~-:.

~
t
;
. /'
12,- S-Li
{f)
Latin Square Design b '.•
.... should contain the same application of treatments a:1d hc:ncc ,he
variation between the row means and between the column means ca.i
be assessed and eliminated from the error rncreas;ng tlie pr.!:1~10:i of
i Q) LATIN SQUARE DESIGN
the estimates. The experimental design which s1,)mhnn..:ollly co1:1mls
the variation in two dircct10n, is as Li.Hin Sq11<1ro1 Dcs1J?,11
l 18.1 Desci·ipt.ion.
If it is fcund that the experimental n,aterial can be divided into
hnmogeneot,s groups of experimental units on any one factor only,'
l?w1Jo111ized Block Des,g;1 is suitable with one 1estriction that e<!_ch
'
J·

'After the experimental are.i is divided int0 pl01-; :;u:h :h.;t th.:
number of plots in each row is the same as the nu:nber 1,f pbls ,n e::ch
column, this number being equal to the number of treatments, Lh.!
plots are then assigned to the difforent trea,rnents such tl,r,t ,;,c-ry
treatment occurs once and only once in each row ,tnd •;:.id, c,,1umn.
The number of replications per treatrne111 i5 als, e41,d to th.; nua.ber
treatment must occur in each group (block) . For instance, in i::ase of
of treatments. Thus, if n is the number of 1reatnitnt,, ,h~, e will oo n
fidd experiment, if the fertility gradient is in one direction, it can be
rows, n columns and, therefore, 112 piots ir. th,: ~quar, .
divided into homogeneous blocks of land, all the creatments being ... • - •
represented m each block. Now if the material can be divided into The shape of the mdividual plots m:1y an)' thrng 1rom a s;iu::,re
homogeneous groups by one factor and also into groups by the second to a long narrow strip and the shape of the Latin Square ~an also be a
factor, Lari,: Square Design is very suitable under the restriction that square or a rectangle accordingly. The term 'Sq,wn' has, thcn.:fore,
tach experimental unit foils into one of the first factor groups and one been used r.ot because its sides :rre equai bu1 bec,mse nlllnb.:rs of
of the seccnd factor groups. For instance, in case of field experimems, rows "nd columns are equal.
it may be possible that the fertility gradient be in two directions, with The design is very rdiable to give precised r..:suh~ wh;!n the
the result that the field may be divided into -homogeneous blocks, in number of treatments is from S to 8 or at the most 12
two ways. The blocks in one direction are commonly known as Rows The following are the'important pt,iuts to be k.:pt 111 mud for
and the blocks in the other direction as Columns. this design :-
Columns Restrictions :
'lo-
l-
Groups on First Factor (I) Number of replications= Numbc:r of tre,1t111cnts
o
1 2 3 4 5
(2) Number of rows== Number of columns = Nu111h.:r 0f trc:unwnb
...
fffffl
-0
~l
i:::
8 f<.2. (3) Randomization of treatmenb is done- 1n such a w.:y tl::lt
\
2
0 <)
3 each treatmem occurs once and only once in each row anJ ea..:11 c0lumn
0
g. 4 l_J--L_LJ_-1 12- ,, Advice:
e L J_I__ J__J_J Ri
l\
5
This design should nol be used fo,· h:s~ thar, S I t•,,,111: :11 s ...
0
c.,-
--
c.., (.1 1 ci;-
l. --,
111 this tl<!jgn it is essential that tach row and each column precised resulls arc needed.
.w,.iil
 @,7
,----
.J
'
, ..,_,__~,
I

®
B-56
18.2 Randomization of the Treatments.
Statistical Method~
Here the necessary condition is that the treatments are to be so

-»- Latin Square Design

4th Step:-
Randomization of treatments.
Random numbers
B
A
A
D
E
C
D
E
C
B
randomized that every treatment occurs once and only once in each 2, 5, 1, 3. 4, C B D A E
column and each row. The easiest way of obtaining such randomization :. Treatments written in this E C A B D
is to start with a Reduced Latin Square (or a Latin Square in the standard order wi ll be
D E B C A
form), which is the one in which the first row and the first column are B, E, A, C, D. /
arranged in alphabetical order, and then reshuffling the rows, columns
fiheY..,are to r:-eplace A, B, C, D and E in order. Now, we get the
lsc;Ef\
and treatments with the help of random numbers.
square· with the restriction that each treatment is to occur once and
(1) only once in each row and each column and with the treatments
allocated in a perfectly random order. After this the expaiment will
1st Step:- (2) B 0 -- D E A
be laid by applying the treatment A to the plots corresponding to
Reduced Latin Square (3) c; _IL E A. B positions of A's in the final square, treaiment B, to the plots
or (4) D E A B C corresponding to positions of B's, and so on.
j Standard Square
(5) E, A B C D
18.3 Structure of Analysis and the Standard Error.
'
In this design the different sources of variation will be (I) Rows,
(1) (2) (3) (4) (5)
(2) Columns, (3) Treatments, and (4) Chance error. lf the number of
2nd Step:- C D E A B treatments is n, the toral numuer of observations will be n". The break-
Randomization of rows :- up of the degrees of freedom on the same lines as ~xpfained earlier
E A B C D
Random numbers has been given in the following structure of analysis of variance.
A B C D E
l
3, 5, 1, 4, 2,
D E A B C
Arranging the rows in this Sources of
B C D ·E A D. F. s. s. M.S. F CA-I .
order we get Variation
. V2. VR./Vf = F',.,....
Rqws n.-_ 1. s-s~ ·
3rd Step:- E D \/L Ve. Jvi' : - t--::.
A C .B Columns
.
rr~
n-1 f.s::c
Randomization of colun,ns :- B A Treatments n.t: l V·r v, / Y£ .· Fr
Random number~
4, I, 5, 3, 2,
C B --E.n-or
·.
(,! - I)(n ::-- ~), ~£ F Vf:
Arranging the columns in this
order we get

C
A
A
D
E
C
Totals .n 2
- I. .
(i) Standard error of the difference between any two trealmt:nl
.L 'I ijK -= µ + Y,· + ~j + tk + e..,:i-k
r.: " 1.. -t-1-o 'hr"vV e.5-fe-c1'-
• 4--h .. I\. • • .. 'f..:, ... , ,_
 318 Agricultural Statistics

of the, s e got in the second step are ~domized,

ln the next step the rows this select a set of four single digit om
keeping the first row as such. E 'be be 4 2 5 3 This · 1cates that
numbers from 2 to 5. The om num rs may , , , .
row 4 should be place ter row 1, row 2 next, and so on. Thus, ave the square

E A C

E .c
A D B
D C B A E

is the final layout of 5 x 5 LSD. The treatme

ayout. are allocated as givon.

23.4 ANALYSIS AND INTERPRETATION OF LSD RESULTS

The analysis of variance model for LSD is
Y··1c
I/
= µ+ r-+ c-+
I I J
tL+
,.
e··L
1/"-

Where, r; = i' row effect

h

ci = j'h column effect

t1c = /ch treatment effect.
Other notations have usual meaning.
The results of LSD will be in the fonn of two-way tables according to rows and
columns.
th
The results have to be arranged according to treatments also. Let R; be the
i row total, Ci be the }'h column total, T1c be the /ch treatment total, and G.T. be the
grand total. The different sums of squares fort x t LSD can be obtained as follows :
CF= (G.~.)2
,.,,
Total SS = I:.>i1c - CF

Row SS = .! LR~ - CF
l '

Column SS = .! I: CJ - CF
t

Treatment SS = .!. I: Tl - CF
l

Error SS = Total SS - RowSS - Column SS - treatment SS

These results can be summarised in the form of analysis of variance table.
 Lalin Square Design
Table 23.1 Analysis ofvariance fort x t LSD.
Source• of ss MS F
variation
d/

Rows t-1 RSS RMS RMS !EMS

Columns. t-1 css CMS CMS !EMS

TreatmcntJ, t-1 TSS TMS TMS !EMS

Error (t-l)(t-2) ESS EMS

Total r-1 TotalSS

If F is not significant for treatments, we can conclude that the treatment effects

I
do not differ significantly among themselves. If Fis significant, the significance
n of any treatment contrast can be tested by using the Cll ~alue, in the same way as
discussed for RRD. The CD is given by
CD = t · SE(d), ¥
where, t = table value oft for a specified level of significance and error df.
SE (d) = 6
r .
where, r = nurr,Jer of rows.
Example 23.1
I
In a varietal trial on paddy to test the yielding ability of five varieties (A, B, C,
D and E), an experiment was laid out in a 5 x 5 latin square design. The net plot size
d was 10 x 5 square metres. The results are given in Table 23.2.
1e
IC Table 23.2 Grain yield of paddy, kg/plot.
D A E B C TOTAL
39.0 24.t 26.t 37.0 42.2 168:4

E B A C D
155.7
21.2 38.1 24.0 39.3 33.t

C E B D A 172.2
35.6 33.5 38.t 40.8 24.2

A C D E B
182.2
30.8 31.1 46.7 28.7 44.9

B D C A E 165.7
44.3
Tow
29.6 41.1 26.3 24.4
-
156.4 176.0 172.t 168.8 844.2
170.9

.
The trcaunentwise arrangement of the results is as follows :
 CJ)
Agricullural Statistics
120
B C D E
A

24.1 37.0 42.2 39.0 26.1

24.0 · . 38.1 · 39.3 33.l 21.2

24.2 38.1 35.6 40.8 33.5

30.8 44.9 31.l 46.7 28.7

26.3 44.3, 41.1 29.6 24.4

Total 129.4 202.4 189.3 189.2 133.9

Mean 25.88 40.48 37.86 37.84 26.78

CF = (~/> 2
= 28506.95

TotalSS = (<39.0)2+ (24.1) 2 + ..... + (24.4)2) - CF

= 29892.86 - 28506.95

= 1385.91

RowSS = (~168.4)2+ (155.7)2+ .......... + (165.7}2)- CF

= 2858 1.44 - 28506.95

= 74.49

Column SS = (~170.9)2 + (156.4)2 + ........ + (168.8)2)- CF

= 28551.12 - 28506.95

= 44.1 7

Variety SS = [ ½(129.4)2 + (202.4)2 + ..... + (133.9) 2] - CF

= 29454.09-28506.95 = 947.14
Error SS = 1385.91- 74.49- 44.17 - 947:14 = 320.09
 Lalin. ~quote Design 321

Table 13.J Anal. ys1S

· of v¢ance for ~e in Table 23.2.

Sources of
variation
df -. ss MS F

Rows 4 74.49 18'.62 <1

Columns 4 44.17 11.04 <1

1
Varictie"i 4 .947.14 236.78 ' 8.~

EITOG' ., 12 . :er, 3~.09 -,. 26.66 :.

Total 24 , • 1385.9~
- '( ' ' .

e treatments differ
-
,.... .,1 • , . 1 . •-.-·1.,.u::.
- significantly~ a.,.. c:l :.;:,.,~!
•--' '-
'

-r. ;.-•• : :, · ",. •... ,.fl

Th
- S'E(d) ~ "---=~-
~:,-~, ( •
_ • · ••,1 2t26!66} i· ") t~ ,. ai·;, '-.;!:,rid.r.oJ ;..t.!. ';)(cla'f
- ·- -·=3:263:
- -- - - - ·- - - - --. 5·. . ·- . . ' .. ,... . "'
The table value oft foLCX - 0.05-an<lll 'it.Lisl l ;~~ :~ .
CD=(2.179)(3.263)=7~11

l 1/ 1 • I -

The bar chart is · .,

.c-- D
• • / ., • - J 1
Based on the bar chart the conclusions can be drawn. ,m, •"''

23.S REPEATEOLATIN.§QUAR~ .
We have seen that -me en:or variance ·is not estimable for 2 x 2, latin square
design amrthe error degrees of freedom are too small in case of 3 x 3 and 4 x 4
LSDr In order to make the latin square design more effective in such ases the latin
square designs may be repeated a number of times. All the squares will be~ same
order. In each square I.he treatments are same. Each square will tiave a separate set
of experimental units. The randomization for each square will be done separately.
Suppose I.here are s latin squares of order , x t. For each square .t separntc
analysis is done in the usual way. The corresponding sums of squares from
different latin squares are then added to give the 'pooled sum o( squares'. The
pooled row sum of squares is also calle4 'rows within squares ss·. Similarly tht>re
will be 'columns within squares SS', 'treatments within squares SS' and 'crro1
within squares ss·.
Next, the ($<1uare x treatment) uible is ronned and lhe following sums ol
squares arr ,...,btained: ·