A Field Guide to Experimental 
Designs

The Split Plot on a CRD

Field marks:

  • Main experimental subjects of a CRD are divided further into additional independent units (subplots) to which another set of treatments are randomly assigned.
  • Replications of treatments are assigned completely at random to main independent experimental subjects, not to the subplots.
  • Adjacent main independent subjects could potentially have the same treatment.

Sample layout:
Different colors represent different treatments. Top colors represent assignment of main effects or treatments; bottom colors subplot treatments. There are 4 replications (1-4) each of 3 main treatments (A-C) divided into 3 further treatments (a-c).
Split plot sample layout

A1a A1b A1c B1c B1b B1a A2b A2c A2a C1a C1c C1b

C2c C2a C2b A3b A3c A3a B2c B2a B2b C3b C3a C3c

B3b B3c B3a A4a A4c A4b C4c C4a C4b B4a B4b B4c

ANOVA table format:

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
Treatments (Tr) t-1 SSQTr SSQTr/(t-1) MSTr/MSEm
Error-main plots (Em) t*(r-1) SSQEm SSQEm/(t*(r-1))  
Subplots (S) s-1 SSQS SSQS/(s-1) MSS/MSEs
Subplots X Treatments (SxT) (t-1)*(s-1) SSQSxT SSQSxT/(t-1)*(s-1) MSSxT/MSEs
Error-subplots (Es) t*(r-1)*(s-1) SSQEs SSQEs/(t*(r-1)*(s-1))  
Total (Tot) t*r*s-1 SSQTot    
awhere t=number of main treatments, r=number of replications per treatment and s=number of subplots.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
Treatments 2 199.08 99.54 10.20a
Error-main plots 9 87.80 9.76  
Subplots 2 193.46 96.73 15.00b
Subplots X Treatments 4 25.38 6.35 0.98c
Error-subplots 18 116.08 6.45  
Total 35 621.81    
aF test with 2,9 degrees of freedom at P=0.05 is 4.26
bF test with 2,18 degrees of freedom at P=0.05 is 3.55
cF test with 4,18 degrees of freedom at P=0.05is 2.93.

Sample SAS GLM statements:

PROC GLM;
  CLASS TREATS REPS SUBPLOTS;
  MODEL WHATEVER = TREATS REPS(TREATS) SUBPLOTS
                   SUBPLOTS*TREATS;
  TEST H = TREATS E = REPS(TREATS);
RUN;

Compare with:

 

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Thursday, August 17, 2000