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Latin Squares in Practice and in Theory I

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3. The statistical analysis of a latin-square experiment

Ronald A. Fisher realized that latin squares could be abstracted from the partition of growing plots and applied to the elimination of systematic error in a much more general context.

In the 2-dimensional plot of land, the systematic error due to variation in soil, etc. can be minimized by a suitable latin square partition of the plot. More generally, whenever there are two independent factors that may introduce systematic error into an experiment, a latin square arrangement in ``experiment space'' can compensate for these errors. (Fisher also showed how graeco-latin and ``hyper graeco-latin" squares could be applied to more complex experiments; see Fisher).

The following example is adapted from what was D. H. Kim's Stat 470 website at the University of Michigan. The data set is taken from The Design and Analysis of Experiments by Douglas C. Montgomery (Wiley).

The experiment is to study the burning rate of five different formulations of a rocket propellant. The formulations are mixed from raw material that comes in batches whose composition may vary. Furthermore, the formulations are prepared by several operators, and there may be differences in the skills and experience of the operators. So in this experiment there are two presumably unrelated sources of systematic error: different batches and different operators.

To compensate for these systematic errors by a latin square design, five operators are chosen at random, and five batches of raw material are selected at random, each one large enough for samples of all five formulations to be prepared. A sample from the one of the five batches (labelled at random I, II, III, IV, V) is assigned to one of the five operators (labelled at random 1, 2, 3, 4, 5) for preparation of one of the five formulations (labelled at random A, B, C, D, E) according to the following latin square arrangement; the table also contains the observed burning rate for that formulation of that sample.

IA   24B   20C   19D   24E   24
IIB   17C   24D   30E   27A   36
IIIC   18D   38E   26A   27B   21
IVD   26E   31A   26B   23C   22
VE   22A   30B   20C   29D   31

The calculational scheme used to analyze these data goes as follows.

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