Graeco-Latin square designs

If there are three types of blocking factors, Graeco-Latin square designs can be used to minimize their effects. The following is a 4 x 4 Graeco-Latin square. What do a, p, y, and o represent ... 

Table 18.2 Experimental design expressed as a Graeco Latin square design ...

Graeco-Latin square an experimental design which permits study of the effects of 4 factors at n levels in n2 runs (n > 4). 

Table 2.68 i Design of experiment of 8 x 8 Graeco-Latin square ... 

The data given below are results of 25 design points performed at five temperatures and with five different time periods, with the idea of establishing effects of the given factors on conversion in a chemical reactor. To avoid inequality effects, five chemical reactors and five operators were included in the experiment. So, 25 design points were done in five reactors with five operators by design of experiment of a 5x5 Graeco-Latin square in such a way that each operator used each reactor only once at each temperature and for a constant conversion time period. Characters denote reactors and numbers the operators. Do the analysis of variance. 

The design can be written in two ways as a Taguchi style experiment (see, for example, Breyfogle, 2000) as shown, in Table 18.1 and as a modified Graeco Latin square (see, for example, Cochran and Cox, 1957) as shown in Table 18.2. Notice that each row and column in Table 18.2 has equal numbers of 2 h and 4 h lower dwell time and wet and dry environments. Other designs (see, for example Metcalfe, 1994, and Wu and Hamada, 2000), such as central composites and star designs exist but would require more trials to obtain the same balance (see Appendix III). 

Example 2, which uses four factors at three levels, is from a family of multi-level designs formally known as hyper-Graeco-Latin squares. A Latin square is an x square grid filled with symbols (letters or numbers) so that each symbol appears once and only once in each row or column. When n = 9 and the symbols ate numbers, the result is the familiar Sudoku puzzle. The concept appears to have been devised by Arabic mathematicians in the Middle Ages. The simplest n = 3 is... 

The more than three-dimensional designs are referred to as hyper-Graeco-Latin squares, or a complete set of mutually orthogonal Latin squares. In Example 2, the use of four factors at three levels adds the prefix hyper. ... 

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