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A Latin Square design is a block design with 2 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 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 treatment factor, the model for this design is given by:

\[ Y_{ijk} = \mu + \rho_i + \beta_j + \tau_k + \epsilon_{ijk} \]where \(i, j, \in 1, \ldots k\) are the levels of the blocking factors, \(k = d(i, j) \in 1, \ldots k\) is the level of the treatment, which is dependant on the levels of the blocking factors, \(Y_{ijk}\) 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 \(X_{3k}\) and \(\epsilon_{ijk}\) is the error term.

- \(k\)
- The number of treatment/blocking levels.
- Treatment Labels (optional)
- A comma seperated list of treatment names. The number of treatment names must equal \(k\).
- 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 "Latin square and related designs"
- Penn State "The Latin Square"