A Field Guide to Experimental 
Designs

The Split-split Plot on a RCB

Sometimes in experiments, subjects are assigned at random to a set of treatments. Then those are subdivided into sub-units to which another set of treatments are applied. And then, those in turn are subdivided again and a third set of treatments are applied. This is the split-split plot, not be confused with a 3-way factorial.

A split-split plot can also be imposed on a CRD, but is not included at this time in the field guide.

Field marks:

  • Main experimental subjects of a RCB are divided further into additional independent units (subplots) to which another set of treatments are randomly assigned. These subplots are additionally split into subunits assigned randomly to yet another set of treatments.
  • Main 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 main treatment can be adjacent to any other treatment, but not to the same treatment within the block.

Sample layout:
Different colors represent different treatments; each horizontal row represents a block. Plot colors represent assignment of main effects or treatments; top plant colors split plots; bottom colors split-split treatments. There are 3 blocks (I-III) each of 2 main treatments (A & B) split into 2 treatments (1 & 2) and further split into 3 treatments (a-c) in split-split plots.

Split-split plot sample layout

Block I        Treatment A    +    Treatment B
            1a 1b 1c 2c 2b 2a + 2b 2c 2a 1a 1c 1b 
Block II       Treatment B    +    Treatment A     
            1c 1a 1b 2b 2c 2a + 2c 2a 2b 1b 1a 1c 
Block III      Treatment A    +    Treatment B
            2b 2c 2a 1a 1c 1b + 1c 1a 1b 2a 2b 2c

 

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/MSEm
Treatments (Tr) t-1 SSQTr SSQTr/(t-1) MSTr/MSEm
Error-main plots (Em) (t-1)*(b-1) SSQEm SSQEm/((t-1)*(b-1))  
Subplots (S) s-1 SSQS SSQS/(s-1) MSS/MSEs
Subplots X Treatments (SxT) (t-1)*(s-1) SSQSxT SSQSxT/(t-1)*(s-1) MSSxT/MSEs
Error-subplots (Es) t*(b-1)*(s-1) SSQEs SSQEs/(t*(b-1)*(s-1))  
Split-subplots (U) u-1 SSQU SSQU/(u-1) MSU/MSEu
Split-subplots X Treatments (UxT) (t-1)*(u-1) SSQUxT SSQUxT/(t-1)*(u-1) MSUxT/MSEu
Split-subplots X Subplots (UxS) (s-1)*(u-1) SSQUxS SSQUxS/(s-1)*(u-1) MSUxS/MSEu
Split-subplots X Subplots X Treatments (UxSxT) (t-1)*(s-1)*(u-1) SSQUxSxT SSQUxSxT/((t-1)(s-1)*(u-1)) MSUxSxT/MSEu
Error-split-subplots (Eu) t*s*(b-1)*(u-1) SSQEu SSQEu/(t*s*(b-1)*(u-1))  
Total (Tot) t*b*s*u-1 SSQTot    
awhere t=number of main treatments, b=number of blocks and s=number of subplots.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
B 2 84.64 42.32 1.12a
T 1 0.70 0.70 0.02b
Error-main 2 75.38 37.69  
S 1 665.77 665.77 13.06c
S X T 1 25.71 25.71 0.50c
Error-subplot 4 203.85 50.96  
U 2 231.09 115.54 20.94d
T X U 2 9.28 4.64 0.84d
S X U 2 0.73 0.37 0.07d
T X S X U 2 8.66 4.32 0.78d
Error-split-subplots 16 88.28 5.52  
Total 35 1394.11    
aF test with 2,2 degrees of freedom at P=0.05 is 19.00
bF test with 1,2 degrees of freedom at P=0.05 is 18.51
cF test with 1,4 degrees of freedom at P=0.05 is 7.71
dF test with 2,16 degrees of freedom at P=0.05 is 3.63.

Sample SAS GLM statements:

PROC GLM;
  CLASS Blk Trt Sub U;
  MODEL WHATEVER = Blk Trt Trt*Blk 
                   Sub Sub*Trt Blk*Sub(Trt)
                   U U*Trt U*Sub U*Trt*Sub;
  TEST H = Blk Trt  E = Trt*Blk;
  TEST H = Sub Sub*Trt  E = Blk*Sub(Trt);
RUN;

Compare with:

Note: a Split-block-split-plot design is possible, but is not illustrated in the field guide.

 

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Thursday, August 17, 2000