Factorial Analysis of a Set of Contingency Tables
The aim of this work is to present a method of joint factorial analysis of several contingency tables. This method that we have called Simultaneous Analysis (SA), is especially appropriate to analyze frequency tables whose row margins are different, for e
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Abstract. The aim of this work is to present a method of joint factorial analysis of several contingency tables. This method that we have called Simultaneous Analysis (SA), is especially appropriate to analyze frequency tables whose row margins are different, for example when the tables are from different samples or different time points. Furthermore, SA may be applied to the joint analysis of more than two data tables in which rows refer to the same entities, but columns may be different. SA allows us to maintain the structure of each table in the overall analysis by centering each table internally with its margins, as is done in Correspondence Analysis (CA) and provides a joint description of the different structures contained within each table. Besides jointly studying the intrastructure of the tables, SA permits an overall comparison of the similarities and differences between the tables.
1 Introduction The need of jointly analyzing several contingency tables has produced several factorial methods. Some of the proposed methods consist in the analysis of the table obtained as sum of the separated contingency tables and/or the analysis of the table obtained as juxtaposition of the initial tables (Cazes (1980) and (1981)) and the Intra Analysis (Escofier (1983)). Nevertheless, in Zárraga and Goitisolo (2002) it is shown that there are situations where none of these methods permits an analysis of the similarities among rows that mantains the similarity in the analyses of the separated tables. The aim of this work is to present a factorial method for the joint analysis of several contingency tables that allows, in a similar way to correspondence analysis, the study of the similarity among the set of rows, of columns and the relations between both sets. Also cite the non symmetrical analysis (D’ Ambra and Lauro (1984) and Lauro and D’ Ambra (1989)) and more recently the Multiple Factor Analysis for Contingency Tables (Pagès and Bécue-Bertaut (2006)).
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Amaya Zárraga and Beatriz Goitisolo
2 Methodology Let T = {1, . . . ,t, . . . , T } be the set of contingency tables to be analyzed. Each of them classifies the answers of n..t individuals with respect to two categorical variables. All the tables have one of the variables in common, in this case the row variable with categories I = {1, . . . , i, . . . , I}. The other variable of each contingency table can be different or the same variable observed at different time points or in different subsamples. On concatenating all these contingency tables, a joint set of columns J = {1, . . . , j, . . . , J} is obtained. The element ni jt corresponds to the total number of individuals who choose simultaneously the categories i ∈ I of the first variable and j ∈ Jt of the second variable, for table t ∈ T. Sums are denoted in the usual way, for example, ni.t = j∈Jt ni jt , and n denotes the grand total of all T tables. In order to maintain the internal structure of each table t, SA begins by obtaining the relative frequencies of each table as usually done in CA: pitj = ni jt /
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