute Inc 2011. SAS/STAT 9.2 User’s Guide, Cary, SAS
Institute Inc., NC.
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number of varieties. J Agric Sci 26: 424-55.
Table 6. Generating α-design from Table 5
Replicate R1 R2 R3
Block B1B2B3B4B5B6B7B8B9B10 B11 B12 B13 B14 B15 B16 B17 B18
123456123456123456
7 8 9 10 11 12 10 11 12 7 8 9 11 12 7 8 9 10
13 14 15 16 17 18 14 15 16 17 18 13 18 13 14 15 16 17
19 20 21 22 23 24 23 24 19 20 21 22 22 23 24 19 20 21