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Three-Factor Experiments Design selection, treatment composition, and randomization remain the same Each additional factor adds a layer of complexity to.

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Presentation on theme: "Three-Factor Experiments Design selection, treatment composition, and randomization remain the same Each additional factor adds a layer of complexity to."— Presentation transcript:

1 Three-Factor Experiments Design selection, treatment composition, and randomization remain the same Each additional factor adds a layer of complexity to the analysis In a 3-factor experiment we estimate and test: –3 main effects –3 two-factor interactions (first order) –1 three-factor interaction (second order)

2 Same Song - Second Verse... Construct tables of means Complete an ANOVA table Perform significance tests Compute appropriate means and standard errors Interpret the results

3 In General... We have ‘a’ levels of A, ‘b’ levels of B, and ‘c’ levels of C –Total number of treatments will be a x b x c –Total number of plots will be a x b x c x r SSTot = –Where i =1 to a j =1 to b k =1 to c l =1 to r

4 Table of Means for Treatments TreatmentMeans over Reps A 1 B 1 C 1 T 111 A 1 B 1 C 2 T 112 A 1 B 1 C...T 11.. A 1 B 1 C c T 11c A 1 B 2 C 1 T 121 A 1 B 2 C 2 T 122 A 1 B 2 C...T 12.. A 1 B 2 C c T 12c.... A a B b C c T abc Compute a mean over replications for each treatment. Total number of treatments = a x b x c

5 Table of Block Means Block I II III Mean R 1 R 2 R 3 If there are blocks…

6 Table for Main Effects Level 12...f Factor A A 1 A 2...A a Factor BB 1 B 2...B b Factor CC 1 C 2...C c Where A 1 represents the mean of all of the treatments involving Factor A at level 1, averaged across all of the other factors.

7 First-Order Interactions Factor B Factor A12...b 1T 11. T 12....T 1b. 2T 21. T 22....T 2b................ aT a1. T a2....T ab. Compute a table such as this for each first order interaction: A x BA x CB x C

8 ANOVA Table (fixed model) SourcedfSSMSF Totalrabc-1SSTot= Blockr-1SSR=MSR=F R SSR/(r-1)MSR/MSE Main Effects Aa-1SSA=MSA=F A SSA/(a-1)MSA/MSE Bb-1SSB=MSB=F B SSB/(b-1)MSB/MSE Cc-1SSC=MSC=F C SSC/(c-1)MSC/MSE

9 ANOVA Table Continued... SourcedfSSMSF First Order Interactions AB(a-1)(b-1)SSAB=MSABF AB AC(a-1)(c-1)SSAC=MSACF AC BC(b-1)(c-1)SSBC=MSBCF BC 3-Factor Interaction ABC(a-1)(b-1)(c-1)SSABC=MSABCF ABC Error(r-1)(abc-1)SSE=MSE SSTot-SSR-SSA-SSB-SSC -SSAB-SSAC-SSBC-SSABC

10 Standard Errors FactorStd Err of MeanStd Err of Difference AMSE/rbc2MSE/rbc BMSE/rac2MSE/rac CMSE/rab2MSE/rab ABMSE/rc2MSE/rc ACMSE/rb2MSE/rb BCMSE/ra2MSE/ra ABCMSE/r2MSE/r

11 Interpretation Depends on the outcome of the F tests for main effects and interactions If the 3-factor (AxBxC) interaction is significant –None of the factors are acting independently –Summarize with 3-way table of means for each treatment combination If 1 st order interactions are significant (and not the 3-factor interaction) –Neither of the main effects are independent –Summarize with 2-way table of means for significant interactions If Main Effects are significant (and not any of the interactions) –Summarize significant main effects with a 1-way table of factor means

12 Example Seed yield of chickpeas Study the effect of three production factors: –Variety (2) –Phosphorus Fertilization (3) None, 25 kg/ha, 50 kg/ha –Weed Control (2) None, Herbicide Using an RBD design in three blocks

13 ANOVA SourcedfSSMSF Total351936.75 Block2270.17135.085.93** Variety (V)1306.25306.2513.44** Phosphorus (P)232.0016.00.70 Herbicide (W)112.2512.25.54 V x P218.679.33.41 V x W1283.36283.3612.44** P x W2468.67234.3310.29** V x P x W244.2222.11.97 Error22501.1622.78

14 Means and Standard Errors Herbicide VarietyNoneSome V 1 56.8952.44 V 2 57.1163.89 *Standard error = 1.59 Mean seed yield (kg/plot) from two varieties of chick- peas with and without herbicide Phosphorus HerbicideNone25 kg/ha50 kg/ha None60.0057.8353.17 Some52.5058.6763.33 *Standard error = 1.95 Mean seed yield (kg/plot) of chick- peas at three levels of phosphorus fertilization with and without herbicide

15 Interpretation The effect of herbicide depended on variety –The addition of herbicide reduced the yield for variety 1 –The yield of variety 2 was increased by the use of herbicide Response to added phosphorus depended on whether or not herbicide was used –If no herbicide, seed yield was reduced when phosphorus was added –However, seed yield increased when phosphorus was added in addition to herbicide

16 A Picture is Worth a Thousand Words Herbicide x Variety Interactions 52 54 56 58 60 62 64 Yield Herbicide V1 V2 None Some 02550 Phosphorus in kg/ha Without Herbicide With Herbicide Phosphorus x Herbicide Interactions Yield 52 54 56 58 60 62 64

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