A Field Guide to Experimental Designs

The Randomized Complete Block (RCB) with subsampling

Field marks:

  • Replications of treatments are assigned completely at random to independent groups of experimental subjects within blocks.
  • Each treatment is repeated more than once per block.

Sample layout:
Different colors represent different treatments. Blocks are arranged in vertical rows. There are 4 blocks (I-IV) of 3 treatments (A-C) with 3 subsamples (a-c).
RCB with subsampling sample layout

Aa Ab Ac  +  Ba Bb Bc  +  Ca Cb Cc  +  Ba Bb Bc
Ba Bb Bc  +  Aa Ab Ac  +  Ba Bb Bc  +  Aa Ab Ac
Ca Cb Cc  +  Ca Cb Cc  +  Aa Ab Ac  +  Ca Cb Cc

Block I   +  Block II  +  Block III +  Block IV

 

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
Treatments (Tr) t-1 SSQTr SSQTr/(t-1) MSTr/MSE
Experimental Error (E) (t-1)*(b-1) SSQE SSQE/((t-1)*(b-1)) MSE/MSS
Sampling Error (S) t*b*(s-1) SSQS SSQS/(t*b*(s-1))  
Total (Tot) t*b*s-1 SSQTot    
awhere t=number of treatments, b=number of blocks and s=number of subsamples.

Sample ANOVA table:

Source of
variation		 Degrees of
freedom		 Sums of
squares (SSQ)		 Mean
square (MS)		 F
Blocks 3 64.32 21.44 2.16a
Treatments 2 201.86 100.93 10.17b
Experimental Error 6 59.53 9.93 1.64c
Sampling Error 24 145.18 6.05  
Total 35 470.89    
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 6,24 degrees of freedom at P=0.05 is 2.51

Sample SAS GLM statements: 

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

Compare with:

 

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Wednesday, August 16, 2000