1 Randomized Complete Block (RCB) Design
We use blocking whenever possible in an experiment to increase power and precision by reducing error
variance. The model for the RCB design is:
yij =µ+αi+βj+εij,
where αiis the treatment effect (usually a fixed effect) and βjis the block effect (usually a random
effect). Thus the analysis of an RCB design is similar to a CRD with 2 factors and no replication, and we
use the treatment-by-block interaction as the error term to test for the treatment effect. If the treatments
and blocks do interact, this lowers the power of the test for the treatment, and complicates interpretation of
the data. We can use our standard residual-by-predicted plot to try to detect some forms of treatment-block
interaction. We can attempt to remove a treatment-by-block interaction by transforming the response, either
via an optimal Box-Cox transformation or using Tukey’s 1 degree of freedom method. See the SAS code
for this lecture for an example. As discussed in the text, we do not formally test for a difference among
blocks, even though an F test will be calculated by most computer programs. Rather than test for a block
difference, we can quantify the effectiveness of blocking using the concept of relative efficiency.
1.1 Relative Efficiency
One way to measure the effectiveness of the blocking is by comparing the variance estimate from our RCB
design to an estimate of what the variance would have been with a CR design. As shown in the text, the
formula for this relative efficiency is:
ERCB:C RD =(νrcb + 1)(νcrd + 3)
(νrcb + 3)(νcrd + 1)
σ2
crd
σ2
rcb
,
where σ2
rcb is estimated by the variance estimate from the RCB analysis, and σ2
crd is estimated using
mean squares from the RCB analysis, as shown in the text. The larger the relative efficiency, the more
effective the blocking has been.
1.2 Related Randomized Block Designs
Variants of the basic RCB design above are often encountered. For one example, suppose we had randomized
our flavored milk subjects (from the example discussed in lecture) also to a type of cookie. Then we would
have a factorial treatment structure in each block, this is often called a randomized block factorial
design.
For a different type of variation on the RCB, suppose that we divide people into age strata (15-25, 26-35,
36-45, ... ). Suppose that we obtain subjects for each stratum, and randomize them so that each subject
drinks one of the three types of flavored milk, and an equal number of subjects in each age stratum drinks
each type of flavored milk. Now we are blocking on age, but we have replication of the treatments in each
block. This type of design is called a randomized block design with replication, or a generalized
randomized block (GRCB) design. The analysis of a GRCB design proceeds like the analysis of a
CRD with factorial treatment structure with replication, including a test for interaction between blocks and
treatments.
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