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# Graeco-Latin squares

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# Graeco-Latin squares

Graeco-Latin squares and hyper Graeco-Latin squares are extensions of the basic Latin square designs where the number of blocking factors is greater than two. With three blocking factors, e.g. days, buses and bus drivers, extending the previous example, a structure is needed to control for the third blocking factor (drivers). By convention this third factor is denoted by Greek lettering and included within (superimposed upon) the basic Latin square design. If we use this approach to extend the fuel efficiency example, we again require four drivers, and the table now becomes:

 4x4 Graeco-Latin square Bus 1 Bus 2 Bus 3 Bus 4 Day 1 Aα Bβ Cγ Dδ Day 2 Bδ Aγ Dβ Cα Day 3 Cβ Dα Aδ Bγ Day 4 Dγ Cδ Bα Aβ

Notice that each cell contains a unique ordered pair of letters, with each separate letter occurring once in each row and column. Thus the Latin letters and Greek letters by themselves each determine a Latin square design. Graeco-Latin squares can be devised for k=3+ treatment levels, except for k=6. The statistical model that applies is an extension of the Latin squares version, with the extra blocking factor added into the equation:

where the new element is the Greek-ed blocking factor, G. Hyper-Graeco-Latin squares follow the same pattern, but include a fourth blocking factor.