RCB

The Randomized Complete Block design (RCB)

The RCB is the standard design for agricultural experiments. The field or orchard is divided into units to account for any variation in the field. Treatments are then assigned at random to the subjects in the blocks-once in each block.

Field marks:

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.
Used to control variation in an experiment by accounting for spatial effects.

Sample layout:
Different colors represent different treatments; each horizontal row represents a block. There are 4 blocks (I-IV) and 4 treatments (A-D) in this example.
RCB sample layout

Block I     A   B   C   D
Block II    D   A   B   C
Block III   B   D   C   A
Block IV    C   A   B   D

ANOVA table format:

Source of variation	Degrees of freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
Blocks (B)	b-1	SSQ_B	SSQ_B/(b-1)	MS_B/MS_E
Treatments (Tr)	t-1	SSQ_Tr	SSQ_Tr/(t-1)	MS_Tr/MS_E
Error (E)	(t-1)*(b-1)	SSQ_E	SSQ_E/((t-1)*(b-1))
Total (Tot)	t*b-1	SSQ_Tot
^awhere t=number of treatments 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	26.36	8.79	1.36^a
Treatments	3	216.23	72.08	11.19^a
Error	9	57.95	6.44
Total	15	300.54
^aF test with 3,9 degrees of freedom at P=0.05 is 3.86

Sample SAS GLM statements:

PROC GLM;
  CLASS BLOCKS TREATS;
  MODEL WHATEVER = BLOCKS TREATS;
RUN;

Compare with:

Complete Randomized Design (CRD): : replications of treatments are assigned completely at random to independent experimental subjects without regard to other subjects.
Randomized Complete Block (RCB) with subsampling: more than one sub-unit per block is assigned to each treatment.
Latin square design: treatments are assigned once per row and once per column.
Experiment with pseudo-replication: multiple measurements made on sub-units, but no replication on independent experimental units.
Two-way Factorial arrangement on a CRD: often confused in texts with RCB! -is an arrangement of two sets of treatments randomly assigned to subjects.
Two-way Factorial arrangement on a RCB: treatments are combinations of two other sets of treatments.
Analysis of covariance--controlling variation with a second measureable factor: additional variation is controlled by measuring a second variable on each subject.
RCB repeated at more than one location: experiment is executed at more than one location.

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Friday, August 25, 2000