Methods for Randomized Complete Block Designs
Randomized complete block (RCB) designs reduce error variation by us-
ing blocks that are homogeneous, have the same number of experimental
units as treatment levels, and by randomizing the treatments to the units
within the blocks. The standard notation for an RCB is illustrated in the
text. The model for an RCB design is:
Xij =µ+ti+bj+εij,
where the terms are defined as usual, in particular, εij are i.i.d. random
variables with median 0. If the errors are normally distributed with mean
0 and common variance σ2, then we test the null hypothesis H0:t1=t2=
... =tkversus Ha: Not all tiare equal, with the F statistic:
F=
b
k
P
i=1
(Xi. −X)2/(k−1)
k
P
i=1
b
P
j=1
(Xij −Xi. −X..j +X)2/[(k−1)(b−1)]
,
which has an Fdistribution with k−1 and (k−1)(b−1) degrees of
freedom under H0.
A Permutation RCB F test
The key to the permutation RCB F test is to have the permutation
method mimic the original randomization, which is within blocks. Thus,
permuted data sets are obtained by permuting the observations separately
within each block. With ktreatments, there are k! permutations possible in
each block, and since there are bblocks, there are a total of (k!)belements in
the permutation distribution for a given data set. The procedure is similar to
permutation tests we have done previously, and is outlined in the text. Note
that alternate statistics such as SST∗=
k
P
i=1
(Xi. −X)2or SSX∗=
k
P
i=1
(Xi.)2
may be used for the test. Generally a random sample from the permutation
distribution will be obtained instead of completely enumerating the distri-
bution. The text also covers a permutation HSD procedure to use with the
RCB, based on the Q∗= maxij ¯
¯Xi. −Xj.¯
¯statistic.
Friedman’s test for RCB designs
Friedman’s test is a rank-based test for an RCB design, which proceeds
by ranking the values separately within each block. In the case where there
are no ties within blocks, the Friedman statistic is:
1
