Estudi de les inèrcies estructurals en anàlisis de correspondències. Aportacions per a una millora de les anàlisis

Resumen

The research work memory is structured in a first chapter with the description of the objectives of the doctoral thesis and the research work organization. Then, the Chapter 2 Descriptive factorial analysis is exclusively dedicated to present the tools used in factorial analysis. A general analysis, the canonical analysis, to compare two groups of variables is introduced, and then it is generalized to more than two questions. We present correspondence analysis, simple and multiple, as a particular case of canonical analysis, but we present both methods also from a more classical point of view. We also introduce graphical techniques -symmetric displays and Biplots- and the inertia concept. Chapter 3 Loglinear and graphical models introduces briefly loglinear models and graphical models, the conditional independence concept and its use. Next, we develop the loglinear model formula and the constraints that characterize the model zero sum or zero corner treatment- and the transition relations. We make a study of the influence of the parameters over the model generation, especially about the significance of interaction terms over principal and independent terms. In this chapter we introduce the deviance, as a likelihood ratio between two models, its expression and its relation with the ?2 statistic and other model divergence indicators. In Chapter 4 Study of the inertias in correspondence analysis we start with the relation among inertias, the contingence coefficient and deviance. Next, we study the inertia decomposition as the contribution of each individual, category or variable. Its applied to study the case of square matrices. With reference to the multiple correspondence analysis of the Burt table, we decompose inertia by blocks and we investigate the influence of the blocks of the diagonal. A summary of several treatment methodologies is done. We propose a new methodology to treat squared skew-symmetric matrices, based on a double decomposition, on the one hand, in the symmetry-skew-symmetry analysis and on the other using the factorial reconstitution of the symmetric part, based on a k-EM algorithm, where k is the reconstitution order. The k-EM reconstitution methodology can be applied to the diagonal tables of a Burt table, which leads a result equivalent to Joint Correspondence Analysis. In Chapter 5 - MCA with respect to a model and conditional MCA- firstly we start presenting MCA with respect to a model. Secondly we discuss conditional MCA, where an external variable plays a partitioning role. The contribution is based on studying the inertia and its decomposition in two parts linked to the external conditioning variables, the inter and intra inertia. Next, we find their distribution function and, by means of this, we can test the significance of the conditioning variable. Therefore, using the conditional MCA and loglinear models introduced before, we study the inertia behaviour with relation to the model and with different relation levels between the variables. The Chapter 6 -Multiconditional MCA- consideres problems generated by the implementation of multiple conditional analysis, since it is not possible to generalize this definition to the simple conditional case. By means of the study of conditional inertias and loglinear models, we develop our proposal for a multiconditional analysis. These results are compared with the results obtained in a loglinear modelling process. We apply our proposal to a data example in Chapter 7. The dissertation ends with a chapter of conclusions which sums up our principal contributions and suggests some future research lines in this field of investigation and with the macros programmed and used.