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Three-Factor Experiments

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)

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Three-Factor Experiments

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  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 plots per replication will be a x b x c • SSTot = • Excel Spreadsheet: = DEVSQ(Range of all observations) • With a calculator: = s2*(n-1) = 2*n, where n = rabc

  4. Table of Block Means • Block I II III Mean • R1 R2 R3

  5. Table for Main Effects Level 1 2 ... f Factor A A1 A2 ... Aa Factor B B1 B2 ... Bb Factor C C1 C2 ... Cc Where A1 represents the mean of all of the treatments involving Factor A at level 1, averaged across all of the other factors.

  6. Table of Means for Treatments Treatment Means over Reps A1B1C1 T111 A1B1C2 T112 A1B1C... T11.. A1B1Cc T11c A1B2C1 T121 A1B2C2 T122 A1B2C... T12.. A1B2Cc T12c .... AaBbCc Tabc Compute a mean over replications for each treatment. Total number of treatments = a x b x c.

  7. First-Order Interactions Factor B Factor A 1 2 ... b 1 T11. T12. ... T1b. 2 T21. T22. ... T2b. ... ... ... ... ... a Ta1. Ta2. ... Tab. Compute a table such as this for each first order interaction: A x B A x C B x C

  8. ANOVA Table (fixed model) • Source df SS MS F • Total rabc-1 SSTot= • Block r-1 SSR= MSR= FR • SSR/(r-1) MSR/MSE • Main Effects • A a-1 SSA= MSA= FA • SSA/(a-1) MSA/MSE • B b-1 SSB= MSB= FB • SSB/(b-1) MSB/MSE • C c-1 SSC= MSC= FC • SSC/(c-1) MSC/MSE

  9. ANOVA Table Continued... • Source df SS MS F • First Order Interactions • AB (a-1)(b-1) SSAB= MSAB FAB • AC (a-1)(c-1) SSAC= MSAC FAC • BC (b-1)(c-1) SSBC= MSBC FBC • 3-Factor Interaction • ABC (a-1)(b-1)(c-1) SSABC= MSABC FABC • Error (r-1)(abc-1) SSE= MSE • SSTot-SSR-SSA-SSB-SSC • -SSAB-SSAC-SSBC-SSABC

  10. Standard Errors Factor Std Err of Mean Std Err of Difference A MSE/rbc 2MSE/rbc B MSE/rac 2MSE/rac C MSE/rab 2MSE/rab AB MSE/rc 2MSE/rc AC MSE/rb 2MSE/rb BC MSE/ra 2MSE/ra ABC MSE/r 2MSE/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 1st 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 • 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 RBD design in three blocks

  13. ANOVA Source df SS MS F Total 35 1936.75 Block 2 270.17 135.08 5.93** Variety (V) 1 306.25 306.25 13.44** Phosphorus (P) 2 32.00 16.00 .70 Herbicide (W) 1 12.25 12.25 .54 V x P 2 18.67 9.33 .41 V x W 1 283.36 283.36 12.44** P x W 2 468.67 234.33 10.29** V x P x W 2 44.22 22.11 .97 Error 22 501.16 22.78

  14. Herbicide Variety None Some V1 56.89 52.44 V2 57.11 63.89 *Standard error = 1.59 Phosphorus Herbicide None 25 kg/ha 50 kg/ha None 60.00 57.83 53.17 Some 52.50 58.67 63.33 *Standard error = 1.95 Mean seed yield (kg/plot) of chick-peas at three levels of phosphorus fertilization with and without herbicide Means and Standard Errors Mean seed yield (kg/plot) from two varieties of chick-peas 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 Phosphorus x Herbicide Interactions V2 64 64 62 62 60 60 Yield 58 Yield 58 56 56 54 54 V1 52 52 Some None 0 25 50 Phosphorus in kg/ha Herbicide Without Herbicide With Herbicide

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