Abstract
This chapter has a methodological objective. We deal with the following issues:
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References
Benzécri, J.P., et al. (1973). L’analyse des données, 2. L’analyse des correspondances. Paris: Dunod.
Benzécri, J.P. (1977). Analyse discriminante et analyse factorielle. Les Cahiers de l’Analyse des Données, 2(4), 369–406.
Benzécri, J.P. (1992). Correspondence analysis handbook. New York: Dekker.
Hardin, R. (2002). Trust and trustworthiness. New York: Russell Sage Foundation.
Le Roux, B. (2014a). Analyse géométrique des données multidimensionnelles. Paris: Dunods.
Le Roux, B. (2014b). Structured data analysis. In J. Blasius & M. Greenacre (Eds.), Visualisation and verbalisation of data (pp. 185–204). London: Chapman & Hall.
Le Roux, B., & Cassor, F. (2015). Assigning Objects to Classes of a Euclidean Ascending Hierarchical Clustering. In International symposium on statistical learning and data sciences (pp. 389–396). Springer.
Le Roux, B., & Perrineau, P. (2011). Les différents types d’électeurs au regard de différents types de confiance (Vol. 54, pp. 5–35). Les Cahiers du CEVIPOF. Retrieved from http://www.cevipof.com/fr/lespublications/lescahiersducevipof/.
Le Roux, B., & Rouanet, H. (2004). Geometric data analysis: From correspondence analysis to structured data analysis. Dordrecht: Kluwer.
Le Roux, B., & Rouanet, H. (2010). Multiple correspondence analysis (Series QASS) (Vol. 163). Thousand Oaks: SAGE.
Rouanet, H., Bernard, J.M., Bert, M.C., Lecoutre, B., Lecoutre, M.P., & Le Roux, B. (2000). New ways in statistical methodology (2nd ed.). Berne: Peter Lang.
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Appendix
Appendix
Doubling Procedure
The doubling procedure consists in assigning two scores per individual instead of a single score (see Benzécri et al. 1973, pp 25–27). We coded the four levels of each scale with (3,0), (2,1), (1,2) and (0,3), respectively, and the two levels of dichotomous question with (1,0), (0,1).
For example, in our study, the table is doubled with a “trust pole” (denoted q+) and a “distrust pole” (denoted q–) for each question.
Weighting Procedure
In order to balance each heading we weight each one here with the inverse of the number of questions the heading contains; consequently, the weight of a question is that of its heading.
Technically, the weighting procedure for CA consists in multiplying the two columns of each question by the weight.
Correspondence Analysis After Doubling

In the CA of a doubled table (Le Roux 2014a: 206–209) all the rows of the table have the same total, hence each individual receives the same weight.

In the CA display, we obtain two points for each question and one point for each individual. The line joining the two poles of one question passes through the origin (as shown in Fig. 19.1 for the question “trust in police”), and the distances of poles to origin are inversely proportional to the means of the scores of the pole.

The distance between two individuals resulting from a question depends on the number of levels between their answers. The global distance is a weighted mean of the distances resulting from questions.

The weight of the question has the following property: The greater the deviation from the mean of the question to the midpoint of the scale, the greater the weight of the question.
Nevertheless, it worth noting that, for a dichotomous question, the formula is exactly the same as that of MCA (Le Roux and Rouanet 2010: 35).

The contribution of a question to the variance of cloud depends on:
the dispersion of responses around the mean;
the number of levels of the scale (questions will preferably be coded on scales with the same number of levels).
To sum up: Questions with extreme answers create greater distances, and questions exhibiting great agreement make a small contribution to the total variance.
For examples of CA after doubling, see Benzécri (1973, three studies in Tome 2, Part C) and Benzécri (1992: chapter 12).
Structured Data Analysis
A structuring factor generates a partition of the cloud of individuals (Le Roux and Rouanet 2004; Le Roux 2014b: Chapter 12). By plotting the mean point of each subcloud, we get a derived cloud of mean points whose variance defines the betweenvariance of the partition. The average variance of the subclouds defines the withinvariance of the partition.
The coefficient η^{2} (eta square: squared coefficient ratio) is equal to the betweenvariance divided by the total variance (between plus within).
Useful geometric summaries of subclouds in the plane are provided by concentration ellipses (Le Roux and Rouanet 2010: 69–71).
Trajectory of Questions (Coordinates of the Poles of Questions in CA)
The transition formulas for doubling CA are similar to the ones of MCA. The location of the pole of a question can be reconstituted from the cloud of individuals.
Given an axis, for each pole and each question, we compute the coordinates as follows. The coordinate of q ^{+} is equal to the weighted mean of the coordinates of individuals divided by the standard deviation of the axis (\( \sqrt{\lambda } \)). The weight of an individual (denoted ) depends on the coded score of the individual to the question, denoted , and of the mean () of the coded scores of the questions. For the pole q ^{−}, the width of the scale (denoted e) intervenes in the formula (here, e = 3).
The coordinates of the two poles of each question are calculated from the individuals of each supplementary survey; we can then depict the trajectory of each question.
Assigning Object to Clusters of a Euclidean AHC
In a Euclidean AHC (Ward’s method), the usual method for allocating a supplementary object to a cluster is based on the geometric distance from the objectpoint to the barycenter of the cluster. The main drawback of this method is that it does not take into consideration that clusters differ with regard to weights, shapes, and dispersions. Neither does it take into account successive dichotomies of the hierarchy of classes. This is why we propose a new ranking rule adapted to geometric data analysis that takes the shape of clusters into account.
From a set of supplementary objects, we propose a strategy for assigning these objects to clusters stemming from an AHC. The idea is to assign supplementary objects at the local level of a node to one of its two successors until a cluster of the partition under study is reached. We define a criterion based on the ratio of Mahalanobis distances from the objectpoint to barycenters of the two clusters that make up the node (Le Roux and Cassor 2015; Le Roux 2014a, b; Benzécri 1977).
R script Interfaced with Coheris SPAD is available from the authors.
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Cassor, F., Le Roux, B. (2019). Assigning Changes Over Time Using Geometric Data Analysis Methods: Application to the French “Barometer of Political Trust”. In: Blasius, J., Lebaron, F., Le Roux, B., Schmitz, A. (eds) Empirical Investigations of Social Space. Methodos Series, vol 15. Springer, Cham. https://doi.org/10.1007/9783030153878_19
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