RCB repeated at locations

RCB repeated at more than one location

Field marks:

Blocks are laid out at more than one location. Treatments are assigned at random to those blocks as below.
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 treatments or locations. Blocks are delineated. There are 3 treatments, 3 locations and 3 blocks per location in this example.
RCB repeated at several locations sample layout

Location 1          Location 2
Block I  II  III    Block I  II  III
      A   B   C           B   C   B
      B   A   B           A   B   C
      C   C   A           C   A   A
              
              Location 3
              Block I  II  III
                    A   C   A
                    C   B   C
                    B   A   B

ANOVA table format:

Source of variation	Degrees of freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
Locations (L)	l-1	SSQ_L	SSQ_L/(l-1)	MS_L/MS_El
Error for Locations (El)	l*(b-1)	SSQ_El	SSQ_El/(l*(b-1))
Treatments (Tr)	t-1	SSQ_Tr	SSQ_Tr/(t-1)	MS_Tr/MS_E
Treatments X Locations (TxL)	(t-1)*(l-1)	SSQ_TxL	SSQ_TxL/((t-1)*(l-1))	MS_TxL/MS_E
Error (E)	l(t-1)(b-1)	SSQ_E	SSQ_E/(l(t-1)(b-1))
Total (Tot)	ltb-1	SSQ_Tot
^awhere l=number of locations, t=number of treatments and b=number of blocks or replications.

Sample ANOVA table:

Source of variation	Degrees of freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
Locations	2	33.63	16.82	1.17^a
Error for Locations	6	86.05	14.34
Treatments	2	186.76	93.38	11.47^a
Treatments X Locations	4	16.84	4.21	0.52^a
Error	12	97.72	8.14
Total	26	421.01
^aF test with 2,6 degrees of freedom at P=0.05 is 5.14; 2,12 is 3.89; 4,12 is 3.26

Sample SAS GLM statements:

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

Compare with:

Randomized Complete Block (RCB): all subjects are at a single location
Two-way Factorial arrangement on a RCB assigned treatments are combinations of treatments from two sets
Split Plot on a RCB: one set of treatments is assigned at random to experimental units; a second set is assigned to at random to sub-units.

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July 7, 2000