We have one more observation for each treatment and the six remaining plots occur two in each of the
three blocks. We therefore have no choice but to add two different treatments to each block. The choice of
which pair of treatments to duplicate together is arbitrary; the treatments that are duplicated together will
be slightly more precisely compared than other treatment pairs. The resulting block-treatment allocation is
then


Block 1 treatments 1 2 3 4 5 6 1 2

Block 2 treatments 1 2 3 4 5 6 3 5

Block 3 treatments 1 2 3 4 5 6 4 6


When the treatments are randomized in each block all the eight "treatments" listed above are considered
equally. The resulting randomization could look like


 Plot 1 2 3 4 5 6 7 8

Block 1 5 1 4 2 1 3 2 6

Block 2 5 3 1 5 3 6 4 2

Block 3 6 4 4 1 2 6 5 3


Randomization always produces some odd-looking patterns and provided the blocking system correctly
identifies the underlying pattern of plot-to-plot variation any randomisation is acceptable. If some
randomisation results make us uncomfortable then the answer is to redefine the blocking system, not to try
another randomisation. For example, in the above case we could decide to work with six blocks of four
plots (a half-row per block) or to impose a column classification as well as a row classification. Neither of
these would be appropriate here, I believe, since the rows do genuinely appear to be the most appropriate
definition of blocks.

The range of standard errors for estimated treatment differences is

0.707 a to 0.718 a with a mean of 0.717 a.

Compared with the minimum possible S.E. of 0.707 c the use of the correct form of blocking has produced
virtually no penalty of variable precision.

2.5 Fourteen Treatments in Blocks of 4, 5 and 6

The experiment includes 14 treatments, being seven varieties combined with two levels of nitrogen. The
important treatment comparisons will be the difference between the two nitrogen levels for each variety
and the differences between varieties for each nitrogen level. The nitrogen main effect is well-known and
does not need to be reconfirmed.