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Latin Squares in Practice and in Theory I

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2. Agronomy experiments: Latin squares in real life.

A field planted with a crop using five different treatments according to a 5 x 5 latin square arrangement.

Suppose we want to test the relative effectiveness of 5 different fertilizer mixtures on a crop of, say, oats. We apply the fertilizer, wait for the crop to mature, harvest it and measure the yield per unit area. But the five experiments cannot be carried out on the same plot of land. Even contiguous plots may vary in fertility because of a moisture gradient, different previous use of the land, or some other reason. Dividing a single plot into a 5 x 5 grid of subplots, and administering the fertilizers (labelled randomly A, B, C, D, E) according to a latin square arrangement like the one used in the figure above:

  A B C D E B D A E C C E D B A D C E A B E A B C D  

will give a fair test to the five.

This idea seems fairly elementary but in fact, even though latin squares had been around since the time of Euler (1707-1783) it was not until 1911 that W. S. Gosset (a research chemist who, publishing under the pen-name of ``Student'', was responsible for great advances in statistics) suggested scattering varieties to be compared in small plots over an experimental field, and not until some ten years or so later that Ronald A. Fisher devised a systematic application of latin and graeco-latin squares to the design of experiments. Fisher and Yates contains a rich set of examples, from which the one above was taken. Gossett worked for the Guiness brewery, a great consumer of barley, and applied his analytical talent to experiments aimed at improving that crop. In 1912 and 1913, working with the maltster E. S. Beaven, he designed an experimental layout with many of the properties of a latin square arrangement:

An experimental layout devised by W. S. Gosset and E. S. Beaven for testing eight varieties of barley. This figure was reproduced in Pearson where the lines separating the several blocks of eight were darkened for clarity.

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