Statistical Methods in Medical Research 4ed

9.4 Latin squares
Suppose we wish to compare the effects of a treatments in an experiment in
which there are two other known sources of variation, each at a levels. A
complete factorial design, with only one observation at each factor combination,
would require a 3 observations. Consider the following design, in which a ˆ 4.
The principal treatments are denoted by A, B, C and D, and the two secondary
factors are represented by the rows and columns of the table.
Column
Row 1 2 3 4
1 D B C A
2 C D A B
3 A C B D
4 B A D C
9.4 Latin squares 257
Only a 2 …ˆ 16† observations are made, since at each combination of a row and a
column only one of the four treatments is used. The design is cunningly
balanced, however, in the sense that each treatment occurs precisely once in
each row and precisely once in each column. If the effect of making an observation
in row 1 rather than row 2 is to add a constant amount on to the
measurement observed, the differences between the means for the four treatments
are unaffected by the size of this constant. In this sense systematic
variation between rows, or similarly between columns, does not affect the
treatment comparisons and can be said to have been eliminated by the choice
of design.
These designs, called Latin squares, were first used in agricultural experiments
in which the rows and columns represented strips in two perpendicular
directions across a field. Some analogous examples arise in medical research
when treatments are to be applied to a two-dimensional array of experimental
units. For instance, various substances may be inoculated subcutaneously over a
two-dimensional grid of points on the skin of a human subject or an animal. In a
plate diffusion assay various dilutions of an antibiotic preparation may be
inserted in hollows in an agar plate which is seeded with bacteria and incubated,
the inhibition zone formed by diffusion of antibiotic round each hollow being
related to the dilution used.
In other experiments the rows and columns may represent two identifiable
sources of variation which are, however, not geographically meaningful. The
Latin square is being used here as a straightforward generalization of a randomized
block design, the rows and columns representing two different systems of
blocking. An example would be an animal experiment in which rows represent