Chapter 4 Supplemental Text Material
S4-1. Relative Efficiency of the RCBD
In Example 4-1 we illustrated the noise-reducing 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 4-4), 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
b
r
b
r
r
b
=
+
+
+
+
⋅
(
)(
)
(
)(
)
1
3
3
1
2
2
σ
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.
σ
r
2
and
σ
b
2
b
σ
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
σ
r
2
and
E
from the RCBD to estimate
, and it may be shown [see
Cochran and Cox (1957), pp. 112-114] that
σ
b
2
±
(
)
(
)
σ
r
Blocks
E
b
MS
b a
MS
ab
2
1
1
1
=
−
+
−
−
is an unbiased estimator of the error variance of a the CRD. To illustrate the procedure,
consider the data in Example 4-1. Since
MS
E
= 7.33, we have
2
ˆ
7.33
b
σ
=
and
2
(
1)
(
1)
ˆ
1
(5)38.45
6(3)7.33
4(6)
1
14.10
Blocks
E
r
b
MS
b a
MS
ab
σ
−
+
−
=
−
+
=
−
=
Therefore our estimate of the relative efficiency of the RCBD in this example is

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