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Details of the FACTEX Procedure |

See FACTEX7A in the SAS/QC Sample Library |

A *q*×*q* Latin square
is an arrangement of *q* symbols, each
repeated *q* times, in a square of side *q* such that each symbol
appears exactly once in each row and in each column. Such arrangements
are useful as designs for *row-and-column* experiments, where it is
necessary to balance the effects of two *q*-level factors
simultaneously.

A Graeco-Latin square is actually a pair of Latin squares; when superimposed, each symbol in one square occurs exactly once with each symbol in the other square. The following is an example of a 5×5 Graeco-Latin square, where Latin letters are used for the symbols of one square and Greek letters are used for the symbols of the other:

Whenever *q* is a power of a prime number, you can
construct up to *q*-1 squares, each
with *q* symbols that are balanced over all the other
factors. The result is called a *hyper-Graeco-Latin Square*
or a complete set of *mutually orthogonal* Latin squares. Such
arrangements can be useful as designs
(refer to Williams 1949), or they can be used to construct other
designs.
When *q* is a prime power, hyper-Graeco-Latin squares are straightforward to
construct with the FACTEX procedure.
This is because *a
complete set of q-1 mutually orthogonal q×q Latin squares is
equivalent to a resolution 3 design for q+1 q-level factors in
q^{2} runs, where two of the factors index rows and columns and each of
the remaining factors indexes the treatments of one of the squares.*

For instance, the following statements generate a complete set of three mutually orthogonal 4×4 Latin squares, with rows indexed by the factor ROW, columns indexed by the factor COLUMN, and the treatment factors in the respective squares indexed by T1, T2, and T3. The first step is to construct a resolution 3 design for five four-level factors in 16 runs.

proc factex; factors row column t1-t3 / nlev=4; size design=16; model resolution=3; output out=graeco t1 cvals=('A' 'B' 'C' 'D') t2 cvals=('A' 'B' 'C' 'D') t3 cvals=('A' 'B' 'C' 'D'); run;

In most cases, the form that appears in the output data set GRAECO is most useful. The form that usually appears in textbooks is displayed in Output 15.10.1, which can be produced using a simple DATA step (not shown here).

Square 1 : A D B C D A C B B C A D C B D A Square 2 : A D B C C B D A D A C B B C A D Square 3 : A D B C B C A D C B D A D A C B |

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