A Field Guide to Experimental 
Designs

Two way Factorial arrangement on a CRD

Field marks:

  • Treatments are combinations of two factors, such as chemical compositions and methods of applying chemicals. This is an arrangement of treatments within a CRD design.
  • Replications of treatments are assigned completely at random to independent experimental subjects.
  • Adjacent subjects could potentially have the same combination of treatments.

Sample layout:
Different colors represent different combinations of treatments; each combination is randomly assigned to a subject. There are 4 replications (1-4) and 4 treatment combinations (Aa, Ba, Ab and Bb) in this example.
CRD factorial sample layout

Aa1  Ba1  Ab1  Aa2

Bb1  Aa3  Bb2  Ab2 

Ba2  Bb3  Ab3  Ba3

Ab4  Aa4  Ba4  Bb4

ANOVA table format:

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
First factor (F1) f-1 SSQF1 SSQF1/(f-1) MSF1/MSE
Second factor (F2) s-1 SSQF2 SSQF2/(s-1) MSF2/MSE
First X Second (FxS) (f-1)*(s-1) SSQFxS SSQFxS/((f-1)*(s-1)) MSFxS/MSE
Error (E) f*s*(r-1) SSQE SSQE/(f*s*(r-1))  
Total (Tot) f*s*r-1 SSQTot    
awhere f=number of treatments in the first factor, s=number of treatments in the second factor and r=number of replications.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
First 1 80.30 80.30 259.43a
Second 1 34.50 34.50 111.47a
First X Second 1 0.22 0.22 0.70a
Error 12 63.71 0.31  
Total 15 118.73    
aF test with 1,12 degrees of freedom at P=0.05 is 4.75

Sample SAS GLM statements:

PROC GLM;
  CLASS FIRST SECOND;
  MODEL WHATEVER = FIRST SECOND FIRST*SECOND;
RUN;

Compare with:

 

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Wednesday, August 16, 2000