May, 1943
Research Bulletin 318
Some Additional Lattice
Square Designs
By W. G.
COCHRAN
AGRICULTURAL EXPERIMENT STATION
IOWA STATE COLLEGE OF AGRICULTURE
AND MECHANIC ARTS
STATISTICAL SECTION
BUREAU OF AGRICULTURAL ECONOMICS.
UNITED STATES DEPARTMENT OF AGRICULTURE
Cooperating
AMES, IOWA
CONTENTS
Page
Introduction ________________________________________________________________________
731
1.
Properties of the designs ____________________________________________
732
2.
Field layout and randomization ________________________________
733
3.
Statistical analysis ______________________________________________________
738
4.
Estimation of the standard errors __________________________
742
5.
Loss of efficiency due to asymmetry ________________________
745
Summary ____________________________________________________________________________
747
References cited __________________________________________________________________
748
Some Additional Lattice Square
Designs
1
By W. G.
COCHRAN'
The latin square design has been extensively used in
field experiments, because of its ability to eliminate from
the experimental errors the effects of soil fertility variations
in two directions. More recently, Yates (1, 2) developed the
lattice square designs for varietal trials conducted under a
plant breeding program. In these designs, the number of
varieties must be an exact square. Within each replication,
the plots are laid out in a square array, in which every row
and column contains the same number of plots. (The physical dimensions of the rows and columns will, of course, be
different unless the plots are square in shape.) In successive
replications the groupings of the varieties into rows and
columns are changed so that with the complete design every
pair of varieties has appeared together once either in the
same row or in the same column. This symmetry makes it
possible to adjust the varietal total or average yields, by
simple calculations, for variations in the fertility of different
rows and columns. In this way the effects of soil fertility
variations in two directions can be eliminated from the experimental errors, just as in the latin square.
These designs will be described as balanced lattice
squares, in order to distinguish them from the designs which
form the subject of this bulletin. In the useful range for
most practical purposes, the available selection of balanced
lattice squares is shown in table 1a.
TABLE la.
Numher of varieties
Numbers of replicates
AVAILABLE SELECTION OF BALANCED
LATTICE SQUARES.
I 16 I 25 I 49 I
I 5, 10 I 3, 6 I 4, 8 I
64
9
I 81 I 121 I 169
I 5, 10 I 6 I 7
The selection is restricted both as to number of varieties
and the number of replicates. The restricted choice of numbers of varieties does not appear to limit seriously the utility
of these designs in plant breeding work, where the number
of varieties introduced into an experiment can usually be
1 Project 514 of the Iowa Agricultural Experiment Station In cooperation
with the Bureau of .<\gricultural Economics, United States Department of
Agriculture.
2 Research Professor, Agricultural Statistics, Iowa Agricultural Experi.
ment Station,
732
varied to some extent. The restrictions on the number of
replicates constitute perhaps a more frequent limitation. In
varietal trials it is convenient to be able to vary the number of replications anywhere between three and nine or more,
according to the labor and field space available and to the
state of development of the breeding program.
The purpose of the present bulletin is to describe an additional group of lattice square designs, which allow a somewhat wider choice of number of replications. In this group,
the numbers of varieties and replicates which are likely to
be suitable for plant breeding experiments are shown in
table lb.
TABLE lb.
SO:'\IE ADDITIONAL LATTICE SQUARE DESIGNS.
Number of varieties
Numbers of replicates
I
49
3
64
3, 4
I
I
81
3, 4
I 121
I 3, 4, 5
I 169
I 3, 4, 5, 6
It will be noted that the designs above all require fewer
replicates than the corresponding balanced lattice squares
in table 1a. In types of experimentation where the balanced
lattice squares have proved successful, these additional designs may therefore be serviceable when it is not practicable
to plant the full number of replications ·necessary for a balanced lattice square. In this respect, the designs in table 1b
are analogous to the simple and triple lattices (3), which
are used when the number of replications is insufficient for
a balanced lattice.
1. PROPERTIES OF THE DESIGNS
The experimental plans, except that for 64 varieties, are obtained simply by selecting the desired number of replicates
from the corresponding plan for a balanced lattice square.
Any set Of replicates may be chosen, provided that they are
all different. The method of constructing balanced lattice
squares has been described by Yates (1, 2) and may be followed until the required number of replicates has been obtained. It is hoped to publish a handbook showing the complete field plans for the balanced lattice squares.
If there are 64 varieties, a particular group of replicates
must be taken from the balanced set. The plan for this
design, with four replicates, is given in table 8. If threefold replication is desired, the first three replicates in table
8 may be used. Since the balanced lattice square with 64
varieties requires nine replications, the new designs for 64
varieties are perhaps the most useful of those in table 1b,
as suggested by Yates (1).
The nature of the groupings into rows and columns may be
733
seen by studying table 8. It will be found that each variety
appears in the same row as 28 other varieties and in the
same column as a different set of 28 other varieties. There
remain seven varieties which do not appear either in the
same row or column as the chosen variety. Thus the additional designs lack the complete symmetry of the balanced
designs.
The statistical analysis follows the same procedure as that
for the balanced lattice square with p' varieties in (p
1) /2
replications, except for minor changes which involve no extra
work. A numerical example is given in section 3 below.
The standard error of the difference between two varietal
means varies according to the relation between the varieties
in the experimental plan. Theoretically, three standard errors are required, one for the difference between two varieties which appear in the same row, one for the difference
between two varieties which appear in the same column,
and one for the difference between two varieties which do not
appear together in a row or column. Formulae are given for
the three types of standard error. It is shown in section 4,
however, that the use of the average standard error for all
varietal comparisons is generally of sufficient accuracy for
practical purposes in experiments involving three or more
replicates.
With designs which are only partially balanced, there is a
slight overall loss of efficiency relative to balanced designs.
As Yates' results (1) indicate, this loss is negligible when
at least three replicates are used (see section 5).
Lattice square designs can be constructed for numbers of
replications other than those in tables 1a and lb, e. g., for
49 varieties in five, six or seven replicates. Since the statistical analysis is more complicated, such designs will not
be discussed here.
+
2. FIELD LAYOUT AND RANDOMIZATION
As we have indicated, the plots within each replication are
laid out in square formation, so that differences among rows
and columns represent fertility variations in two directions at
right angles.
For many field crops the plots normally used for experimental purpose are long and narrow. If the narrow sides of
the plots are parallel to the rows of the square, the rows
may be fairly compact in shape, but the columns will be
extremely long and narrow. For instance, in corn experiments with plots 2 hills by 10 hills, the dimensions of each
row in a 9 x 9 lattice square are 18 hills by 10 hills, while
those of each column are 2 hills by 90 hills. It may be
734
doubted in such cases whether the extra control of variation
among columns will materially reduce the experimental error.
Some information on this question was obtained by examining the results of 16 corn experiments with 2 x 10 hill
plots, involving 49, 81 or 121 varieties. These experiments
were carried out at various locations in Iowa during the
1941 season, as part of the corn breeding program of the
Iowa Agricultural Experiment Station, in cooperation with the
Division of Cereal Crops and Diseases, Bureau of Plant Industry, U. S. D. A: In each of these experiments it is possible to estimate what the experimental error variance would
have been if the variation among columns had not been
eliminated, i. e., if lattice designs (3) had been used instead
of lattice squares. The results were expressed in terms of
the relative accuracy of the two designs, this quantity being
the inverse ratio of their experimental error variances.
In 10 of the 16 experiments, no appreciable increase in
precision was obtained from the elimination of column differences, the accuracy of the lattice squares relative to the
lattices varying from 96 percent to 107 percent. The relative accuracies in the remaining six experiments were 120,
131, 132, 137, 193 and 199 percent. In three experiments,
the variation among columns was actually greater than the
variation among rows. The results indicate that although
the extra control may be inoperative in the majority of cases,
there are occasionally substantial fertility variations parallel
to the narrow sides of the plots. For this reason the ability
to adjust for variations in both directions is useful, particularly in experiments on outlying farms where the fertility
contours may not be well known.
It must not be concluded from these comparisons that lattice designs would have been inferior or at best equal to
lattice square designs in the experiments discussed above.
With lattice designs, the best field layout, on the average,
is obtained by making the incomplete blocks compact in
shape, and arranging these blocks so that the replications are
also as compact as is practicable. With 81 varieties, the incomplete block might measure 18 hills by 10 hills, this being
the same as the row of the lattice square, while the replication
might measure 54 hills by 30 hills, as compared with 18 hills
by 90 hills for the lattice square. A reasonably compact replication (32 hills by 40 hills) is also possible with 64 varieties,
though designs with 25, 49 or 121 varieties are less convenient in this respect. A comparison of lattice and lattice square
designs, with each design arranged in the most accurate
3 r am indebted to Dr. G. F. Sprague and :\Yr. L. A. Tatum for permission to use these data, and the data In the numerical example below.
735
layout, could be obtained only from uniformity data. Such
a comparison, however, might not be entirely relevant to
the conditions under which field experiments are usually laid
down. In most experiment stations, the sites available for
experiments in any year are fully occupied, leaving as little
waste space as possible. It is frequently impractical to ar·
range every experiment in its most suitable layout. In these
circumstances it is convenient to be able to use either a lat·
tice or a lattice square, according to which can most appro·
priately be fitted into the available field space.
Some comparisons of lattice and lattice square designs
nave been made by Johnson and Murphy (5) on oats uni·
formity data, with plots 3 feet by 16 feet and 4 feet by 8
feet. When the plots within each replication were arranged
in a square, Johnson and Murphy obtained a substantial re·
duction in the error mean square from the control of varia·
tion among columns, even in cases where a column measured
3 feet by 88 feet. Further, lattice square designs proved on
the average superior in accuracy to lattice designs, although
the incomplete blocks in the latter were arranged into repli·
cations as compact as possible.
While lattice squares are expected to give best results
when the plots within each replicate are laid out in a square,
they may occasionally be worthwhile when practical consider.
ations necessitate a different field arrangement. For in·
stance, varietal trials are frequently planted with the plots
in one continuous line. Where there is a fertility gradient
along this line, part of the resulting error will be eliminated
by the replications, and a further part by grouping the plots
within each replicate into incomplete blocks. If a lattice
square is used with this layout, these incomplete blocks might
constitute the rows of the square. The first column of the
lattice square would then contain the first plot in each block,
and so on. Thus variation among columns corresponds to
the fertility gradient within the blocks, though only insofar
as that gradient is in the same direction within all blocks
of a replicate. Thus when the plots are to be laid out in a
straight line, a lattice square may be advisable if there is
reason to suspect a consistent fertility gradient along the
line. Even if the columns prove ineffective, the loss of ac·
curacy a& compared with a lattice design is small. These
remarks apply also to latin squares which have sometimes
proved successful under similar conditions, for example where
the plots lay side by side down a sloping field.
The designs described here are randomized in the same
way as balanced lattice squares. A separate random rear·
rangement should be made of the rows and columns of each
replication.
736
For reasons which will now be described, it is also advisable to assign the varieties to the variety numbers at random. The groups of variety numbers which form the rows
and columns of any replication are decided by the experimental plan, and these groups are not changed by randomizing the rows and columns of each square. .Further, all
varieties which fall in the same row or column of a replication receive the same adjustment for that row or column,
and if an adjustment is over-estimated, all varieties in the
row or column are favored. Consequently, if the differences
among rows are large, the variance per plot of the difference between two groups of varieties which lie almost entirely in different rows of one replicate may be considerably
larger than the average error variance. In the numerical
example below, where the mean square for columns happens
to be relatively though not exceptionally large, the error
variance of a column mean is 7.19 per plot, as against an
average variance of ·6.18 per plot. The additional randomization suggested above gives every group of p varieties an
equal chance of forming a row or column in the field layout
and thus helps to ensure that the average error variance
may be used for comparisons among groups of varieties, as
well as between individual pairs of varieties. These considerations apply to all lattice and lattice square designs, and
not merely to the designs discussed in this bulletin.
TABLE 2.
PLOT YIELDS IN POUNDS.
(Variety numbers are shown in boldface type)
26.5
24
34.9
10
38.7
19
33.7
33
35.3
47
27.6
40
33.5
5
34.9
26
35.6
12
33.5
229.0
-64.1
- 3.8
234.1
-51.5
- 3.0
17
38.2
31
34.4
45
24.9
38
31.4
3
Total
1\1
E
Replication I
21
18
15
30.7
33.3
32.9
32
35
29
33.2
30.5
31.7
46
49
43
29.4
32.6
32.S
42
39
36
30.4
30.1
24.9
4
7
1
36.0
37.6
34.1
28
25
22
'32.4
34.6
30.5
14
8
11
36.S
34.5
36.2
231.1
-51.4
- 3.0
229.1
-74.8
- 4.4
225.0
-38.0
- 2.2
20
33.6
34
33.6
48
29.3
41
30.6
6
31.5
27
37.7
13
31.6
227.9
-71.5
- 4.2
16
33.4
30
35.2
44
30.6
37
29.4
2
35.7
23
26.7
Total
/)
235.8
-44.4
-1.0
233.9
-67.3
-1.1>
207.2
-58.6
-1.3
210.3
-49.2
-1.1
236.3
-86.8
-2.~
232.4
-40.0
-0.9
246.7
-59.3
-1.3
9
35.4
L
226.4 1602.6
-405.6
-54.3
- 3.2
737
TABLE 2 (continued)
Total
M
e
Total
1\1
e
29
31.4
39
33.1
16
30.8
49
35.1
3
28.3
26
36.1
13
32.3
48
30.0
2
36.0
35
31.0
12
32.2
15
34.3
38
27.2
25
35.3
227.1
-57.3
- 3.4
226.0
-46.0
- 2.7
1
29.6
23
26.6
40
29.5
35
28.3
18
30.9
45
23.8
13
31.8
48
25.4
21
30.1
31
32.2
26
32.0
200.5
+27.S
+ 1.6
210.4
+31.8
+ 1.9
Replication II
23
42
5
33.5
27.2
31.2
45
8
33
28.6
39.2
34.9
22
41
10
35.2, 30.5
31.2
6
18
36
31.1
33.9
26.9
9
28
46
34.6
28.1
26.0
32
44
20
26.5
28.4
25.5
19
31
7
32.5
30.7
31.1
221.6
-39.1
- 2.3
9
33.8
36
26.1
4
30.8
36.8
21
34.8
47
29.0
24
32.3
34
31.4
1
30.0
37
29.3
223.6
-43.3
- 2.5
19
28.7
41
15.8
2
24.6
46
17.8
29
18.2
14
19.8
24
26.5
17
34.5
27
36.9
4
30.6
30
34.5
40
33.8
.14
33.2
43
29.7
Total
L
-43.9
-1.0
243.5
-67.3
-1.5
218.3
-34.5
-0.8
226.0
-46.2
-1.0
216.5
-40.6
-0.9
206.9
- 9.6
-0.2
220.9
-25.8
-0.6
233.2 1556.7
-267.9
-24.3
- 1.4
Total
L
39
21.6 187.9 +75.5
12
21.5 167.9 +91.9
22
25.4 190.8 +91.5
17
30.3 188.0 +82.0
7
9.3 160.6 +135.0
34
16.1 . 169.9 +100.7
44
14.8 177.8 +96.9
VARIETY TOTALS (UNADJUSTED>.
624.9
3
4
74.2
97.4
10
11
95.6 103.S
18
17
103.0
95.5
24
25
93.1
98.7
31
32
97.3
88.2
39
38
81.8
85.1
45
46
77.3. 73.2
5
95.3
12
87.2
19
94.9
26
103.7
33
97.8
40
96.8
47
75.1
87.2
13
95.7
20
85.3
27
101.3
34
81.1
41
76.9
48
84.7
7
78.0
14
87.5
21
98.2
118
86.3
35
89.8
42
81.3
49
91.4
641.9
650.8
6i2.2
612.5
622.9
/j
224.6
189.6 179.1 172.9 151.4 139.0 1242.9
+673.5
+71.1 +109.9 +102.0 +149.6 +181.3
+ 4.2 + 6.5 + 6.0 + 8.8 +10.6
Grand total
4402.2
2
1
93.-7
96.3
8
9
107.2 103.8
16
15
93.5
92.6
22
23
93.0
SO.5
29
30
81.3
98.9
36
37
77.9
81.9
44
43
90.4 - 70.9
637.0
203.8
-23.1
- 1.4
Replication III
28
10
30
27.7
25.7
29.2
43
32
3
27.9
26.6
19.4
49
20
11
30.2
23.7
25.2
6
8
37
24.6
23.2
31.8
38
27
47
26.7
18.5
23.2
16
5
25
28.4
26.9
28.8
15
33
42
27.6
26.3
20.0
TABLE 3.
Total
221.4
-34.8
- 2.0
11
6
Total
622.1
680.8
663.0
657.2
634.4
581.1
563.0
4402.2
II
+1.7
+2.1
+2.1
+1.9
+3.1
+2.3
+2.2
738
TABLE 4.
1
90.1
8
108.1
15
95.4
22
99.5
29
85.1
36
76.4
43
91.2
a
97.8
9
101.1
16
90.7
23
77.7
30
99.5
37
82.9
44
77.0
VARIETY TOTALS (ADJUSTED).
3
72.2
10
96.5
17
108.3
24
96.5
31
93.4
38
81.3
45
72.3
4
94.3
11
102.3
18
93.2
25
99.8
32
89.8
39
88.4
46
77.3
Grand total
6
83.8
13
90.0
20
86.6
27
102.9
34
84.9
41
79.7
48
79.1
5
96.1
113
91.9
19
98.5
26
100.0
33
96.8
40
94.1
47
76.6
7
83.3
14
91.3
131
92.8
28
84.0
35
83.9
42
80.7
49
89.9
4405.0
3. STATISTICAL ANALYSIS
The statistical analysis follows the same general· pro1) /2 replicedure as for a balanced lattice square with (p
cations (2). In order to present a numerical example, the
fourth replication was omitted from a lattice square experiment on corn, with 49 varieties, carried out at Cresco, Iowa,
in 1940. Each plot measured 4 hills by 5 hills, the hills being spaced 3 feet 6 inches apart in each direction. Individual
plot yields, in pounds, are given in table 2.
In the algebraic formulae p' is the number of varieties and r
the number of replicates. The computing instructions are
as follows:
+
1. Calculate the row and column totals for each replication, the replication totals and the grand total, inserting
these in table 2 as shown. The variety totals are then obtained and inserted, in the form of a square, in table 3. It is
also worthwhile to insert the row and column totals of this
table as shown.
2. Calculate and insert the quantities L for each row of
the experiment.
L =
(Total yield of all varieties appearing in the row) r (Row total).
The sum inside the first bracket is obtained by addition from
table 3. For rows of the first replication, this quantity is
simply one of the row totals of table 3 and has been calculated and written down in step 1. Thus for the first row
of replication I,
.
L
=
663.0 -
3 (235.8)
=
-44.4.
739
In the other .replications, time is probably saved by mentally arranging the variety symbols in each row in increasing
order before summing their yields. If this is done, the first
variety will be found to lie in the first row of table 3, the
second variety in the second row, and so on. For the first
row of replication II, for instance, the variety totals are
summed in the order 5, 11, 17, 23, 29, 42, 48. This procedure
facilitates the location of varieties in table 3 and reduces the
likelihood of errors.
Similarly the quantities M are calculated and. entered at
the foot of each column, where
M = (Total yield of all varieties appearing in the column) r (column total).
The replication totals (-405.6, -267.9, +673.5) of Land
M are obtained and inserted in table 2. As a check, these
numbers should be equal to
(Grand total of individual yields) - r (Replication total of
individual yields)
while the total of the r numbers should be exactly zero.
3. The analysis of variance may now be calculated. The
total sum of squares and the sums of squares for replications
and varieties are obtained just as in an ordinary randomized
blocks design. The sum of squares for rows, after eliminating varietal effects, is the sum of squares of deviations of
the quantities L from their replication means, divided by
pr(r - 1). In the example, p = 7, r = 3, this gives
12 [(44.4)2 + (67.3)' + ... + (96.9)"{(405.6)2 + (267.9)' + (673.5)2} /7] = 134.19.
The sum of squares for columns is calculated from the quantities M by a similar rule. The error sum of squares is found
by subtraction. All results should be checked by re-calculation.
TABLE 5.
ANALYSIS OF YARIANCE: POUNDS PER PLOT.
Sums of squares
lIIean square
2
48
18
18
60
1,564.37
1,383.11
134.19
515.65
305.63
28.815
7.455
28.647
5.094
146
3,902.95
d. f.
Replications
Varieties
Rows
Columns
Error
Total
740
4. If R, C and E are the mean squares for ~ows, columns
and error, respectively, we now calculate the factors,
)..' =
(R - E) _
p(r-1)R -
(7.455 - 5.094) _ 02262
14 X 7.455
-.
' _ (C - E) _ (28.647 - p.094) _ 05873
p(r-1)C 14 X 28.647
-.
p. -
(These· quantities correspond to Yates' AlP and p.lp for the
balanced lattice square.) They provide the multipliers necessary to convert Land M into adjustments for row and column effects, applicable to the varietal totals in table 3. If
either R or C is less than E, the corresponding factor is put
equal to zero, no adjustment being made for the corresponding row or column effects. Similarly, should both Rand C
be less than E, the' unadjusted varietal totals are used.
Each quantity L is multiplied by)..', giving the quantities
8 (table 2), and each quantity M by p.', giving the quantities
( (table 2). It is usually sufficient to carry the same number
of decimal places in 8 and ( as in the varietal totals.
The totals of all the 8's and of all the t:'s should be_ zero,
apart from rounding-off errors. If one or both of these totals
appear suspicious, a rough check may be obtained from
th~ result that the absolute magnitude of the rounding-off
error should be less than h Ph + ~ P: ), except abou! once
in 20 times, where II is the rounding-off interval. In the
present. example, the 8's total
0.3 and the ('s total
0.1.
(It is worth recording these totals, as they provide a check
in step 5 below.) The rounding interval is 0.1, and the
check formula above gives (0.1) (0.5 + y7) = 0.31, slightly
above the higher discrepancy.
5. To adjust the varietal totals in table 3, the quantities
8 and t: are added for each row and column in which the variety appears. Thus the adjusted total yield for variety 1 is
+
93.7 -
2.0 -,2.2 -
,
0.2 - 2.5
+
+ 1.7 + 1.6 =
,
90.1
These values are shown in table 4 above. For a check by
summation, it may be noted that the total of the adjusted
yields, minus the total of the unadjusted yields, should exactly equal p times the sum of the 8's and £'s. Thus,
4405.0 - 4402.2 = 2.8 = 7 (+ 0.3 + 0.1)
For tests of significance, the error mean square in the
analysis of variance must be increased in order to take into
741
account the sampling error of the row and column adjustments. The average experimental error variance per plot
is given by
E { 1
=
(5.094)
+ p ~~\
0:
+ p.') }
{1 + (2.625) (.0~262 + .05873) } = 6.18
This quantity is multiplied, as usual, by 2r to give the average
variance of the difference between two varietal totals, and
by 2/r to give the average variance of the difference between
two varietal means.
As indicated previously, the average error will usually be
sufficiently accurate for tests of significance of the difference
between any pair of varieties. Individual formulae for the
three types of comparisons are as follows:
Two varieties in the same row:
(r - 1) ):
rp.'} = 6.22
E {1
Two varieties in the same column:
E {1
r): + (r -1) p.'} = 6.04
Two varieties not appearing together:
E {1 + rA'
rp.'} = 6.34
+
+
+
+
Like the lattices and balanced lattice squares, these designs
can be analyzed alternatively as if they were in randomized
blocks, the rows and columns being ignored. The error sum
of squares in this case is equal to the total sum of squares
for rows, columns and error in table 5, and gives a mean
square of 9.953. A z- or F-test of the unadjusted varietal
totals can thus readily be obtained from table 5, by forming
the ratio of the varieties mean square to the mean square
9.953 above. Varietal differences are highly significant. The
varieties mean square must not be compared directly with
the error mean square in table 5, since the former has not
been adjusted for row and column effects.
A comparison of the experimental errors 9.95 and 6.18
indicates that the adjustments for row and column differences
produced a marked increase in accuracy.
The randomized blocks analysis is useful where preliminary
results must be summarized under pressure of time, where
subsidiary measurements on the plots are little affected by
local soil variations or are not required with high accuracy,
or where it is necessary to omit the yields of some varieties
from the results.
742
It may be worth noting that the analysis described above
is also valid for a balanced lattice square in which one or more
replications must be omitted through field damage or for
'Other reasons.
4. ESTIMATION OF THE STANDARD ERRORS
The theory of the statistical analysis follows very closely
that given by Yates (2) for the balanced lattice square with
(p + 1) /2 replications. In particular, the structure of the
:analysis of variance and the method of calculating the row
.and column adjustments are exactly similar, except for variations introduced by the change from (p + 1) /2 to r repli-cations, and need not be reproduced here. The derivation
'Of the varietal standard errors will, however, be indicated,
-since these are required in further discussion and since the
balanced design contains no case in which two varieties do
not appear within the same row or column.
As Yates (2) has pointed out, it is convenient, in a discus-sion of the theory, to estimate first the total yields of the
groups of p varieties which constitute the rows and columns.
The relation between these yields and the yields of the individual varieties may be illustrated by considering the case
'Of nine varieties (p = 3).
'TABLE 6.
ORTHOGONAL SUBDIVISION OF THE EIGHT DEGREES OF
FREEDOll AllONG NINE VARIETIES.
Set II
Set I
12
3
1417
5
8
6
9
1213
14 5 6
7
8 9
Set III
15 I 62 1 34
9
7
8
Set IV
1
6
1213
4
5
897
The table shows 12 groups of three varieties each. Every
variety belongs to four of the groups. Further, in the remaining members of these four groups, every other variety appears
once. For example, in the four groups containing variety 1,
its companions are 2, 3; 4, 7; 5, 9; and 6, 8. It follows that the
total yield of all the groups to which a variety belongs is equal
to three times the yield of the variety plus the total yield of
all varieties.
Groups possessing these properties can be constructed for
any value of p for which a completely orthogonalized latin
- 'Square (4) exists; these values include p = 3, 4, 5, 7, 8, 9, 11,
13. In the general case, a variety belongs to (p + 1) groups,
which together contain all the other varieties once. Thus the
total yield of the groups is equal to 11 times the yield of the
variety plus the grand total. These relations enable the individual varietal yields to be estimated if the yields of each
group total are known.
743
In these designs, r of the groups to which a variety belongs
form rows of the experiment, a further r groups form columns, while the remaining (p + 1 - 2r) groups are unconfounded with rows or with columns. In the case of a group
which forms a row in one of the replications, two estimates
of the group total are available, one from the replication in
question, and one from the remaining (r - 1) replications, in
which the group is unconfounded with rows or columns. If
the error and inter-row variances per plot are l/wj and l/wn
respectively, so that Wi and Wr represent the error and interrow weights, the variances of these two estimates are respectively p/wr and p/ (r -l)wj. The factor p is included
because each group contains p varieties. If the two estimates.
are combined by weighting them inversely as their variances,
the variance of the weighted mean, by a well-known statistical
theorem, is
p
wr
+ (r -1)
(1)
Wi
For a group which constitutes a column of one of the replicates, we need only replace Wr by We in the above expression,
where We is the reciprocal of the inter-column variance per
plot. The remaining group totals, which are unconfounded in
all replications, are estimated each with variance p/rwj.
From the discussion above of the relation between the
groups and the individual varieties, the difference between
the means of any two varieties is equal to the difference
between the totals of their corresponding groups, divided by p.
Every pair of varieties has, however, one group in common.
Consequently, if two varieties occur together in a row, the
variance of the difference between their means is
(r - 1)
+
2p [
p' Wr
(r - l)wj We
+
r
+ (r -l)Wi
+ (p + 1 -
2r) ]
rWj
(2}
After some algebraic manipulation, this becomes
2
[
1
+
(r -
1»),'
(WI wr )
) ,1
h
were
. =_.
p (r - 1) WI + Wr '
,
p.
+ rIL'
]
1
= _.
P
(r-1)wi
(3}
+ We'
The corresponding result for a pair of varieties which have a
744
column in common is found, by interchanging Wr and We in
the above result, to be
r!1 [
1
~ rA' .+ (r -1)fL']
(4)
For a pair of varieties having no row or column in common,
the common group must be one of the (p + 1 - 2r) groups
which are unconfounded. The variance of the difference between the varietal means is therefore
2P [
if
Wr
+
r
(r -
+
1) WI
which reduces to
[
r
+ (p-2r) ]
We + (r - 1)wl
rWi
•
(1
+ rA' + rfL')
]
(5)
(6)
To find the average variance of the difference between two
varietal means, we may note that any variety appears in the
same row as r(p -1) other varieties and in the same column
as an equal number of other varieties. Hence, there remain
(p - 1) X (p + 1 - 2r) other varieties which do not
occur in the same row or column anywhere in the design.
The three types of variance must therefore be weighted in
the ratios, r, r, (p + 1 - 2r) respectively. This gives
"'!1
[1+ P~1 O:+fL')]
(7)
In practice, the values of WI> Wr and We must be estimated
from the analysis of variance. The values of A' and fL' given
in section 3 are the maximum likelihood estimates of the
theoretical values in the formulae above. In general, the use
of the same symbol for the theoretical value and its estimate is not advisable, but should cause no confusion here. The
substitution of estimated weights results in a slight underestimation of the standard errors. This bias is negligible in
the larger designs where considerable numbers of degrees of
freedom are available for estimating the weights. Even with
the 7x7 design in three replicates, the underestimation of
the average standard error does not appear to ex;ceed 2 percent for any values of the true weights. With a 2 percent
underestimation of the standard error, the apparent 5 percent
point has a true probability of about 4.6 percent.
We may now consider the justification for the use of the
745
average error variance for all types of comparison between
pairs of varieties. Upon investigation of the formulae given
above for the three error variances, it will be found that the
greatest difference between the average variance and anyone
of the three variances, when expressed as a fraction of the
average variance, cannot exceed the larger of
.
2r
p {r (p + 3) -
(p + 1)}
(p
and
p {r (p
+ 1-r)
+ 2) -
(p + 1)}
For the relevant values of p and r, the greater of these two
quantities is shown below.
TABLE 7.
~fAxnIU:\1
PERCENT ERROR RESULTING FROM THE
USE OF THE AVERAGE VARIANCE.
Number of varieties
169
121
81
64
49
3
U~
2.56
3.13
3.90
I
Number of repllcations
4
I
5
I
U~
U~
2.34' I
2.86
6
1.13
The corresponding percent errors in the standard deviations are about half the values above. Thus for the 7 x 7
design, an apparent 5 percent probability obtained by using
the average error would represent a true probability varying,
at the most, between 4.6 and 5.4 percent. For the other designs this variation is still smaller. These maximum errors
occur either in the case where both the inter-row and intercolumn variations are large relative to the error variation,
or in the case when one is large while the other is no larger
than the error variation.
5.
The
justed
factor
comes
LOSS OF EFFICIENCY DUE TO ASYMMETRY
average error variance per plo,f applicable to the advarietal yields may be obtained .by deleting the
~ in formula (7) above. Written in full, this ber
~[1
+ _ l ' _ { (WI - wr ) + (WI-We) }]
P + 1 (r - 1)wl + Wr (r - 1)wl + We
(8)
The corresponding value for a balanced design, in (p + 1) /2
Wi
replicates, is obtained by putting r = (p + 1) /2 in (8).
For given values of Wi> Wr and We, 1. e., for given field condi.
tions, the variance is always slightly greater for the partially
balanced design than for the balanced design. This increase
in variance represents a loss of efficiency due to asymmetry
of the design.
746
The loss of efficiency is relatively greatest when Wr = We
= 0, that is, when the design is highly effective in removing
variation among rows and columns. In this case, the factor
inside the square brackets reduces to
rep
+ 3) - (p + 1)
+ 1) (r -1)
(9)
(p
+
and to (p
l)/{p - 1) for the balanced design. Under the
same field conditions, the accuracy per replication of the
partially balanced design relative to the balanced design cannot therefore be less than
.
+
(p
(p -1) {rep
l)"(r - 1)
3) - (p
+
+ 1)}
(10)
These ratios are shown below
Number of replicates
Number of varieties
169
121
81
64
49
3
4
.960
.960
.961
.964
.969
.980
.982
.986
.992
I
5
6
.990
.994
.996
The loss of accuracy does not amount to more than 4 percent in any of the above designs, and of course approaches
zero as the designs become more symmetrical with increasing replication.
It may be noted that the formula (9) gives the reciprocal
of the efficiency factor of these designs, as defined by
Yates (1).
TABLE 8.
LATTICE SQUARE DESIGN FOR 64 VARIETIES IN FOUR
REPLICATES
Replication 1
Replication II
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
64
23
3
25
26
27
28
29
30
31
32
55
40
33
34
35
36
37
38
39
40
28
41
42
43
44
45
46
47
48
37
49
50
51
52
53
54
55
56
19
57
58
59
60
61
62
63
64
10
1
46
27
39
18
13
16
53
60
31
25
54
12
45
34
29
4
58
41
11
22
9
50
63
5
24
38
43
51
15
42
20
6
32
57
61
48
14
33
26
7
52
30
36
21
47
59
49
8
44
2
62
17
56
35
747
Replication III
1
11
20
Replication IV
30
34
48
53
63
- 1 32 47 61 22 50 12 35
15
2
56
45
59
28
22
33
29
2
14
49
39
64
43
20
21
52
3
39
32
58
9
46
42
13
3
24
60
33
30
55
26
47
38
4
17
13
64
51
59
54
18
4
48
31
87
9
40
62
31
19
5
49
42
12
. 23
36
57
46
5
11
56
26
43
25
61
16
55
6
36
18
52
63
40
27
10
6
17
45
54
24
10
57
44
35
7
29
16
41
28
34
51
21
7
62
60
37
41
50
14
23
27
8
38
19
53
15
25
44
58
8
SUMMARY
Some additional experimental designs are described for
varietal trials in which a large number of varieties are compared. The plots in each replication are usually arranged in
square formation, the arrangement allowing the varietal
yields to be adjusted for fertility or other differences among
the rows and columns of each replicate. The designs are
similar to the balanced lattice squares developed by Yates,
except that a smaller number of replications are used.
The statistical analysis, which differs only in minor details from that of a balanced lattice square, is illustrated by
means of a numerical example. Formulae for the standard
errors of the adjusted varietal yields are derived and discussed.
Some discussion is given of the field layout and of the
relative accuracy of lattice and lattice square designs.
748
REFERENCES CITED
(1) Yates, F. A further note on the arrangement of variety
trials: quasi-latin squares. Ann. Eugenics, 7 : 319·331. 1937.
(2)
Yates, F.
Lattice squares.
Jour. Agr. Sci., 30 : 672·687.
1940.
(3) Cox, G. M., Eckhardt, R. C., and Cochran, W. G. The analysis
of lattice and triple lattice experiments in corn varietal tests. Iowa
Agr. Exp. Sta., Res. Bu!. 281 : 1·66. 1940.
(4) Fisher, R. A. and Yates, F. Statistical tables for biological,
agricultural and medical research. Oliver and Boyd, Edinburgh. 1938.
(5) Johnson, I. J. and Murphy, H. C. Lattice and lattice square
designs with oat uniformity data and in variety trials. Jour. Amer.
Soc. Agron., 35 : 291·305. 1943.