Latin Squares and Two-Level Fractional Factorial Designs

Abstract

In an industrial setting one is often challenged to find minimum run designs that will result in maximal information and can be readily understood by all involved. The following discussion considers an example where the design was constructed such that it was both a Latin square and a fractional factorial in order to optimize analysis options as well as presentation options. For an M × M Latin square if M = 2k, then the three factors (rows, columns, and treatments) can also be studied using the same number of runs and a 23k–k fractional factorial design. The advantage to this is that the results can be easily analyzed using Yates' algorithm and the confounding pattern is readily available. While every Latin square corresponds to a subset of the points in a complete 23k design, only one standard square corresponds to a 23k–k fractional factorial design, and all the others correspond to designs where the contrasts are not all orthogonal. Thus, it may not always be desirable to choose a Latin square at random. An example using a Latin square/fractional factorial design to study a fermentation process is discussed.