ST 524
NCSU - Fall 2007
Incomplete block Design
Incomplete block designs
Blocking
 Remove the among-block variation from experimental error.
 Comparison of treatments under similar conditions,
 As the number of treatments in the experiment increases, the size of the block to contain a full replication also increases.
 Increase precision of these conditions: Precision decreases as block size increases. Small blocks are favored.
Setting
Large number of treatments and want all comparisons among pairs of treatments with equal precision.
 Plant breeders: comparisons among a large number of selections in a single trial.
 Use of incomplete blocks designs: Experimental units are grouped into blocks which are smaller than a complete replication of the
treatments.
o Balanced Block Designs: Each treatment occurs together in the same block with every other treatment an equal number of times.
May be constructed for any number of treatments and any number of units per block. Numbers of replications required may
become too large
o
o Partially Balanced Designs: lack the symmetry of balanced incomplete block designs.
 PBIB Designs with two associated classes: Some pairs of treatments happen together 1 times, while some other pairs

happen together 2 times, with 1 and 2 whole numbers.
Some comparisons between some pairs of treatments are made with greater precision than among other pairs, when block
variation is large.
Example
Balanced Incomplete design: Four treatments in blocks of size two:
Block
(1)
(2)
Rep I
A
C
Treatment
A
B
C
D


Rep II
B
D
Block (1)
1
1
0
0
(3)
(4)
Block (2)
0
0
1
1
A
B
Rep III
C
D
Block (3)
1
0
1
0
(5)
(6)
Block (4)
0
1
0
1
A
B
Block (5)
1
0
0
1
D
C
Block (6)
0
1
1
0
Each treatment occurs together with any other treatment once.
Each pair of treatment will be compared with about the same precision
Tuesday October 30, 2007
1
ST 524
NCSU - Fall 2007
Incomplete block Design
 Three replicates, each with two blocks of size 2 each.
Example
Partially Balanced Incomplete design: Nine treatments in blocks of size three:
Block
(1)
(2)
(3)
A
D
G
Treatment
A
B
C
D
E
F
G
H
I
Rep I
B
E
H
Block (1)
1
1
1
0
0
0
0
0
0
C
F
I
(4)
(5)
(6)
Block (2)
0
0
0
1
1
1
0
0
0
A
B
C
Block (3)
0
0
0
0
0
0
1
1
1
Rep II
D
E
F
Block (4)
1
0
0
1
0
0
1
0
0
G
H
I
Block (5)
0
1
0
0
1
0
0
1
0



A occurs together in the same block once with B, C, D and G
A does not occur together in same block with E, F, H and I
1 = 1 and 2 = 0.

Incomplete blocks can be grouped together to form a complete replication.
Block (6)
0
0
1
0
0
1
0
0
1
Lattice design
 Number of treatments must be an exact square.
 Number of units in each block is the square root of the number of treatments
 These incomplete blocks combined in groups to create a complete replication.
 May be analyze as randomized block design
Balanced Lattices
 Each treatment occurs together in the same block with every other treatment once.
 Each pair of treatments are compared with the same precision.
 To obtain balance: if k is the block size (k2 is the number of treatments), then k + 1 replicates are required.
Example: 3 x 3 Balanced Design
Tuesday October 30, 2007
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ST 524
NCSU - Fall 2007
Incomplete block Design
t=9 treatments, k=3 block size, r = 4 replicates, b=12 blocks (number of blocks)
Block
(1)
(2)
(3)
A
D
G
Rep I
B
E
H
C
F
I
(3)
(4)
(5)
Rep II
D
E
F
A
B
C
G
H
I
(6)
(7)
(8)
Rep II
E
B
H
A
G
D
I
F
C
A
D
G
Rep IV
H
B
E
F
I
C
Field Arrangement
 Units (plots) within Blocks should be as homogeneous as possible
 Blocks within the same replication should be as similar as possible: maximize variation among replications.
Randomization
 Randomize the order of the blocks within replications: separate randomization within each replication.
 Randomize treatment code numbers separately in each block.
 Randomize the assignment of treatments to the code numbers.
ANOVA of a Balanced Lattice Design
Source
Total
Replication
Treatment(unadjusted)
Block(adjusted)
Intrablock Error
d.f.
k2 + k2 + 1
k
(k2 – 1)
(k2 - 1)
(k2 – 1(( k2 – 1)
SS
SSTotal
SSR
SST
SSB
SSE
MS
Eb
Ee
If Eb > Ee then blocking has been effective, and adjusted treatments totals are calculated as T j  T j  AW j , where W j  kTj   k  1 B j  G ,
and G is the Grand Total, Bj is the sum of the Block Totals for all blocks in which the jth treatment occurs and Tj is the total for jth treatment.

yj 
Adjusted treatment means are calculated as
Tj
 k  1
 
Ee
V  yj  
   k  1
Adjustment factor A 
 Eb  Ee 
k 2 Eb
Relative precision of the balanced lattice relative to that of a randomized block design
Tuesday October 30, 2007
3
ST 524
NCSU - Fall 2007
Incomplete block Design
pooled error Erb 
 SSB  SSE 
k  k 2  1
Effective error mean square Ee  Ee 1  A 
% relative precision 
Tuesday October 30, 2007
with  k  1  k 2  1 d . f .
Erb
100
Ee
4