A Field Guide to Experimental 
Designs

The Completely Randomized Design (CRD) with subsampling

Subsampling, such as several limbs per tree or fruit per box, provides the researcher with better error control when experimental units might otherwise be limited. However, it also requires a different analysis than is used for the CRD design without subsampling.

Field marks:

  • Replications of treatments are assigned completely at random to independent groups of experimental subjects, such as adjacent trees, plants wihin the same pot or leaves on the same tree.
  • Adjacent groups could potentially have the same treatment.
  • A group of experimental subjects is considered a single independent experimental unit.

Sample layout:
There are 4 replications (1-4) of 3 treatments (A-C) with 3 subsamples (a-c) per replication. Different colors represent different treatments.
CRD with subsampling sample layout

A1a A1b A1c B2a B2b B2c C3a C3b C3c B4a B4b B4c
B1a B1b B1c A2a A2b A2c C2a C2b C2c A4a A4b A4c
C1a C1b C1c B3a B3b B3c A3a A3b A3c C4a C4b C4c

 

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/MSE
Experimental Error (E) t*(r-1) SSQE SSQE/(t*(r-1)) MSE/MSS
Sampling Error (S) t*r*(s-1) SSQS SSQS/(t*r*(s-1))  
Total (Tot) t*r*s-1 SSQTot    
awhere t=number of treatments, r=number of replications per treatment and s=number of subsamples per replication.

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
Experimental Error 9 87.80 9.76 1.61b
Sampling Error 24 145.18 6.05  
Total 35 621.81    
aF test with 2,9 degrees of freedom at P=0.05 is 4.26
bF test with 9,24 degrees of freedom at P=0.05 is 2.30

Sample SAS GLM statements:

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

Compare with:

 

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Friday, August 25, 2000