Relative Efficiency of a
Design A to Design B
lecture - 10
A commonly used index for
comparing the efficiency of two different designs is the inverse ratio of the
variance per unit, i.e., the MSE's. Since different designs may have different
degrees of freedom for error, a correction factor, suggested by Fisher, which
multiplies the inverse ratio of variances, will give a better measure of the
relative efficiency (RE).
.png)
Where MSE
If RE > 1
If RE < 1
If RE = 1
Relative Efficiency of RCBD vs.
CRD
If a RCB design (say,
design-A) is used, one may want to estimate the relative efficiency compared
with a CR design (say, design-B). This is possible by using the following
equation to estimate the MSE of CRD (MSE B
.png)
Example:
The
six treatments in each block were randomly assigned to the six plots by drawing
random numbers from Appendix Table A-1 in the manner described in Chapter 7.
Note in this case that there are only six random numbers (1 - 6) to be drawn
for each block, e.g., for block 1 the random sequence was 3, 6, 5, 2, 1, and 4.
Assigning treatments A-F to numbers 1-6 results in the block 1 treatment
sequence:
|
Block 1 |
40.9 (C) |
40.6 (F) |
39.7 (E) |
38.8 (B) |
31.3 (A) |
40.9 (D) |
|
Block 2 |
33.4 (A) |
41.7 (D) |
37.5 (B) |
41.0 (F) |
40.6 (E) |
39.3 (C) |
|
Block 3 |
37.4 (B) |
39.5 (C) |
39.4 (D) |
39.2 (E) |
41.5 (F) |
29.2 (A) |
|
Block 4 |
40.1 (D) |
38.6 (C) |
38.7 (E) |
32.2 (A) |
41.1 (F) |
35.8 (B) |
|
Block 5 |
39.8 (C) |
40.0 (D) |
33.9 (A) |
38.4 (B) |
41.9 (E) |
39.8 (F) |
Treatments A-F is levels
of nitrogen fertilizer from 0 - 250 lbs/acre in 50 lb increments. The number in
parenthesis is the root yield per plot in tons/acre. Test the significance of
six treatments and compare the efficiency to CRD.
Solution:
Proceed the hypothesis testing and compute ANOVA table for RCBD:
|
Block |
A |
B |
C |
D |
E |
F |
Ti. |
Ti.^2 |
|
1 |
31.3 |
38.8 |
40.9 |
40.9 |
39.7 |
40.6 |
232.2 |
|
|
2 |
33.4 |
37.5 |
39.3 |
41.7 |
40.6 |
41.0 |
233.4 |
|
|
3 |
29.2 |
37.4 |
39.5 |
39.4 |
39.2 |
41.5 |
226.2 |
|
|
4 |
32.2 |
35.8 |
38.6 |
40.1 |
38.7 |
41.1 |
226.5 |
|
|
5 |
33.9 |
38.4 |
39.8 |
40.0 |
41.9 |
39.8 |
233.8 |
|
|
T.j |
160 |
187.9 |
198 |
202.1 |
200.1 |
204 |
1152.1 |
265523.53 |
|
T.j^2 |
|
|
|
|
|
|
222610.83 |
|
.png)
.png)
|
SV |
df |
SS |
MS |
F |
|
Treatment |
5 |
277.69 |
55.54 |
46.28 |
|
Block |
4 |
9.44 |
2.36 |
1.97 |
|
Error |
20 |
24.00 |
1.20 |
|
|
Total |
29 |
331.13 |
|
|
Now to
compute the MSE of CRD from RCBD ANOVA:
.png)
It means
that RCBD is 11.4 % efficient than CRD for this experiment.
CR Design vs. RCB Design
Let we have
“t” treatments and each treatment is replicated “b” times.
ANOVA Table
of CRD is given below:
|
SV |
df |
SS |
MS |
F |
|
Treatment |
t-1 |
SST |
MST |
|
|
Error |
tb - t |
SSE |
MSE |
|
|
Total |
tb - 1 |
|
|
|
ANOVA Table
of RCBD is given below:
|
SV |
df |
SS |
MS |
F |
|
Treatment |
t-1 |
SST |
MST |
|
|
Block |
b-1 |
SSB |
MSB |
|
|
Error |
(t - 1) (b - 1) |
SSE |
MSE |
|
|
Total |
tb-1 |
SSTotal |
|
|
From both
ANOVA Table, it is observed that
.png)
RCBD can be a very
effective noise-reducing technique if SS block is large.
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