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Title: Consequences of combining Fisher's optimal scores with biadditive models Authors:  Niel Le Roux - Stellenbosch University (South Africa) [presenting]
John Gower - The Open University (United Kingdom)
Sugnet Lubbe - University of Cape Town (South Africa)
Abstract: Fisher's optimal scores quantify a dependent categorical variable of a two-way table. The quantifications maximising the additive part of a biadditive model are found as the eigenvector associated with the largest eigenvalue satisfying an eigenequation. The non-additive part of the model can then be inspected by constructing a biadditive biplot based on a singular value decomposition (svd) of the residuals. However, this biplot is not optimal. It can be shown that the non-additive part of the model can be optimally represented using the eigenvector that is associated with the smallest eigenvalue in the above eigenequation suggesting the construction of an improved biplot for representing the non-additive part of the biadditive model. An important question is what about the intermediate solutions to the eigenvalue problem. The quantifications found from all possible solutions can be arranged as the columns of a matrix leading to a Long form of the problem parallel to the Short form version in terms of the original two-way table. Thus two routes for analysis become available: finding all solutions of the eigenequation arising from the Short form or an svd of the Long form. Links between these two routes are discussed and some extensions to existing biadditive biplots are proposed.