Experimental Design. So Far In Experimental Design  Single replicate Designs.  Completely Randomized Blocks.  Randomized Complete Blocks.  Latin Square.

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Presentation transcript:

Experimental Design

So Far In Experimental Design  Single replicate Designs.  Completely Randomized Blocks.  Randomized Complete Blocks.  Latin Square Designs.  Lattice Square Designs.  Rectangular Lattice Designs.

Two, or more, factor designs

Multiple Factor Designs  An organism may vary in one factor according to conditions set by another factor.  Single factor experiments have limitations as they only relate to the conditions under which the factor is examined.

Multiple Factor Designs  Examine the effect of differences between factor 1 levels.  Examine the effect of differences between factor 2 levels.  Examine the interaction between factor 1 levels and factor 2 levels.

Interactions GenotypeNitrogen 1Nitrogen 2 A3,4684,088 B2,5044,791 A B

Interactions Low NHigh N A B Yield

Interactions Low NHigh N A B Yield

Interactions Low NHigh N A B Yield

Split-Plot Designs Factorial Designs Strip-plot Designs

Factorial Experimental Designs  Experimental design where all possible combinations of levels from two (or more) factors is called a factorial design.  Factorial designs are usually balanced, but unbalanced designs are possible (but not advised).

Factors  Species  Genotype  Nutrient  Water  Soil type  Seeding time  Seeding rate  Intercepted radiation  Day length  Location  Temperature  Feed stock  Tillage  Machinary

Factorial Experimental Design Irrigation Days between defoliation dayI1.D0I1.D4I1.D8I1.D12 2 dayI2.D0I2 D4I2D8I2D12 3 dayI3.D0I3.D4I3D8I3D12

3 t21 t13 t32 t41 t33 t4 3 t12 t21 t42 t11 t22 t3 3 t31 t22 t21 t33 t43 t1 2 t33 t21 t12 t44 t11 t4 1 t23 t12 t11 t12 t43 t2 1 t33 t32 t23 t41 t42 t3 I II III Factorial Experimental Design

Two-Factor Factorial Model Y ijk =  + r i + d j + w k + dw jk + e ijk Where Y ijk is the performance of the the j th replicate, and the j th d factor and k th w factor;  in the overall mean; r j is the effect of the j th replicate; d i is the effect of the i th d-factor; w k is the effect of the k th w-factor; dw jk is the interaction effect between d j and w k ; and e ijk is the error term.

Three-Factor Factorial Model Y ijk =  +r i +d j +w k +g l +dw jk +dg jl +wg kl +dwg jki +e ijk Where Y ijk is the performance of the the j th replicate, and the j th d factor and k th w factor and l th g-factor;  in the overall mean; r j is the effect of the j th replicate; d i is the effect of the i th d-factor; w k is the effect of the k th w-factor; gl is the effect of the l th g-factor ; dw jk is the interaction effect between d j and w k ; dg jl is the interaction effect between d j and w l ; wg jl is the interaction effect between w k and g l ; dwg jkl is the interaction effect between d j, w k and g l ; and e ijk is the error term.

Factorial Experimental Designs  Can be used with any number of factors and factor levels.  Gives equal precision to estimating all factors and levels.  Greatest mistake by researchers is to include too many factors where interpretation of three-way interactions can be difficult.

Split-Plot Designs  Three irrigation treatments  Four cultivars.  Two seeding rates (HIGH and low)

Split-Plot Designs III

III

aCdB AcDb III

aCdBDCbA AcDbdcba CbaDDcBA cBAddCba dabCbcDA DABcBCda III

 Greater precision of measurement is required on one of the factors (assigned to sub-plots).  Less precision required on the other factor (assigned to main-plots).  The relative size of the main effect of two factors is different.  Management practices do not allow factorial designs.

Main Plots Split-Plot Design

1 423 Main Plots Split-Plot Design

Main Plots Split-Plot Design

1B1B 1A1A 1C1C 1D1D 4B4B 4C4C 4A4A 4D4D 2D2D 2C2C 2A2A 2B2B 3D3D 3B3B 3C3C 3A3A Main Plots Sub-Plots Split-Plot Design

AABA BBAB ABBA BAAB BBAA AABB BABB ABAA I II III IV Split-Plot Design

I B A C II C A B III B A C Split-Plot Design

Split-Split-Plot Design

MP.3 MP.4 MP.1 MP.2 Split-Split-Plot Design

MP.3 MP.4 MP.1 MP.2 SP.1 SP.2 SP.1 SP.2 SP.1 SP.2 Split-Split-Plot Design

SSP.3SSP.1SSP.2SSP.4 MP.3 MP.4 MP.1 MP.2 SP.1 SP.2 SP.1 SP.2 SP.1 SP.2 Split-Split-Plot Design

SSP.3SSP.1SSP.2SSP.4 SSP.1SSP.4SSP.3SSP.2 SSP.3SSP.4SSP.1 SSP.4SSP.1SSP.3SSP.2 SSP.3SSP.2SSP.1SSP.4 SSP.2SSP.4SSP.3SSP.1 SSP.4SSP.1SSP.2SSP.3 SSP.4SSP.2SSP.3SSP.1 MP.3 MP.4 MP.1 MP.2 SP.1 SP.2 SP.1 SP.2 SP.1 SP.2 Split-Split-Plot Design

Split-Plot Design Model Y ijk =  + r i + g j + e(1) ij + t k + gt jk + e(2) ijk

Split-Plot Design Model Y ijk =  + r i + g j + e(1) ij + t k + gt jk + e(2) ijk

Split-Plot Design Model Y ijk =  + r i + g j + e(1) ij + t k + gt jk + e(2) ijk Where Y ijk is the performance of the the j th replicate, and the j th main-plot and k th sub-plot;  in the overall mean; r j is the effect of the j th replicate; g i is the effect of the i th main-plot; e(1) ij is the main-plot error; t k is the effect of the k th sub-plot; gt jk is the interaction effect between g j and t k ; and e(2) ijk is the sub-plot error term.

B A C A C B Strip-Plot Design I II III IV

B A C II A C B B A C IV A C B Strip-Plot Design III IV I II

B A C A C B IV III I II B A C A C B Strip-Plot Design

B A C A C B IV III I II B A C A C B Strip-Plot Design

B A C A C B IV III I II B A C A C B Strip-Plot Design 1

Strip-Plot Design Model Y ijk =  +r i +g j +e(g) ij +t k +e(t) ij +gt jk +e(gt) ijk Where Y ijk is the performance of the the j th replicate, and the j th strip and k th strip;  in the overall mean; r j is the effect of the j th replicate; g i is the effect of the i th strip-plot; e(g) ij is the g- factor error; t k is the effect of the k th strip-plot; e(t) ij is the t-factor error; dw jk is the interaction effect between g j and t k ; and e(gt) ijk is the sub- plot error term.

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