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

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 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

Statistical Model/Formulation 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 Y

Parameter Estimates Units: Water: Irrigation Totals (mm) Corn: Harvest yield (100s kg/hectare) “Extreme” Effects: Var W-MaxW-MinC-MaxC-Min Year6,1454 Quad1414 Irr (Dir)1321 Irr (Co)1431

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

Results for Irrigation Data 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 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