Three-Factor Experiments Design selection, treatment composition, and randomization remain the same...
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![Page 1: Three-Factor Experiments Design selection, treatment composition, and randomization remain the same Each additional factor adds a layer of complexity to.](https://reader036.fdocuments.in/reader036/viewer/2022072006/56649cf75503460f949c744c/html5/thumbnails/1.jpg)
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)
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Same Song - Second Verse...
Construct tables of means
Complete an ANOVA table
Perform significance tests
Compute appropriate means and standard errors
Interpret the results
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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
2ijkl(Y Y)
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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
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Table of Block Means
Block I II III Mean
R1 R2 R3 Y
If there are blocks…
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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.
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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
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ANOVA Table (fixed model)
Source df SS MS F
Total rabc-1 SSTot= Block r-1 SSR= MSR= FR
SSR/(r-1)MSR/MSEMain Effects A a-1 SSA= MSA= FA
SSA/(a-1)MSA/MSEB b-1 SSB= MSB= FB
SSB/(b-1)MSB/MSEC c-1 SSC= MSC= FC
SSC/(c-1)MSC/MSE
2
ijkl ijklY Y
2
l ...labc Y Y
2
i i...rbc Y Y
2
j . j..rac Y Y
2
k ..k.rab Y Y
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ANOVA Table Continued...
Source df SS MS F
First Order InteractionsAB (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
2
ij ij..rc Y Y SSA SSB
2
ik i.k.rb Y Y SSA SSC
2
jk . jk.ra Y Y SSB SSC
2
ijk ijk.r Y Y SSA SSB SSC
SSAB SSAC SSBC
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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
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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
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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
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ANOVA
Source df SS MS F
Total 35 1936.75Block 2 270.17 135.08 5.93**
Variety (V) 1 306.25 306.25 13.44**Phosphorus (P) 2 32.00 16.00 .70Herbicide (W) 1 12.25 12.25 .54
V x P 2 18.67 9.33 .41V 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
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Means and Standard Errors
Herbicide
Variety None Some
V1 56.89 52.44
V2 57.11 63.89
*Standard error = 1.59
Mean seed yield (kg/plot) from two varieties of chick-peas with and without herbicide
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
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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
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A Picture is Worth a Thousand Words
Herbicide x Variety Interactions
52
54
56
58
60
62
64
Yie
ld
Herbicide
V1
V2
None Some 0 25 50
Phosphorus in kg/ha
Without Herbicide With Herbicide
Phosphorus x Herbicide Interactions
Yie
ld
52
54
56
58
60
62
64