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# Greaco-Latin Squares Design

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.

## Design Parameters

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## Design Parameter Explanation

$$k$$
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.

## Design

No design has been generated yet.

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