When might this occur? Let's consider the factory example again. In this factory you have four machines and four operators to conduct your experiment. You want to complete the experimental trials in a week. Use the animation below to see how this example of a typical treatment schedule pans out.

As the treatments were assigned you should have noticed that the treatments have become confounded with the days. Days of the week are not all the same, Monday is not always the best day of the week! Just like any other factor not included in the design you hope it is not important or you would have included it into the experiment in the first place.

What we now realize is that two blocking factors is not enough! We should also include the day of the week in our experiment. It looks like day of the week could affect the treatments and introduce bias into the treatment effects, since not all treatments occur on Monday. We want a design with 3 blocking factors; machine, operator, and day of the week.

One way to do this would be to conduct the entire experiment on one day and replicate it four times. But this would require 4 × 16 = 64 observations not just 16. Or, we could use what is called a Graeco-Latin Square.

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Graeco-Latin Squares
Section* *

We write the Latin square first then each of the Greek letters occurs alongside each of the Latin letters. A Graeco-Latin square is a set of two orthogonal Latin squares where each of the Greek and Latin letters is a Latin square and the Latin square is orthogonal to the Greek square. Use the animation below to explore a Graeco-Latin square:

The Greek letters each occur one time with each of the Latin letters. A Graeco-Latin square is orthogonal between rows, columns, Latin letters and Greek letters. It is completely orthogonal.

How do we use this design?

We let the row be the machines, the column be the operator, (just as before) and the Greek letter the day, (you could also think of this as the order in which it was produced). Therefore the Greek letter could serve the multiple purposes as the day effect or the order effect. The Latin letters are assigned to the treatments as before.

We want to account for all three of the blocking factor sources of variation, and remove each of these sources of error from the experiment. Therefore we must include them in the model.

Here is the model for this design:

\(Y_{ijkl}= \mu + \rho _{i}+\beta _{j}+\tau _{k}+ \gamma _{l}+e_{ijkl}\)

So, we have three blocking factors and one treatment factor.

and* i*, *j*, *k* and *l* all go from *1*, ... , *t* , where *i* and *j* are the row and column indices, respectively, and *k* and *l* are defined by the design, that is, *k* and *l* are specified by the Latin and Greek letters, respectively.

This is a highly efficient design with \(N = t^2\) observations.

You could go even farther and have more than two orthogonal latin squares together. These are referred to a Hyper-Graeco-Latin squares!

Fisher, R.A. *The Design of Experiments*, 8th edition, 1966, p.82-84, gives examples of hyper-Graeco-Latin squares for *t* = 4, 5, 8 and 9.

**NOTE!**It is impossible to have a 6 × 6 Graeco-Latin square! So in designing your experiment with a Graeco-Latin Square - don't have 6 treatments! Add another, or drop one!