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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.
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:
Washington State University
Tree Fruit Research and Extension Center
1100 N. Western Ave.
Wenatchee WA 98801
phone: 509-663-8181
fax: 509-662-8714
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Friday, August 25, 2000
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