Doctor of Philosophy
School of Mathematics and Applied Statistics
Beh, Eric John, Correspondence analysis using orthogonal polynomials, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 1998. https://ro.uow.edu.au/theses/2044
Simple correspondence analysis is a multivariate statistical technique and is a method of visualising the categories of a two-way contingency table, while multiple correspondence analysis is a method of visualising the categories of a multi-way contingency table. However, the technique completely ignores any ordinality that may exist within a set of categories. Likewise, the classical Pearson chi-squared test ignores this ordinal structure.
This thesis presents a decomposition of the Pearson chi-squared statistic for multi-way contingency fables by generalising the decomposition of the two-way Pearson chi-squared statistic developed in the past. The advantage of these new statistics is that they enable a detailed investigation of the nature of the association between two or more categorical variables, of which one or more has an ordinal structure.
A new method of graphically displaying the categories of a two-way contingency table is developed and is conducted by using the decomposition of its Pearson chi-squared statistic. This method takes into consideration the ordinal nature of any underlying variable and enables a more informative and easier interpretation than the classical approach. This new method of correspondence analysis is then extended so that the categorical variables of a three-way and more generally any multi-way contingency table, with one or more of these variables being ordinal, can be analysed. The new Pearson chi-squared decompositions are employed for this analysis.
The new technique of correspondence analysis is shown to be applicable in a broader context, by analysing data sets that are not in the form of a contingency table. For example, this thesis focuses on the application of the new technique to rank type data, which enables the researcher to visualise the association between the products tested and the rankings they received.
A new approach to parameter estimation using orthogonal polynomials for log-linear analysis is also presented and is a much simpler method of model-fitting than the widely used technique. The advantage of this n e w approach is that parameter estimates higher than the first order can be easily calculated, thereby providing the researcher with a better fitting method than by using the techniques used in the past.