Split block

Split-block design

Field marks:

Two sets of treatments are randomized across each other in strips in an otherwise RCB design.
Used where logistics make it necessary to run treatments completely across each block.
The number of blocks is the number of replications.

Sample layout:
In this example there are 3 blocks (I-III) de-lineated by colored lines; and 3 treatments (A-C) in one set indicated by color gradients either blue to green or yellow to brown and 2 treatments (1 & 2) in a second set indicated by blue-green or yellow-orange colors
Split-block sample layout

Block I   Block II   Block III
 A1  A2    C2  C1     B1  B2
 B1  B2    A2  A1     C1  C2
 C1  C2    B2  B1     A1  A2

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_TrxB
Treatments (Tr)	t-1	SSQ_Tr	SSQ_Tr/(t-1)	MS_Tr/MS_TrxB
Treatments X Blocks (TrxB)	(t-1)*(r-1)	SSQ_TrxB	SSQ_TrxB/((t-1)*(r-1))
Cross Treatments (C)	c-1	SSQ_C	SSQ_C/(c-1)	MS_C/MS_CxB
Cross X Blocks (CxB)	(c-1)*(r-1)	SSQ_CxB	SSQ_CxB/((c-1)*(r-1))
Cross X Treatments (CxT)	(c-1)*(t-1)	SSQ_CxT	SSQ_CxT/((c-1)*(t-1))	MS_CxT/MS_E
Error (E)	(t-1)(c-1)(b-1)	SSQ_E	SSQ_E/ ((t-1)(c-1)(b-1))
Total (Tot)	tcb-1	SSQ_Tot
^awhere t=number of treatments, c=number of cross 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	2	17.99	8.99	8.47^a
Treatments	2	168.62	84.31	79.43^a
Treatments X Blocks	4	4.25	1.06
Cross Treatments	1	209.21	209.21	128.95^b
Cross X Blocks	2	3.24	1.62
Cross X Treatments	2	22.49	11.25	2.85^c
Error	4	15.77	3.94
Total	17	441.57
^aF test with 3,4 degrees of freedom at P=0.05 is 6.94 ^bF test with 1,2 degrees of freedom at P=0.05 is 18.51 ^cF test with 2,4 degrees of freedom at P=0.05 is 6.94

Sample SAS GLM statements:

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

Compare with:

Randomized Complete Block (RCB): only a single set of treatments.
Two-way Factorial arrangement on a RCB: combinations of treatments are applied at random to experimental units within blocks.
Split-plot on a RCB: experimental units are assigned to treatments from one set and then subdivided into units for the second set.
RCB repeated in time: effects have a correlation with time
RCB repeated at more than one location: one set of treatments is examined at several different locations.

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Thursday, October 3, 2002