### Document

```Kuswanto-2012
Rancangan Bujur Sangkar Latin:
RBL adalah pengembangan dari RAK.
Dimana RBL diterapkan untuk lahan yang
mempunyai 2 arah gradien penyebab heterogenitas
Sangat tepat untuk penelitian dengan
gradien kemiringan dan kelembaban tanah
Imagine a field with a slope and fertility gradient:
fertility
slope
B
C
A
D
E
B
C
D
E
B
A
C D
A
A
B
C
D
E
B
C
D
E
A
C
D
E
A
B
D
E
A
B
C
E
A
B
C
D
C
E
D B
E
A
Imagine a field with a slope and fertility gradient:
fertility
slope
B
C
A
D
E
B
C
D
E
B
A
C D
A
A
B
C
D
E
B
C
D
E
A
C
D
E
A
B
D
E
A
B
C
E
A
B
C
D
C
E
D B
E
A
Imagine a field with a slope and fertility gradient:
fertility
slope
B
C
A
D
E
B
C
D
E
B
A
C D
A
A
B
C
D
E
B
C
D
E
A
C
D
E
A
B
D
E
A
B
C
E
A
B
C
D
C
E
D B
E
A
We refer to Latin Squares as 3x3 or 5x5 etc.
A Latin square requires the same number of
replications as we have treatments.
Degrees of freedom are calculated as follows
(6x6 example):
Total
= (6x6) – 1 = 35
Rows
= r -1 = 6 – 1 = 5
Columns
=c–1=6–1=5
Treatments = k – 1 = 6 – 1 = 5
Error
= 35 – 5 – 5 – 5 = 20
or (r-1)(c-1) – (k – 1) = (5x5) – 5 = 20
Example:
We are interested in the effect of 4 fertilizers
(A,B,C,D) on corn yield. We have seed which was
stored under four conditions and we have four
fields in which we are conducting the experiment.
stor1
stor2
stor3
stor4
Field1
B
D
A
C
Field2
C
A
B
D
Field3
A
C
D
B
Field4
D
B
C
A
stor1
stor2
stor3
stor4
fld1
B
D
A
C
fld2
C
A
B
D
fld3
A
C
D
B
fld4
D
B
C
A
Each treatment appears in each row and column once.
Treatments are assigned randomly, but as each is
assigned, constraints are placed on the next
treatment to be assigned.
How to randomizing??
1
2
3
4
5
1
A
B
C
D
E
2
B
C
D
E
A
C
D
E
A
B
4
D
E
A
B
C
5
E
A
B
C
D
3
Then randomize the rows:
1
2
3
4
5
2
B
C
D
E
A
5
E
A
B
C
D
4
D
E
A
B
C
3
C
D
E
A
B
A
B
C
D
E
1
Pay attention the row position!
Then randomize the rows:
1
2
3
4
5
2
B
C
D
E
A
5
E
A
B
C
D
4
D
E
A
B
C
3
C
D
E
A
B
A
B
C
D
E
1
Pay attention the row position!
Then Randomize columns,
then randomly assign treatments to letters:
5
3
2
4
1
1
E
C
B
D
A
2
A
D
C
E
B
B
E
D
A
C
4
C
A
E
B
D
5
D
B
A
C
E
3
Then Randomize columns,
then randomly assign treatments to letters:
5
3
2
4
1
1
E
C
B
D
A
2
A
D
C
E
B
B
E
D
A
C
4
C
A
E
B
D
5
D
B
A
C
E
3
The LS design is most often used with a field to
account for gradients in soil, fertility, or moisture.
In a greenhouse, plants on different shelves
(rak) and benches (bangku) may be blocked.
Latin Squares are also useful when we know (or
suspect variation) of a linear nature, but do not know
the direction it will take (eg bark beetle study).
The Latin Square design is only useful if both rows and
columns vary appreciably. If they do not, a RCBD (RAK)
or Completely randomized design (RAL) would be better
(more degrees of freedom in the error term for F test)
How to analysis of a Latin Square:
Three way model, treatment fixed effect, rows
and columns are both random effects.
No replication so same problem as RCB design
(RAL) with experimental error. Must remove
interaction from model – assume no interaction.
Model  Source of Variability
Treatment (fixed)
Row (random)
Column (random)
Example: We want to compare effect of 5
different fertilizer on yield of potatoes.
B
D
C
A
C
A
D
B
A
C
B
D
D
B
A
C
Contoh : Hasil pipilan 4 varietas jagung
Lajur
Baris
1
2
1
3
4
3
4
Jlh baris
1,64 (B) 1,21(D)
1,42(C)
1,34(A)
5,62
1,47(C)
1,18(A)
1,40(D)
1,29(B)
5,35
1,67(A)
0,71(C)
1,66(B)
1,18(D)
5,225
1,56(D)
1,29(B)
1,65(A)
0,66(C)
5,17
4,395
6,145
4,475
21,365
Jlh lajur 6,35
2
Hitung jumlah perlakuan (P) dan rata-ratanya
Jumlah perlakuan dan rerata
Perlakuan
Jumlah
Rerata
A
5,855
1,464
B
5,885
1,471
C
4,270
1,068
D
5,355
1,339
Hitung JK
FK = (21,365)²/16 = 28,529
 JKt = {(1,640)² + …+ 0,660)² -FK = 1,4139
 JKb = (5,62)² + …+ (5,170)² -FK = 0,03015
 JKl = (6,350)² +…+ (4,475)² -FK = 0,8273
 JKp = (5,855)² + …+ (5,355)² -FK = 0,4268
 JKe = JKt-JKb-JKl-JKp = 0,1295


Masukkan ke tabel ANOVA 
Tabel Anova
SK
DB
JK
KT
Baris
3
0,03015
0,01005
Lajur
3
0,8273
0,2757
Perlakuan 3
0,4268
0,1422
Galat
6
0,1295
0,0215
Total
15
1,4139
F hit
Ft5% Ft1%
6,59*
4,76
Kesimpulan : Perlakuan  berbeda nyata
9,78
Interpretasi


F hitung perlakuan berbeda nyata berarti 4
perlakuan tersebut secara statistik berbeda
nyata
Perbedaan antar perlakuan menyebabkan
keragaman, dan keragaman yang
disebabkan oleh perlakuan lebih besar
daripada keragaman yang disebabkan oleh
faktor sesatan percobaan (faktor lain)
```