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Split Plot Design
(Rancangan Petak Terbagi)
Kuswanto dan Rizali 2014
Split Plot Design:
In RCB designs or Latin Squares, the
experiment is broken up into
homogenous blocks – every block
gets every treatment combination
The idea is to have the variation
within each block as small as
possible (reduces experimental error)
The greater the number of treatment
combinations, the bigger each block must be
- therefore the variation within each block
increases
To overcome this increased variation within
large blocks, we may use the split plot design
– but we pay a price
We sacrifice precision in estimating the effects
of one of the variables, but often gain precision
in estimating effects of the second factor.
The split plot design examining irrigation (4 levels
and variety (4 levels):
We begin by assigning levels of one factor
randomly to blocks as in a RCB design: eg
irrigation levels
H
L
M
0
L
0
H
M
0
M
L
H
Block 1
Block 2
Block 3
Each of these is referred to as a main plot
Next, we assign treatment levels of the second
factor randomly to sub-plots within each main
plot. eg variety
H
a
c
L
d
c
b
a
M
d
c
c
d
0
a
b
b
d
a
c
Block 1
L
b
d
0
c
a
c
a
H
b
d
a
b
M
c
d
b
a
c
d
Block 2
0
Block 3
d
c
a
M
b
d
b
a
L
c
c
a
d
H
b
a
c
b
d
main-plot
sub-plots
H
a
c
L
d
c
b
a
M
d
c
c
d
0
a
b
b
d
a
c
Block 1
L
b
d
0
c
a
c
a
H
b
d
a
b
M
c
d
b
a
c
d
Block 2
0
Block 3
d
c
a
M
b
d
b
a
L
c
c
a
d
H
b
a
c
b
d
In a split plot design, we sacrifice precision for the
factor assigned to main plots, but we gain
precision for factor assigned to subplots
(compared to RCB design)
main-plot
sub-plots
H
a
Block 1
c
L
d
c
b
a
M
d
c
c
d
a
0
b
b
d
a
c
error between main plots should be larger than
error for sub-plots
main-plot
sub-plots
H
a
Block 1
c
L
d
c
b
a
M
d
c
c
d
a
0
b
b
d
a
c
We may use a split plot when we are
interested in testing the effect of one
factor over a wider variety of conditions
(in our example, test variety over a range
of water conditions)
Here we are more interested in the effect
of variety than water.
We might also use a split plot design
when we are interested in adding a factor
to an existing experiment – ie blocks
already set up, then add sub-plots.
requires enough size in main-plots to be
split into sub-plots
Another use of a split plot design is for
treatments which require a certain area
to be applied (eg types of harvesting
equipment, pesticide applications etc.)
These factors can be assigned to main
plots, with a second factor assigned to
sub-plots
Remember the idea is that variation between
main plots is greater than within sub-plots.
Other possible situations include parts of plants
– variation between plants may be greater than
within leaves of the same plant so plants could
be main plots with leaves as sub-plots
Age of animals may add variation to response
to a treatment – litters of animals could form
main-plots, while individuals within a litter
could form sub-plots
Analysis of a split plot design:
In a split plot design, we first look at the effect of
main plot treatment, then sub-plot treatment
Block 1
H
L
M
0
Block 2
L
0
H
M
Block 3
0
M
L
H
so in our main plot, we have effect of water and block
Block 1
H
L
M
0
Block 2
L
0
H
M
Block 3
0
M
L
H
so in our main plot, we have effect of water and block
GLM model for this is just like a RCB design:
block
irrigation
block*irrigation (remainder)
H
L
M
0
a c d c b a d c c d a b b d a c
Block 1
Block 2
Block 3
L
0
H
M
b d c a c a b d a b c d b a c d
0
M
L
H
d c a b d b a c c a d b a c b d
But we must still deal with sub-plots:
GLM for sub plots looks like:
variety
variety*irrigation
error (really block*irrigation*variety)
So the whole model looks like:
block
main plot effects
irrigation
block*irrigation (remainder a)
variety
variety*irrigation
sub plot effects
error (remainder b)
remainder a is error term used to test main plot effects
remainder b is error term to test sub plot effects
a generalized model looks like:
block
main-plot treatment
mp treat * block (remainder a)
sub-plot treatment
sp treat*mp treat
error (remainder b)
CONTOH SOAL
Sebuah penelitian menguji 4 (V) varietas kedele dengan
dosis pengairan (P). Data dibawah adalah hasil pengamatan
terhadap hasil biji (ku/ha) (Yitnosumarto, 1987)
Perlakuan
1
P1
P2
Total
Ulangan
2
Total
3
V1
4,60
6,27
4,51
15,38
V2
2,63
2,26
1,67
6,56
V3
6,06
4,01
5,10
15,17
V4
6,44
6,35
4,26
17,05
V1
6,19
8,61
7,44
22,24
V2
7,36
9,78
6,44
23,58
V3
12,21
11,12
9,20
32,53
V4
11,64
14,46
11,19
37,29
57,13
62,86
49,91
169,80
1. Analisis 2 arah untuk petak
utama
Petak
utama
Ulangan
1
2
Total
3
P1
19,73
18,89
15,54
54,16
P2
37,70
43,97
43,37
115,64
Total
57,03
62,86
49,91
169,80
2. Anova untuk petak utama
Dari tabel 2 arah antara petak utama dengan
ulangan, dianalisis untuk menghitung jumlah
kuadrat petak utama dan jumlah kuadrat
ulangan.
Perhatikan, tabel tersebut adalah tabel untuk
data RAK faktor tunggal dengan 3 ulangan.
Cara penghitungan jumlah kuadrat juga sama
dengan RAK faktor tunggal, dan akan dihasilkan
JK untuk ulangan, petak utama, galat percobaan
(galat a) dan JK total petak utama.
Hasil analisis ditunjukkan pada tabel anova
petak utama.
Sumber
Keragaman
Db
JK
KT
Fhit
Ulangan
2
10,516
5,258
2,600
Petak Utama 1
(P)
Galat a
2
157,491
157,491
77,888*
4,044
2,022
Total PU
172,051
5
3. Tabel 2 arah interaksi
PUxAP
Untuk menghitung JK anak petak dan interaksinya,
dibuatkan tabel dua arah interaksi
antara petak utama dengan anak petak.
Petak
Utama
Anak petak
Total
V1
V2
V3
V4
P1
15,38
6,56
15,17
17,05 54,16
P2
22,24
23,58
32,53
37,29 115,64
Total
37,62
30,58
47,70
54,34 169,80
4. Tabel anova lengkap


Dari tabel tersebut dapat dihitung JK untuk
anak petak (V), interaksi (PV), galat b. JK
total dihitung sebagaimana rancangan yang
lain, yaitu jumlah kuadrat dari semua data
yang ada dikurangi Faktor Koreksi.
Perhatikan, JK total berbeda dengan JK
total petak utama. JK total adalah jumlah
kuadrat keseluruhan data sedang JK total
petak utama hanya dihitung dari nilai masingmasing petak utama.
Dengan demikian JK total akan digunakan untuk
menghitung JK galat b. Hasil perhitungan secara
lengkap adalah sbb :
Sumber
Keragaman
Db
JK
KT
F hit
Ulangan
2
10,516
5,258
2,600
Petak Utama (P)
1
157,491
157,491
77,888*
Galat a
2
4,044
2,022
Anak Petak (V)
3
57,3
19,1
17,945**
Interaksi PuxAP
3
17,137
5,712
5,372
Galat b
12
12,759
1,063
Total
23
259,248
T tabel
5%
1%


Misal, petak utama dan anak petak
memberikan pengaruh nyata dan
sangat nyata, maka pembahasan
ditekankan pada masing-masing
faktor.
Untuk uji perbandingan berganda masingmasing faktor menggunakan masing-masing
KT galat. Apabila akan membedakan antar
petak utama, digunakan KT galat a, sedang
untu membedakan antar anak petak
digunakan KT galat b.
Analisis ragam di r
aov(Hasil ~ Blok+Pengairan*Varietas + Error(Pengairan:Blok),
data=split.data)
Uji Tukey dari interaksi
Tugas

Lakukan analisis data hasil penelitian
dengan rancangan petak terbagi

Petak utama adalah aplikasi pestisida

Anak petak adalah varietas


Lakukan analisis untuk menghitung anova
dan menguji masing-masing perlakuannya
Kumpulkan Minggu Depan
Soal

Factor A (main plot): spraying at two levels
A1 = no insecticide
 A2 = insecticide


Factor B (subplot): variety
B1 =variety 1
 B2 =variety 2
 B3 =variety 3
