Generalized Latin squares

For comparing two treatments t and ti (such as tissue types or cell lines), both Kerr et al. (2000) and Hsu et al. (2002) recommended the design of two-channel microarray experiments as generalized Latin squares with an even number a of arrays and two dyes, as follows. 

In this plan, the effects of both automobile and wheel position are controlled by blocking. It should, however, be kept in mind that for the Latin square design, as for other blocking plans, it is generally assumed that the blocking variables do not interact with the primary variable to be evaluated. 

Meals were consumed by the subjects following a Latin Square Design. Statistical analysis was performed by a General Linear Models Procedure (20) using split-plot in time analysis with the following non-orthogonal contrasts ... 

There is the possibility of carry-over effects. This is more crucial in Latin square and other cross-over designs. Knowledge of pharmacokinetics and metabolism of a compound under study generally helps in avoiding this problem. 

The fractional factorial designs, including the Latin squares, are generally used for screening possible experimental variables in order to find which are the most important for further study. Their use is subject to some fairly severe assumptions which should be known and taken into consideration when interpreting the data ... 

Although this direct method is more adequate for the given example, because the number of the values that are not available are smaller than the sum of rows and columns, the constant method has also been demonstrated for the case of comparison. It should be noted that both methods are generally used in two-way classification such as designs of completely randomized blocks, Latin squares, factorial experiments, etc. Once the values that are not available are estimated, the averages of individual blocks and factor levels are calculated and calculations by analysis of variance done. The degree of freedom is thereby counted only with respect to the number of experimental values. Results of analysis of variance for this example are... 

The model for a general mxm Latin square with one observation in the cell is ... 

The design of experiment written in this form is a reconstructed Latin square design where one of the diagonals has been left out. Generally speaking, Youdens square is a symmetrically balanced incomplete random block where each factor level appears once and only once in each block position. 

Sample size and treatment choice are key design questions for general multifactor experiments. Authors have proposed the use of standard factorial experiments in completely randomized designs, block designs, or Latin squares (see, for example, Chapter 6 and Churchill, 2003). However, the unusual distribution of gene expression data makes one question the relevance of standard orthogonal factorial experiments in this context. 

The variance analysis for a plan with Latin squares is not different from the general case previously discussed in Section 5.6.3. Therefore we must compute the following sums ... 

We have discussed the 4x4 Latin Square, but there is in general no restriction upon square size. Designs are available for si2es up to 12 X 12. 

It can be seen that both the Greek and Latin characters form Latin squares. Every Graeco-Latin square consists of two Latin squares that are orthogonal, that is, that every pair occurs exactly once. (Graeco-Latin alphanumeric Sudoku puzzles have appeared, but are not as yet popular.) In general, there are n — 1 solutions above, where = 3, one has the A, B, C and the a, /3, y sets. With = 4, there are three solutions ... 

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