A Field Guide to Experimental Designs

The Split Plot on a RCB

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.
  • 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 vertical row represents a block. Top colors represent assignment of main effects or treatments; bottom colors subplot treatments. There are 4 blocks (I-IV) each of 3 main treatments (A-C) divided into 3 further treatments (a-c).
Split plot sample layout

Block-I   Block-II   Block-III   Block-IV
Aa Ab Ac  Bc Bb Ba   Ab Ac Aa    Ca Cc Cb
Cc Ca Cb  Ab Ac Aa   Bc Ba Bb    Aa Ac Ab
Bb Bc Ba  Cb Ca Cc   Cc Ca Cb    Ba Bb Bc

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))  
Total (Tot) t*b*s-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
Blocks 3 64.33 21.44 2.16a
Treatments 2 201.86 100.93 10.17b
Error-main plots 6 59.53 9.92  
Subplots 2 193.45 96.73 15.00c
Subplots X Treatments 4 25.38 6.35 0.98d
Error-subplots 18 116.08 6.45  
Total 35 660.63    
aF test with 3,6 degrees of freedom at P=0.05 is 4.76
bF test with 2,6 degrees of freedom at P=0.05 is 5.14
cF test with 2,18 degrees of freedom at P=0.05 is 3.55
dF test with 4,18 degrees of freedom at P=0.05 is 2.93

Sample SAS GLM statements: 

PROC GLM;
  CLASS BLOCKS TREATS SUBPLOTS;
  MODEL WHATEVER = BLOCKS TREATS TREATS*BLOCKS SUBPLOTS
                   SUBPLOTS*TREATS;
  TEST H = BLOCKS TREATS E = TREATS*BLOCKS;
RUN;

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Thursday, August 17, 2000