The Splitsplit Plot on a RCB
Sometimes in experiments, subjects are assigned at random to a set of treatments. Then those are subdivided into subunits to which another set of treatments are applied. And then, those in turn are subdivided again and a third set of treatments are applied. This is the splitsplit plot, not be confused with a 3way factorial.
A splitsplit plot can also be imposed on a CRD, but is not included at this time in the field guide.
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. These subplots are additionally
split into subunits assigned randomly to yet another set of
treatments.
 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 horizontal row represents a block. Plot colors represent
assignment of main effects or treatments; top plant colors split plots; bottom colors splitsplit treatments. There are 3 blocks (IIII) each of 2 main treatments (A & B) split into 2 treatments (1 & 2) and further split into 3 treatments (ac) in splitsplit plots.
Block I Treatment A + Treatment B
1a 1b 1c 2c 2b 2a + 2b 2c 2a 1a 1c 1b
Block II Treatment B + Treatment A
1c 1a 1b 2b 2c 2a + 2c 2a 2b 1b 1a 1c
Block III Treatment A + Treatment B
2b 2c 2a 1a 1c 1b + 1c 1a 1b 2a 2b 2c
ANOVA table format:
Source of variation 
Degrees of freedom^{a} 
Sums of squares (SSQ) 
Mean square (MS) 
F 
Blocks (B) 
b1 
SSQ_{B} 
SSQ_{B}/(b1) 
MS_{B}/MS_{Em} 
Treatments (Tr) 
t1 
SSQ_{Tr} 
SSQ_{Tr}/(t1) 
MS_{Tr}/MS_{Em} 
Errormain plots (Em) 
(t1)*(b1) 
SSQ_{Em} 
SSQ_{Em}/((t1)*(b1)) 

Subplots (S) 
s1 
SSQ_{S} 
SSQ_{S}/(s1) 
MS_{S}/MS_{Es} 
Subplots X Treatments (SxT) 
(t1)*(s1) 
SSQ_{SxT} 
SSQ_{SxT}/(t1)*(s1) 
MS_{SxT}/MS_{Es} 
Errorsubplots (Es) 
t*(b1)*(s1) 
SSQ_{Es} 
SSQ_{Es}/(t*(b1)*(s1)) 

Splitsubplots (U) 
u1 
SSQ_{U} 
SSQ_{U}/(u1) 
MS_{U}/MS_{Eu} 
Splitsubplots X Treatments (UxT) 
(t1)*(u1) 
SSQ_{UxT} 
SSQ_{UxT}/(t1)*(u1) 
MS_{UxT}/MS_{Eu} 
Splitsubplots X Subplots (UxS) 
(s1)*(u1) 
SSQ_{UxS} 
SSQ_{UxS}/(s1)*(u1) 
MS_{UxS}/MS_{Eu} 
Splitsubplots X Subplots X Treatments (UxSxT) 
(t1)*(s1)*(u1) 
SSQ_{UxSxT} 
SSQ_{UxSxT}/((t1)(s1)*(u1)) 
MS_{UxSxT}/MS_{Eu} 
Errorsplitsubplots (Eu) 
t*s*(b1)*(u1) 
SSQ_{Eu} 
SSQ_{Eu}/(t*s*(b1)*(u1)) 

Total (Tot) 
t*b*s*u1 
SSQ_{Tot} 


^{a}where 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 
B 
2 
84.64 
42.32 
1.12^{a} 
T 
1 
0.70 
0.70 
0.02^{b} 
Errormain 
2 
75.38 
37.69 

S 
1 
665.77 
665.77 
13.06^{c} 
S X T 
1 
25.71 
25.71 
0.50^{c} 
Errorsubplot 
4 
203.85 
50.96 

U 
2 
231.09 
115.54 
20.94^{d} 
T X U 
2 
9.28 
4.64 
0.84^{d} 
S X U 
2 
0.73 
0.37 
0.07^{d} 
T X S X U 
2 
8.66 
4.32 
0.78^{d} 
Errorsplitsubplots 
16 
88.28 
5.52 

Total 
35 
1394.11 


^{a}F test with 2,2 degrees of freedom at P=0.05 is 19.00
^{b}F test with 1,2 degrees of freedom at P=0.05 is 18.51
^{c}F test with 1,4 degrees of freedom at P=0.05 is 7.71
^{d}F test with 2,16 degrees of freedom at P=0.05 is 3.63. 
Sample SAS GLM statements:
PROC GLM;
CLASS Blk Trt Sub U;
MODEL WHATEVER = Blk Trt Trt*Blk
Sub Sub*Trt Blk*Sub(Trt)
U U*Trt U*Sub U*Trt*Sub;
TEST H = Blk Trt E = Trt*Blk;
TEST H = Sub Sub*Trt E = Blk*Sub(Trt);
RUN;
Compare with:
Note: a Splitblocksplitplot design is possible, but is not illustrated in the field guide.
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Thursday, August 17, 2000
