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)

Same Song - Second Verse...

Construct tables of means

Complete an ANOVA table

Perform significance tests

Compute appropriate means and standard errors

Interpret the results

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)

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

Table of Block Means

Block I II III Mean

R1 R2 R3 Y

If there are blocks…

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.

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

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

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

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

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

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

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

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

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

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