On the addition of further treatments to Latin Square designs
Statisticians have made use of Latin Squares for randomized trials in the design of comparative experiments since the 1920s. Through cross-disciplinary use of Group theory, Statistics and Computing Science the author looks at the applications of the Latin Square as row-column design for scientific comparative experiments. The writer presents his argument, based on likelihood theory, for an F-test on Latin Square designs. A distinction between the combinatorial object and the row-column design known as the Latin Square is explicitly presented for the first time. Using statistical properties together with the tools of group actions on sets of block designs, the author brings new evidence to bear on well known issues such as (i) non-existence of two mutually orthogonal Latin Squares of size six and (ii) enumeration and classification of combinatorial layouts obtainable from superimposing two and three symbols on Latin Squares of size six. The possibility for devising non-parametric computer-intensive permutation tests in statistical experiments designed under 2 or 3 blocking constraints seems to have been explored by the author over the candidate's research period - See Appendix V: Part 2 - for the first time. The discovery that a projective plane does not determine all FIZ-inequivalent complete sets of Mutually Orthogonal Latin Squares is proved by fully enumerating the possibilities for those of size p < 7. The discovery of thousands of representatives of a class of balanced superimpositions of four treatments on Latin Squares of size six through a systematic computer search is reported. These results were presented at the 16th British Combinatorial Conference 1997. Indications of openings for further research are given at the end of the manuscript.