Split-Plot Designs - Crop and Soil Science

Split-Plot Designs
 Usually used with factorial sets when the assignment of
treatments at random can cause difficulties
– large scale machinery required for one factor but not
another
• irrigation
• tillage
– plots that receive the same treatment must be
grouped together
• for a treatment such as planting date, it may be necessary to
group treatments to facilitate field operations
• in a growth chamber experiment, some treatments must be
applied to the whole chamber (light regime, humidity,
temperature), so the chamber becomes the main plot
Different size requirements
 The split plot is a design which allows the levels
of one factor to be applied to large plots while
the levels of another factor are applied to small
plots
– Large plots are whole plots or main plots
– Smaller plots are split plots or subplots
Randomization
 Levels of the whole-plot factor are randomly
assigned to the main plots, using a different
randomization for each block (for an RBD)
 Levels of the subplots are randomly assigned within
each main plot using a separate randomization for
each main plot
A2
One Block
A1
A3
Main Plot Factor
B2
Sub-Plot Factor
B4
B1
B3
Randomizaton
Block I
Block II
T3
T1
T2
T1
T3
T2
V3
V4
V2
V1
V2
V3
V1
V1
V4
V3
V1
V4
V2
V3
V3
V2
V3
V1
V4
V2
V1
V4
V4
V2
Tillage treatments are main plots
Varieties are the subplots
Experimental Errors
 Because there are two sizes of plots, there are
two experimental errors - one for each size plot
 Usually the sub-plot error is smaller and has
more degrees of freedom
 Therefore the main plot factor is estimated with
less precision than the subplot and interaction
effects
 Precision is an important consideration in
deciding which factor to assign to the main plot
Advantages
 Permits the efficient use of some factors that
require different sizes of plot for their application
 Permits the introduction of new treatments into
an experiment that is already in progress
Disadvantages
 Main plot factor is estimated with less precision
so larger differences are required for
significance – may be difficult to obtain adequate
degrees of freedom for the main plot error
 Statistical analysis is more complex because
different standard errors are required for
different comparisons
Uses
 In experiments where different factors require
different size plots
 To introduce new factors into an experiment that
is already in progress
Data Analysis
 This is a form of a factorial experiment so the
analysis is handled in much the same manner
 We will estimate and test the appropriate main
effects and interactions
 Analysis proceeds as follows:
– Construct tables of means
– Complete an analysis of variance
– Perform significance tests
– Compute means and standard errors
– Interpret the analysis
Split-Plot Analysis of Variance
Source
df
SS
MS
F
Total
rab-1
Block
r-1
SSR
MSR
FR
A
a-1
SSA
MSA
FA
(r-1)(a-1)
SSEA
MSEA
B
b-1
SSB
MSB
FB
AB
(a-1)(b-1)
SSAB
MSAB
FAB
a(r-1)(b-1)
SSEB
MSEB
Error(a)
Error(b)
SSTot
Main plot error
Subplot error
Computations
 Only the error terms are different from the usual
two- factor analysis
SSTot
SSR
SSA
SSEA
SSB

 i  j  k Yijk  Y

rb   Y
ab  k Y..k  Y
i
i..

Y


2
2
b  i  k Y i.k  Y

ra  j Y. j.  Y


2

2


2
2
 SSA  SSR
SSAB
r  i  j Y ij.  Y
 SSA  SSB
SSEB
SSTot - SSR - SSA - SSEA - SSB - SSAB
F Ratios
 F ratios are computed somewhat differently
because there are two errors
 FR=MSR/MSEA
tests the effectiveness of blocking
 FA=MSA/MSEA
tests the sig. of the A main effect
 FB=MSB/MSEB
tests the sig. of the B main effect
 FAB=MSAB/MSEB tests the sig. of the AB interaction
Standard Errors of Treatment Means
 Factor A Means
MSEA
rb
 Factor B Means
MSEB
ra
 Treatment AB Means
MSEB
r
SE of Differences
 Differences between 2 A means
2 * MSE A
rb
with (r-1)(a-1) df
 Differences between 2 B means
2 * MSEB
ra
with a(r-1)(b-1) df
 Differences between B means at same level of A
e.g., YA1B1 ‒ YA1B2
2 * MSEB
r
with a(r-1)(b-1) df
 Difference between A means at same or different level of B
2 * b  1 MSEB  MSE A 
e.g., YA1B1 ‒ YA2B1 or YA1B1 ‒ YA2B2
critical tA has (r-1)(a-1) df
critical tB has a(r-1)(b-1) df
use critical t’ to compare means
rb
t 
b  1 MSEBtB  MSEA t A
b  1 MSEB  MSEA
Interpretation
Much the same as a two-factor factorial:
 First test the AB interaction
– If it is significant, the main effects have no meaning
even if they test significant
– Summarize in a two-way table of AB means
 If AB interaction is not significant
– Look at the significance of the main effects
– Summarize in one-way tables of means for factors
with significant main effects
For example:
 A wheat breeder wanted to determine the effect
of planting date on the yield of four varieties of
winter wheat
 Two factors:
– Planting date (Oct 15, Nov 1, Nov 15)
– Variety (V1, V2, V3, V4)
 Because of the machinery involved, planting
dates were assigned to the main plots
 Used a Randomized Block Design with 3 blocks
Comparison with conventional RBD
 With a split-plot, there is better precision for sub-plots than
for main plots, but neither has as many error df as with a
conventional factorial
 There may be some gain in precision for subplots and
interactions from having all levels of the subplots in close
proximity to each other
Factorial in RBD
Split plot
Source
Total
Block
Date
Error (a)
Variety
Var x Date
Error (b)
df
35
2
2
4
3
6
18
Source
Total
Block
Date
Variety
Var x Date
Error
df
35
2
2
3
6
22
Raw Data
Block
I
II
III
D1 D2 D3
D1 D2 D3
D1 D2 D3
Variety 1
25 30 17
31 32 20
28 28 19
Variety 2
19 24 20
14 20 16
16 24 20
Variety 3
22 19 12
20 18 17
17 16 15
Variety 4
11 15 8
14 13 13
14 19 8
Construct two-way tables
Date
I
II
III
1
19.25
19.75
18.75
19.25
2
22.00
20.75
21.75
21.50
3
14.25
16.50
15.50
15.42
Mean 18.50
19.00
18.67
18.72
Date
Variety x Date
Means
Mean
Block x Date
Means
V1
V2
V3
V4
Mean
1
28.00
16.33
19.67
13.00
19.25
2
30.00
22.67
17.67
15.67
21.50
3
18.67
18.67
14.67
9.67
15.42
Mean 25.56
19.22
17.33
12.78
18.72
ANOVA
Source
df
SS
Total
Block
Date
Error (a)
Variety
Var x Date
Error (b)
35
2
2
4
3
6
18
1267.22
1.55
227.05
14.12
757.89
146.28
120.33
MS
.78
113.53
3.53
252.63
24.38
6.68
F
0.22
32.16**
37.82**
3.65*
Report and Summarization
Variety
Date
1
2
3
4
Mean
Oct 15
28.00
16.33
19.67
13.00
19.25
Nov 1
30.00
22.67
17.67
15.67
21.50
Nov 15
18.67
18.67
14.67
9.67
15.42
Mean
25.55
19.22
17.33
12.78
18.72
Standard errors: Date=0.542; Variety=0.862; Variety x Date=1.492
Interpretation
 Differences among varieties depended on
planting date
 Even so, variety differences and date differences
were highly significant
 Except for variety 3, each variety produced its
maximum yield when planted on November 1
 On the average, the highest yield at every
planting date was achieved by variety 1
 Variety 4 produced the lowest yield for each
planting date
Visualizing Interactions
Mean Yield (kg/plot)
30
V1
25
V2
20
V3
15
V4
10
5
1
2
Planting Date
3
Variations
 Split-plot arrangement of treatments could be
used in a CRD or Latin Square, as well as in an
RBD
 Could extend the same principles to include
another factor in a split-split plot (3-way factorial)
 Could add another factor without an additional
split (3-way factorial, split-plot arrangement of
treatments)
– ‘axb’ main plots and ‘c’ sub-plots
or
– ‘a’ main plots and ‘bxc’ sub-plots