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A Greaco-Latin Square design is the combination of two orthoganal Latin Squares. This is done to create a block design with 3 blocking factors. Each blocking factor has the same number of levels as there are treatments, \(k\).

This design assumes that: the number of levels of each blocking factor equals the number of treatments (\(k\)) and that there is no interaction between the blocking variables or between the treatment and each blocking variable.

If your experiment satisfies these assumptions, the Greaco-Latin Squares design provides you with a very efficient method requiring only \(k^{2}\) observations.

If we let \(X_{1}\) be the first blocking factor, \(X_{2}\) be the second blocking factor, and \(X_{3}\) be the third blocking factor, and \(X_{4}\) be the treatment level, the model for this design is given by:

\[ Y_{ijkl} = \mu + \rho_{i} + \beta_{j} + \tau_{k} + \gamma_{l} + \epsilon_{ijkl} \]where \(i, j, \in 1, \ldots k\) are the levels of the first two blocking factors, \(k, l \in 1, \ldots k\) are the levels of the third blocking factor and the treatment, which are dependant on the levels of the blocking factors, \(Y_{ijkl}\) is the response, \(\mu\) is the overall mean, \(\rho_{i}\) is the effect of \(X_{1i}\), \(\beta_{j}\) is the effect of \(X_{2j}\), \(\tau_{k}\) is the effect of \(X_{3k}\), \(\gamma_{l}\) is the effect of \(X_{4l}\), and \(\epsilon_{ijkl}\) is the error term.

- \(k\)
- The number of treatment/blocking levels.
- Seed (optional)
- If the design is to be randomized, the seed for the random number generator. Use this to ensure repeatability of the randomization.

- Box, G. E. P., Hunter, W. G., and Hunter, S. J. (1972),
*Statistics for Experimenters,*New York: John Wiley & Sons, Inc.. - National Institute of Standards and Technologies "Graeco-Latin square designs"
- Penn State "What do you do if you have more than 2 blocking factors?"