Split-Plot Designs
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Transcript Split-Plot Designs
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
Split-Plot: Pros and Cons
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
bi k Yi.k Y
ra j Y. j. Y
2
2
2
2
SSA SSR
SSAB
r i j Yij. 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
2 * MSEB
r
e.g., YA3B2 ‒ YA3B4
A2
One Block
A1
with a(r-1)(b-1) df
A3
Main Plot Factor
B2
Sub-Plot Factor
B4
B1
B3
SE of Differences
Difference between A means at same or different level of B
e.g., YA1B1 ‒ YA3B1 or YA1B1 ‒ YA3B2
A2
A1
A3
B2
B1
B4
Comparison of two A means at
the same or different levels of B
involves both the main effect of
A and interaction AB
B1
B3
sed
2 * b 1 MSEB MSE A
rb
One Block
critical tA has (r-1)(a-1) df
critical tB has a(r-1)(b-1) df
use critical t’ to compare means
b 1 MSEB tB MSE A t A
t
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
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
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