Efficiency of incomplete split-plot designs
A compromise between traditional split-plot designs
and randomised complete block design
Kristian Kristensen,
Federica Bigongiali and Hanne Østergård
IAMFE Denmark 2008
Koldkærgård, June 30th to July 3rd 2008
AARHUS
UNIVERSITET
Faculty of Agricultural Sciences
Outline
 Introduction
 What is an incomplete split-plot
 Compared to traditional split-plot and
randomised complete block design
 Performed experiments
 Efficiency of incomplete split-plot designs
 Compared to traditional split-plot and
randomised complete block design
 Discussion and conclusions
Introduction
 Example of trial to be performed
 2-factorial design
 Treatment factor 1 with few levels (e.g. ±
Herbicides)
 Treatment factor 2 with many levels (e.g. a
large number of varieties)




Some possible designs
Split-plot
Randomised complete block designs
Incomplete split-plot
Introduction
 Split-plot
 Very convenient
 Easy to apply herbicides to many plots in one
run
 Needs only guard area around each whole-plot
 Inefficient comparison of treatments
 Herbicides: few and large whole plots, large
replicates and thus large distance between
whole plots
 Varieties: large whole plots and thus large
distance between some sub-plots
Introduction
 Randomised complete block
 Inconvenient
 Difficult to apply herbicides to each individual
plot
 May need guard area around each plot
 Efficiency of treatment comparisons
 Herbicides: many whole plots increase efficiency
but large replicates and thus large distance
between most plots decrease efficiency
 Varieties: large replicates and thus large
distance between most plots decrease efficiency
What is an incomplete split-plot
Small example: ±Herbicide, 9 varieties
Randomised complete block design
9 5 2 8 9 5 4 6 4 1 1 8 7 3 2 7 3 6
5 7 2 9 4 4 8 3 5 6 7 1 6 2 8 1 3 9
Traditional split-plot
2 9 5 8 1 4 7 6 3 9 8 5 6 4 1 2 3 7
7 2 9 3 8 4 1 6 5 7 1 6 5 2 8 9 3 4
Incomplete split-plot
8 2 5 8 2 5 1 4 7 7 4 1 3 6 9 9 6 3
9 7 8 7 8 9 4 6 5 6 5 4 2 1 3 2 1 3
What is an incomplete split-plot
 Incomplete split-plot
 Practical compromise
 Easier than RCB, more difficult than split-plot
 May require guard-area around each pair
(group) of incomplete blocks
 Efficiency
 Herbicides: several whole plots, comparison
within pair (group) of incomplete block and thus
moderate distance between incomplete “wholeplots”: More efficient than split-plot
 Varieties: few plots within each incomplete
“whole plot” and thus small distance between
sub-plots: More efficient that RCB and split-plot
Incomplete split-plot
 Construction
 Can be based on different types of incomplete
block designs
 We choosed to use to use -designs
(generalised lattice)

-designs
 Are resolvable
 Are available for almost any number of
varieties and replicates in combination with a
broad range of block sizes
Performed experiments
Trial
no
A
B
C
D
E
Replicates
r
2
2
2
3
3
Number of
Treat- Varie- Blocks
ments
ties
t
v
s
2
2
2
2
2
48
48
48
35
8
6
8
6
5
2
Plots
per
block
k
8
6
8
7
4
Trial A-D: From the project “Characteristics of spring barley
varieties for organic farming (BAR-OF)“
Trial E: From the project “Screening of the potential
competitive ability of a mixture of winter wheat cultivar
against weeds”
Performed experiments, trial A
Replicate 1
45
20
35
18
24
21
33
40
35
24
18
20
40
45
33
21
03
25
04
16
37
10
23
11
16
37
10
04
25
23
03
11
42
08
46
12
47
06
13
39
47
08
39
13
06
46
42
12
34
27
32
19
05
41
38
44
19
44
32
05
38
34
41
27
17
30
29
43
02
28
22
26
17
28
43
22
29
30
26
02
Replicate 2
14
31
15
09
48
07
01
36
14
31
15
09
01
36
48
07
38
08
45
22
40
28
04
07
40
45
08
07
28
38
04
22
42
25
46
09
21
19
15
30
25
15
09
19
21
42
46
30
36
24
17
10
27
03
33
13
03
27
13
10
17
36
33
24
29
14
18
41
05
37
43
39
37
29
39
14
41
18
43
05
31
02
12
11
20
32
34
06
34
20
12
32
11
06
02
31
16
23
44
01
48
47
26
35
35
01
26
48
16
44
47
23
Each plot is
1.5 m × 11.0 m
Each block is
12.0 m × 11.0 m
Performed experiments, trial E
plotno
6
6.0
3.1
6.1
7.1
1.1
18
19
30
31
42
43
2.1
4.1
5.1
5.0
2.0
8.0
4.0
8
17
20
29
32
41
44
6.1
5.1
8.1
7.0
8.0
6.0
5.0
9
16
21
28
33
40
45
Variety 2.0
4.0
3.0
1.0
2.1
4.1
3.1
1.1
plotno
10
15
22
27
34
39
46
Variety 4.0
1.0
7.0
5.0
4.1
1.1
7.1
5.1
plotno
11
14
23
26
35
38
47
Variety 3.0
8.0
6.0
2.0
8.1
6.1
3.1
2.1
plotno
12
13
24
25
36
37
48
plotno
5
Variety 7.1
Rep 2
3.0
7
Variety 8.1
Rep 1
1.0
plotno
4
3
2
1
Each plot is
2.5 m × 12.5 m
Each block is
10.0 m × 12.5 m
<----- 12.50m ----->
Rep 3
Variety 7.0
Measure of efficiency
 Depends on the comparisons of interest
-Herbicide
+Herbicide
Mean
1
2
3
4
5
6
7
8
9
Mean
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Comparisons:
1. Varieties within + or – herbicides i.e. within a row of the
interaction table
2. Treatments within a given variety, i.e. within a given column of
the interaction table
3. Mean of treatments
4. Mean of varieties
Trial
no.
Efficiency
of the
designs,
Yield
A
B
C
D
E
Design1:
Comparison
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Plan
6.5
6.7
2.2
4.6
8.4
8.4
1.7
6.2
4.9
5.3
2.4
3.5
2.0
2.2
1.0
1.4
5.8
5.9
3.7
4.2
Grain yield, hkg ha-1
LSD
Relative
reduction, %
SplitRCB
Split
RCB
plot
plot
7.9
8.04
18
19
10.1
8.04
34
17
11.3
1.16
81
-90
5.6
5.69
18
19
8.5
9.19
1
9
16.9
9.19
50
9
23.2
1.33
93
-28
6.0
6.50
-3
5
7.9
8.92
38
45
19.7
8.92
73
41
27.0
1.29
91
-86
5.6
6.31
37
45
2.8
2.86
29
30
3.1
2.86
29
23
1.7
0.48
41
-108
2.0
2.02
30
31
5.7
7.4
-2
21
7.6
7.4
23
21
7.8
2.6
53
-40
4.1
5.2
-4
19
Efficiency
of the
designs,
%Mildew
Trial
no.
A
C
D
Design1:
Comparison
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Variety:Treat
Treat:Variety
Treat
Variety
Powdery mildew severity, transformed %
LSD
Relative
reduction, %
Plan
SplitRCB
Split
RCB
plot
plot
0.609 0.625 0.625
3
3
0.609 0.695 0.625
12
3
0.086 0.577 0.090
85
4
0.440 0.442 0.442
0
0
0.950 1.032 1.032
7
7
0.970 1.149 1.032
16
6
0.263 0.954 0.149
72
-77
0.672 0.730 0.730
8
8
0.437 0.437 0.438
0
0
0.437 0.455 0.438
4
0
0.074 0.175 0.074
58
0
0.309 0.309 0.310
0
0
Variable
Efficiency
of the
designs,
other
variables
Wheat
June 19
g 0.25m-2
Log Weed
June 19
Log g 0.25m-2
Vegetation
cover (%)
June 5
LAI
June 21
MTA
June 21
Log DIFN
June 21
LSD
Design:
Comparison
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Variety:Herbicide
Herbicide:Variety
Herbicide
Variety
Plan
161
174
119
115
2.8
2.8
1.0
2.0
.052
.055
.032
.036
0.59
0.59
0.26
0.42
5.7
5.7
2.0
4.1
0.52
0.53
0.26
0.37
Splitplot R
167
210
204
118
2.8
2.9
2.1
2.0
.055
.057
.041
.039
0.60
0.62
0.44
0.42
5.8
6.0
4.3
4.1
0.52
0.57
0.45
0.37
RCB
195
195
69
138
2.8
2.8
1.0
2.0
.077
.077
.027
.055
0.59
0.59
0.21
0.42
5.8
5.8
2.1
4.1
0.53
0.53
0.19
0.38
Relative
reduction, %
SplitRCB
plot R
3
17
17
11
42
-73
3
17
1
0
4
0
53
0
1
0
6
33
3
30
21
-17
6
33
2
1
4
0
41
-24
2
1
0
2
4
2
53
2
0
2
-0
1
7
0
43
-38
-1
1
Discussion and conclusions
 Efficiency
 Compared to randomised complete block
design
 Incomplete split-plot were most often less efficient
when comparing the main effect of treatments
 Larger number of independent plots/smaller blocks
 Incomplete split-plot most often more efficient for
other comparisons
 Compared to traditional split-plot
 Incomplete split-plot were most often more
efficient for all types of comparisons
 Especially for comparing treatment means (many more
degrees of freedom and smaller blocks)
Discussion and conclusions
 Increase in efficiency
 In most cases larger for grain yield than for
mildew
 Probably because mildew is less sensible to soil fertility
 Small for trial E when comparing mean of
varieties and varieties within treatment
 Relative small reduction in block sizes
 Small for trial B when comparing mean of
varieties and varieties within treatment
 Reason unknown
Discussion and conclusions
 Practical considerations
 Treatment applications
 Easier than randomised complete block design
 More difficult than split-plot design
 Guard areas
 Less than for randomised complete block design
 More than for split-plot design
 Design and statistical analysis
 More complex than both randomised complete block
design and split-plot design
 Appropriate software are available and with today's
computer power this should not be a problem