A Field Guide to Experimental Designs

Two way Factorial arrangement on a RCB

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 RCB design.
  • Treatments are assigned at random within blocks of adjacent subjects, each treatment once per block.
  • The number of blocks is the number of replications.
  • Any treatment can be adjacent to any other treatment, but not to the same treatment within the block.

 

Sample layout:
Different colors represent different combinations of treatments; each horizontal row represents a block. There are 4 blocks and 4 treatment combinations (Aa, Ba, Ab and Bb) in this example.
RCB sample layout

Block IV     Aa  Ba  Ab  Bb

Block III    Bb  Aa  Ba  Ab 

Block II     Ba  Bb  Ab  Aa

Block I      Ab  Aa  Ba  Bb

ANOVA table format:

Source of
variation		 Degrees of
freedoma		 Sums of
squares (SSQ)		 Mean
square (MS)		 F
Blocks (B) b-1 SSQB SSQB/(b-1) MSB/MSE
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-1)*(b-1) SSQE SSQE/((f*s-1)*(b-1))  
Total (Tot) f*s*b-1 SSQTot    
awhere f=number of treatments in the first factor, s=number of treatments in the second factor and b=number of blocks or replications.

Sample ANOVA table:

Source of
variation		 Degrees of
freedom		 Sums of
squares (SSQ)		 Mean
square (MS)		 F
Blocks 3 17.92 5.97 8.35a
First 1 109.75 109.75 153.40b
Second 1 49.64 49.64 69.38b
First X Second 1 0.24 0.24 0.33b
Error 9 6.44 0.72  
Total 15 183.99    
aF test with 3,9 degrees of freedom at P=0.05 is 3.86
bF test with 1,9 degrees of freedom at P=0.05 is 5.12

Sample SAS GLM statements: 

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

Compare with:

 

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Thursday, August 17, 2000