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/DataYear\Quad SW SE NE NW

1990 A A A A1991 A D C B1992 B A D C1993 D C B A1994 C B A D1995 C B A 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 quad sched water corn1 1 1 193 93.52 1 1 199 90.43 1 2 138 58.44 1 4 91 50.85 1 3 108 115.06 1 3 155 95.41 2 1 192 92.92 2 4 134 90.43 2 1 80 57.84 2 3 52 42.15 2 2 209 119.06 2 2 132 99.21 3 1 197 83.52 3 3 117 83.53 3 4 51 50.24 3 2 40 36.45 3 1 152 96.06 3 1 207 75.31 4 1 179 81.62 4 2 104 79.73 4 3 98 45.84 4 1 43 34.55 4 4 131 96.06 4 4 205 81.0

Statistical Model/Formulation

6

1

4

1

4

1

1,...,6 1,..., 4

where:

Overall Mean

Effect of Year 0

Effect of Quad 0

Direct Effect of Schedule 0

Carryover Effect of Schedule

ij i j k l ij

i ii

j jj

k kk

l

y i j

i

j

k

l

4

1

2

0

Random error term ~ 0,

ll

ij ij NID

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

X0 X1 X2 X3 X4 1 2 3 4 5 1 2 3 1 2 3 1 2 31 1 0 0 0 0 1 0 0 1 0 0 0 0 01 0 1 0 0 0 1 0 0 1 0 0 1 0 01 0 0 1 0 0 1 0 0 0 1 0 1 0 01 0 0 0 1 0 1 0 0 -1 -1 -1 0 1 01 0 0 0 0 1 1 0 0 0 0 1 -1 -1 -11 -1 -1 -1 -1 -1 1 0 0 0 0 1 0 0 11 1 0 0 0 0 0 1 0 1 0 0 0 0 01 0 1 0 0 0 0 1 0 -1 -1 -1 1 0 01 0 0 1 0 0 0 1 0 1 0 0 -1 -1 -11 0 0 0 1 0 0 1 0 0 0 1 1 0 01 0 0 0 0 1 0 1 0 0 1 0 0 0 11 -1 -1 -1 -1 -1 0 1 0 0 1 0 0 1 01 1 0 0 0 0 0 0 1 1 0 0 0 0 01 0 1 0 0 0 0 0 1 0 0 1 1 0 01 0 0 1 0 0 0 0 1 -1 -1 -1 0 0 11 0 0 0 1 0 0 0 1 0 1 0 -1 -1 -11 0 0 0 0 1 0 0 1 1 0 0 0 1 01 -1 -1 -1 -1 -1 0 0 1 1 0 0 1 0 01 1 0 0 0 0 -1 -1 -1 1 0 0 0 0 01 0 1 0 0 0 -1 -1 -1 0 1 0 1 0 01 0 0 1 0 0 -1 -1 -1 0 0 1 0 1 01 0 0 0 1 0 -1 -1 -1 1 0 0 0 0 11 0 0 0 0 1 -1 -1 -1 -1 -1 -1 1 0 01 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

12345123123123

6 1 2 3 4 5 4 1 2 3 4 1 2 3 4 1 2 3

Note by the formulation:

Water Corn193 93.5199 90.4138 58.491 50.8

108 115155 95.4192 92.9134 90.480 57.852 42.1

209 119132 99.2197 83.5117 83.551 50.240 36.4

152 96207 75.3179 81.6104 79.798 45.843 34.5

131 96205 81

X

Y

Parameter Estimates 1

' 'X X X Y Parm Water Corn 128.7 78.21 45.8 12.62 -4.0 11.93 -37.0 -25.14 -72.2 -37.25 21.3 28.31 17.0 6.92 -0.9 6.13 -8.9 -5.71 15.7 -2.92 2.5 2.53 -19.7 0.11 13.8 -4.12 0.0 0.93 -3.7 2.76 46.0 9.54 -7.2 -7.34 1.5 0.34 -10.0 0.5

6 1 2 3 4 5 4 1 2 3 4 1 2 3 4 1 2 3

Note by the formulation:

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

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

1

0 0 0 0 0 0

1

01 01 01 01 01 01 01 0 1

1

012 012 012 012 012 012 012 0 1 2

1

0123 0123 0123 0123 0123 0123 0123 0 1 2

Total Uncorrected '

' ' '

, ' ' ' |

, , ' ' ' | |

, , , ' ' ' | | |

SS Y Y

R Y PY P X X X X

R Y P Y P X X X X X X X

R Y P Y P X X X X X X X X

R Y P Y P X X X X X X X X

3

1

0124 0124 0124 0124 0124 0124 0124 0 1 2 4

1

01234 01234 01234 01234 01234 01234 01234 0 1 2 3 4

, , , ' ' ' | | |

, , , , ' ' ' | | | |

Type I SS:

(Year) ,

Quad , , ,

Irr Direct,U

X

R Y P Y P X X X X X X X X X

R Y P Y P X X X X X X X X X X

SS R R

SS R R

SS

nadj , , , , ,

Irr Carryover, Adj , , , , , , ,

Irr Carryover, Unadj , , , , ,

Irr Direct, Adj , , , , , , ,

Error Total Uncorrected , , , ,

Total Corrected Total U

R R

SS R R

SS R R

SS R R

SS SS R

SS SS

ncorrected R

Results for Irrigation DataSum of Squares Water Corn

Total Uncorrected 501041 155900R() 428535 142358R(, ) 480101 154589R(,,) 481758 155678R(,,,) 484863 155748R(,,, ) 482970 155779R(,,,, ) 486075 155849Type I SSYear 51565 12231Quad 1657 1089Irr Direct, Unadj 3105 71Irr Carr, Adj 1213 101Irr Carryover, Unadj 1213 101Irr Direct, Adj 3105 71Error 14966 51Total Corrected 72506 13543

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

ANOVA-Water Source df SS MS F P-valueYear 5 51565 10313.08 6.20 0.0093Quad 3 1657 552.26 0.33 0.8025Irr Direct 3 3105 1035.01 0.62 0.6181Irr Carryover 3 1213 404.17 0.24 0.8642Error 9 14966 1662.88Total (Corr) 23 72506

ANOVA-Corn df SS MS F P-valueYear 5 12231 2446.20 432.78 0.0000Quad 3 1089 363.03 64.23 0.0000Irr Direct 3 71 23.54 4.17 0.0417Irr Carr 3 101 33.73 5.97 0.0160Error 9 51 5.65Total (Corr) 23 13543

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