A Field Guide to Experimental 
Designs

Three way Factorial arrangement on a RCB

Sometimes the researcher will want to look at the effect of three different factors superimposed on each other. This is often handled with a 3-way factorial arrangement (but also look at the split-split plot design). The 3-way can also be applied to the CRD and Latin square designs.

Field marks:

  • Treatments are combinations of three 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 drawn box represents a block. There are 3 blocks (I-III) and combinations of 3, 2, & 2 treatments (A-C, changes in red intensity; 1-2, changes in green intensity; and a-b, changes in blue intensity) in this example.
RCB 3 way factorial sample layout

Block I     Block II     Block III
C1a  B2a    C2a  C1a     B2a  B1a
B1a  A2a    C1b  B1a     B2b  A2a
A2b  A1a    B2b  A2a     A2b  A1a
B1b  B2b    A1b  A1a     C1b  C2b
A1b  C2a    B1b  A2b     C1a  A1b
C2b  C1b    C2b  B2a     B1b  C2a

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 (X) x-1 SSQX SSQX/(x-1) MSX/MSE
Second factor (Y) y-1 SSQY SSQY/(y-1) MSY/MSE
Third factor (Z) z-1 SSQZ SSQZ/(z-1) MSZ/MSE
First X Second (XxY) (x-1)*(y-1) SSQXxY SSQXxY/((x-1)*(y-1)) MSXxY/MSE
First X Third (XxZ) (x-1)*(z-1) SSQXxZ SSQXxZ/((x-1)*(z-1)) MSXxZ/MSE
Second X Third (YxZ) (y-1)*(z-1) SSQYxZ SSQYxZ/((y-1)*(z-1)) MSYxZ/MSE
First X Second X Third (XxYxZ) (x-1)*(y-1)*(z-1) SSQXxYxZ SSQXxYxZ/((x-1)*(y-1)*(z-1)) MSXxYxZ/MSE
Error (E) (x*y*z-1)*(b-1) SSQE SSQE/((x*y*z-1)*(b-1))  
Total (Tot) x*y*z*b-1 SSQTot    
awhere x=number of treatments in the first factor, y=number of treatments in the second factor, z=number of treatments in the third 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 2 146.38 73.19 117.43a
First 2 41.97 20.98 33.67a
Second 1 195.70 195.70 313.99b
Third 1 66.84 66.84 107.23b
First X Second 2 0.04 0.02 0.03a
First X Third 2 0.23 0.12 0.18a
Second X Third 1 6.28 6.28 10.08b
First X Second X Third 2 0.69 0.35 0.55a
Error 22 13.71 0.62  
Total 35 471.85    
aF test with 2,22 degrees of freedom at P=0.05 is 3.44
bF test with 1,22 degrees of freedom at P=0.05 is 4.30

Sample SAS GLM statements:

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

Compare with:

 

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Thursday, August 24, 2000