Crossover Design in a Modified Latin
Square Design
Irrigation Water Usage and Corn Growth
over 6 Seasons in 4 Quadrants for 4
Irrigation Schedules
D.D. Steele, E.C. Stegman, and R.E. Knighton (2000). "Irrigation
Management for Corn in the Northern Great Plains, USA," Irrigation
Science, Vol19, pp.107-114.
Experimental Summary
• Goal: Compare the effects of 4 Irrigation Schedules in
terms of Water Usage and Corn Growth over 6
Seasons on 4 quadrants.
• Irrigation Schedules/Methods:
 A=Tensiometer & Infra-red, B&C=H2O Balance, D=CERES
• Seasons: Years 1=1900 to 6=1995
• Quadrants: 1=SW, 2=SE, 3=NE, 4=NW
• Modified Latin Square (Rows=Years,Cols=Quads):
 Year 1: All Quadrants receive schedule A
 Years 2-5: Traditional Latin Square
 Year 6: Repeat of Year 5
Design Summary/Data
Year\Quad
1990
1991
1992
1993
1994
1995
SW
A
A
B
D
C
C
SE
A
D
A
C
B
B
NE
A
C
D
B
A
A
NW
A
B
C
A
D
D
• Modifications allow for each treatment to follow
each treatment (including itself) at least once, and
for independent estimates of direct and carryover
effects of approximately equal precision.
• Effects to be estimated/tested:
• Year (6 levels, 5 degrees of freedom)
• Quad (4 levels, 3 degrees of freedom)
• Direct Scheduling Effect (4 levels, 3 df)
• Carryover Scheduling Effect (4 levels, 3 )df
year
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
quad
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
sched
1
1
2
4
3
3
1
4
1
3
2
2
1
3
4
2
1
1
1
2
3
1
4
4
water
193
199
138
91
108
155
192
134
80
52
209
132
197
117
51
40
152
207
179
104
98
43
131
205
corn
93.5
90.4
58.4
50.8
115.0
95.4
92.9
90.4
57.8
42.1
119.0
99.2
83.5
83.5
50.2
36.4
96.0
75.3
81.6
79.7
45.8
34.5
96.0
81.0
Statistical Model/Formulation
yij     i   j   k   l   ij
i  1,..., 6 j  1,..., 4
where:
  Overall Mean
 i  Effect of Year i
 j  Effect of Quad j
6

i 1
i
0
4

j 1
j
0
 k  Direct Effect of Schedule k
4

k 1
 l  Carryover Effect of Schedule l
k
0
4

l 1
l
0
 ij  Random error term  ij ~ NID  0,  2 
Note that the indices (i,j) refer to year and quad. Only one schedule appears in each
year/quad (see previous slide), and only one schedule appears in the previous
year/same quad. There are no carryover effects in year 1.
Matrix Form – Y = X  

X

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
-1
1
0
0
0
0
-1
1
0
0
0
0
-1
1
0
0
0
0
-1
2
0
1
0
0
0
-1
0
1
0
0
0
-1
0
1
0
0
0
-1
0
1
0
0
0
-1
3
0
0
1
0
0
-1
0
0
1
0
0
-1
0
0
1
0
0
-1
0
0
1
0
0
-1
4
0
0
0
1
0
-1
0
0
0
1
0
-1
0
0
0
1
0
-1
0
0
0
1
0
-1
5
0
0
0
0
1
-1
0
0
0
0
1
-1
0
0
0
0
1
-1
0
0
0
0
1
-1
X4
X3
X2
X1
X0
Y
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
2
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
-1
-1
-1
-1
-1
-1
3
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
1
1
1
0
-1
0
0
1
-1
1
0
0
0
1
0
-1
0
1
1
1
0
0
1
-1
-1
2
0
0
1
-1
0
0
0
-1
0
0
1
1
0
0
-1
1
0
0
0
1
0
0
-1
-1
3
0
0
0
-1
1
1
0
-1
0
1
0
0
0
1
-1
0
0
0
0
0
1
0
-1
-1
1
0
1
1
0
-1
0
0
1
-1
1
0
0
0
1
0
-1
0
1
0
1
0
0
1
-1
2
0
0
0
1
-1
0
0
0
-1
0
0
1
0
0
0
-1
1
0
0
0
1
0
0
-1
3
0
0
0
0
-1
1
0
0
-1
0
1
0
0
0
1
-1
0
0
0
0
0
1
0
-1

1
2
3
4
5
1
2
3
1
2
3
1
2
3
Water
193
199
138
91
108
155
192
134
80
52
209
132
197
117
51
40
152
207
179
104
98
43
131
205
Corn
93.5
90.4
58.4
50.8
115
95.4
92.9
90.4
57.8
42.1
119
99.2
83.5
83.5
50.2
36.4
96
75.3
81.6
79.7
45.8
34.5
96
81
Note by the formulation:
6   1   2  3   4  5  4    1  2  3   4    1   2   3   4   1   2   3 
Parameter Estimates
   X ' X  X 'Y
1
Units:
Water: Irrigation Totals (mm)
Corn: Harvest yield (100s kg/hectare)
“Extreme” Effects:
Var
W-Max
W-Min
C-Max
C-Min
Year
6,1
4
5
4
Quad
1
4
1
4
Irr (Dir)
1
3
2
1
Irr (Co)
1
4
3
1
Note by the formulation:
Parm

1
2
3
4
5
1
2
3
1
2
3
1
2
3
6
4
4
4
Water
128.7
45.8
-4.0
-37.0
-72.2
21.3
17.0
-0.9
-8.9
15.7
2.5
-19.7
13.8
0.0
-3.7
46.0
-7.2
1.5
-10.0
Corn
78.2
12.6
11.9
-25.1
-37.2
28.3
6.9
6.1
-5.7
-2.9
2.5
0.1
-4.1
0.9
2.7
9.5
-7.3
0.3
0.5
6   1   2  3   4  5  4    1  2  3   4    1   2   3   4   1   2   3 
Analysis of Variance
• Goal: Test for Direct and Carryover Effects for Irrigation
Methods.
• Problem (often, as opposed to traditional Latin Square):
Treatment Factors are not orthogonal.
• Solution: Use Type I (Sequential) Sums of Squares





SS(Year|)
SS(Quad|Year,)
SS(Trt Direct|Quad,Year,)
SS(Trt Carryover|Trt Direct,Quad,Year,)
SS(Trt Direct|Trt Carryover,Quad,Year,)
• Due to Modified Latin Square, Direct and Carryover:
 SS(Trt Direct|Quad,Year,) =
SS(Trt Direct|Trt Carryover,Quad,Year,)
Computation of Sums of Squares
SS  Total Uncorrected   Y ' Y
R     Y ' P0Y
P0  X 0  X 0 ' X 0  X 0 '
R   ,    Y ' P01Y
1
P01  X 01  X 01 ' X 01  X 01 ' X 01   X 0 | X 1 
R   ,  ,    Y ' P012Y
1
P012  X 012  X 012 ' X 012  X 012 ' X 012   X 0 | X 1 | X 2 
1
R   ,  ,  ,    Y ' P0123Y
P0123  X 0123  X 0123 ' X 0123  X 0123 ' X 0123   X 0 | X 1 | X 2 | X 3 
R   ,  ,  ,    Y ' P0124Y
P0124  X 0124  X 0124 ' X 0124  X 0124 ' X 0124   X 0 | X 1 | X 2 | X 4 
1
R   ,  ,  ,  ,    Y ' P01234Y
1
P01234  X 01234  X 01234 ' X 01234  X 01234 ' X 01234   X 0 | X 1 | X 2 | X 3 | X 4 
Type I SS:
SS (Year)  R   ,    R   
SS  Quad   R   ,  ,    R   ,  
SS  Irr Direct,Unadj  R   ,  ,  ,    R   ,  ,  
SS  Irr Carryover, Adj  R   ,  ,  ,  ,    R   ,  ,  ,  
SS  Irr Carryover, Unadj  R   ,  ,  ,    R   ,  ,  
SS  Irr Direct, Adj  R   ,  ,  ,  ,    R   ,  ,  ,  
SS  Error   SS  Total Uncorrected   R   ,  ,  ,  ,  
SS  Total Corrected   SS  Total Uncorrected   R   
1
Results for Irrigation Data
Sum of Squares
Total Uncorrected
R()
R(, )
R(, ,)
R(, ,,)
R(, ,, )
R(, ,,, )
Type I SS
Year
Quad
Irr Direct, Unadj
Irr Carr, Adj
Irr Carryover, Unadj
Irr Direct, Adj
Error
Total Corrected
Water
501041
428535
480101
481758
484863
482970
486075
Corn
155900
142358
154589
155678
155748
155779
155849
51565
1657
3105
1213
1213
3105
14966
72506
12231
1089
71
101
101
71
51
13543
ANOVA-Water
Source
Year
Quad
Irr Direct
Irr Carryover
Error
Total (Corr)
df
5
3
3
3
9
23
SS
MS
F
51565 10313.08
1657
552.26
3105 1035.01
1213
404.17
14966 1662.88
72506
ANOVA-Corn
Year
Quad
Irr Direct
Irr Carr
Error
Total (Corr)
df
5
3
3
3
9
23
SS
MS
F
P-value
12231 2446.20
432.78
0.0000
1089
363.03
64.23
0.0000
71
23.54
4.17
0.0417
101
33.73
5.97
0.0160
51
5.65
13543
P-value
6.20
0.0093
0.33
0.8025
0.62
0.6181
0.24
0.8642
There is no evidence of direct or carryover
effects with respect to water usage. Both type of
effects are significant with respect to corn yield
Note that SS(Irr Direct) and SS(Irr Carryover) are the same whether or they have been
adjusted for the other, due to the modified design. In a traditional latin square, they
would not have been