Statistics Review - Mathematics and Statistics for the Environment - page 145

There is usually no reason for testing blocks, but the size of the block mean square relative to the
mean square for error does give an indication of how much precision was gained by blocking. If the
block mean square is large (at least two or three times as large as the error mean square) the test is
more sensitive than it would have been with complete randomization. If the block mean square is
about equal to or only slightly larger than the error mean square, the use of blocks has not improved
the precision of the test. The block mean square should not be appreciably smaller than the error mean
square. If it is, the method of conducting the study and the computations should be re-examined.
In addition to the assumption of homogeneous variance and normality, the randomized block
design assumes that there is no interaction between treatments and blocks; that is, that differences
among treatments are about the same in all blocks. Because of this assumption, it is not advisable to
have blocks that differ greatly, as they may cause an interaction with treatments.

7.16.4 l atin s quare d esign
In the randomized block design the purpose of blocking is to isolate a recognizable extraneous
source of variation. If successful, blocking reduces the error mean square and gives a more sensi-
tive test than could be obtained by complete randomization. In some situations, however, we have
a two-way source of variation that cannot be isolated by blocks alone. In an agricultural field, for
example, fertility gradients may exist both parallel to and at right angles to plowed rows. Simple
blocking isolates only one of these sources of variation, leaving the other to swell the error term and
reduce the sensitivity of the test.
When such a two-way source of extraneous variation is recognized or suspected, the Latin square
design may be helpful. In this design, the total number of plots or experimental units is made equal
to the square of the number of treatments. In forestry and agricultural experiments, the plots are
often (but not always) arranged in rows and columns, with each row and column having a number
of poles equal to the number of treatments being tested. The rows represent different levels of one
source of extraneous variation while the columns represent different levels of the other source of
extraneous variation. Thus, before the assignment of treatments, the field layout of a Latin square
for testing five treatments might look like this:

Columns

1 2 3 4 5

1

2

3

4

5

Treatments are assigned to plots at random, but with the very important restriction that a given
treatment cannot appear more than once in any row or any column. An example of a field layout of
a Latin square for testing five treatments is given below. The letters represent the assignment of five
treatments (which here are five species of hardwoods). The numbers show the average 5-year height
growth by plots. The tabulation below shows the totals for rows, columns, and treatments: