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Chapter 4 Supplemental Text Material
S41. Relative Efficiency of the RCBD
In Example 41 we illustrated the noisereducing property of the randomized complete
block design (RCBD). If we look at the portion of the total sum of squares not accounted
for by treatments (302.14; see Table 44), about 63 percent (192.25) is the result of
differences between blocks. Thus, if we had run a completely randomized design, the
mean square for error
MS
E
would have been much larger, and the resulting design would
not have been as sensitive as the randomized block design.
It is often helpful to estimate the relative efficiency of the RCBD compared to a
completely randomized design (CRD).
One way to define this relative efficiency is
R
df
df
df
df
br
r
b
=
++
⋅
()
13
31
2
2
σ
where
are the experimental error variances of the completely randomized and
randomized block designs, respectively, and
are the corresponding error
degrees of freedom. This statistic may be viewed as the increase in replications that is
required if a CRD is used as compared to a RCBD if the two designs are to have the same
sensitivity. The ratio of degrees of freedom in
R
is an adjustment to reflect the different
number of error degrees of freedom in the two designs.
r
2
and
b
2
b
b
2
df
df
r
and
To compute the relative efficiency, we must have estimates of
.
We can use
the mean square for error
MS
r
2
and
E
from the RCBD to estimate
, and it may be shown [see
Cochran and Cox (1957), pp. 112114] that
b
2
±
r
Blocks
E
bM
S
b
aM
S
ab
2
11
1
=
−
+
−
−
is an unbiased estimator of the error variance of a the CRD. To illustrate the procedure,
consider the data in Example 41. Since
MS
E
= 7.33, we have
2
ˆ
7.33
b
=
and
2
(1
)
)
ˆ
1
(5)38.45
6(3)7.33
4(6)
1
14.10
Blocks
E
r
S
b
S
ab
−+
−
=
−
+
=
−
=
Therefore our estimate of the relative efficiency of the RCBD in this example is
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2
(1
)
(3
)
)
)
(15
1)(20
3) 14.10
3)(20
1)
7.33
1.87
br
df
df
R
df
df
r
b
σ
++
=⋅
=
This implies that we would have to use approximately twice times as many replicates
with a completely randomized design to obtain the same sensitivity as is obtained by
blocking on the metal coupons.
Clearly, blocking has paid off handsomely in this experiment. However, suppose that
blocking was not really necessary. In such cases, if experimenters choose to block, what
do they stand to lose? In general, the randomized complete block design has (
a –
1)(
b 
1)
error degrees of freedom. If blocking was unnecessary and the experiment was run as a
completely randomized design with
b
replicates we would have had
a
(
b 
1) degrees of
freedom for error. Thus, incorrectly blocking has cost
a
(
b 
1)
–
(
a 
1
)(b 
1)
= b 
1
degrees of freedom for error, and the test on treatment means has been made less
sensitive needlessly. However, if block effects really are large, then the experimental
error may be so inflated that significant differences in treatment means could possibly
remain undetected. (Remember the incorrect analysis of Example 41.) As a general rule,
when the importance of block effects is in doubt, the experimenter should block and
gamble that the block means are different.
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 Spring '10
 fandia

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