Taxicab Correspondence Analysis of Ratings and Rankings

Choulakian, Vartan

Taxicab Correspondence Analysis of Ratings and Rankings
[Analyse des Correspondances du Taxi de Notes et de Rangs]

Choulakian, Vartan

Journal de la société française de statistique, Tome 155 (2014) no. 4, pp. 1-23.

Résumé
Abstract

Soit $Y$ un tableau de notes sur $I \times Q;$ où $I$ est un ensemble d’individus, et la $i$ ème ligne représente les notes attribuées par l’individu $i$ sur $Q$ variables ou attributs. Dans cet article nous étudions deux codages du tableau $Y$ avant de le traiter par analyse des correspondances (AC) ou analyse des correspondances du taxi (ACT), ACT étant une variante robuste de AC basée sur la norme L $_{1}$ : Le tableau dédoublé $Y_{D}$ de dimension $I \times 2 Q,$ et le tableau Y $_{n e g a}$ de dimension $I \times (Q + 1)$ où une colonne nommée $n e g a$ est ajoutée à $Y$ representant la note complémentaire globale. L’interprétation des diagrammes du tableau $Y_{D}$ par AC ou ACT est basée sur le principe du bras de levier. Nous utilisons la loi de contradiction pour interpréter les diagrammes du tableau Y $_{n e g a}$ par AC ou ACT. Une condition nécessaire et suffisante pour que l’analyse du tableau Y $_{n e g a}$ par ACT et l’analyse du tableau Y $_{D}$ par ACT soient equivalentes est que le 1er facteur est une fonction affine du total de notes. Et si cette condition est satisfaite, suivant Cox, nous utilisons le 1er facteur comme un résumé de la variable latente. Cette inférence peut être de deux sortes, faible ou forte. Dans le cas de données des rangs représentant des préférences individuelles, la méthode correspond à la règle de Borda ou une version modifiée. Deux exemples de natures différentes sont exposés.

Let Y be an $I \times Q$ ratings data set, where $Q$ represents the number of items, and $I$ represents the number of rated objects or the number of individuals expressing their opinions on the $Q$ items. This paper considers two kinds of data codings before the application of correspondence analysis (CA) or taxicab correspondence analysis (TCA), where TCA is a L $_{1}$ variant of CA: the doubled data set Y $_{D}$ of size $I \times 2 Q,$ and the data set Y $_{n e g a}$ of size $I \times (Q + 1)$ where a column named nega is added representing the cumulative complementary columns. The interpretation of maps in CA of Y $_{D}$ is based on the lever principle. We use the law of contradiction to interpret maps of CA and TCA of Y $_{n e g a}$ . We provide necessary and sufficient conditions for TCA of Y $_{n e g a}$ or Y $_{D}$ so that the first factor score is an affine function of the sum score of the ratings; and, if this is true for a dataset, then following Cox we suggest the use of the sum score of ratings either to reduce the $Q$ ratings into a single index, or to summarize the underlying latent variable. This ordinal inference can be of two types: weak or strong. In the case of a rankings dataset, the proposed approach corresponds to Borda count rule or modified Borda count rule. Examples are provided.

Zbl

Keywords: Sum score, nega, doubling, lever principle, law of contradiction, personal equation, response styles, rogue items, strategic voters, Borda count, Nishisato mapping, taxicab correspondence analysis, IRT
Mot clés : Total de notes, nega, dédoublement, principe du bras de levier, loi de contradiction, équation personnelle, point aberrant, règle de Borda, codage de Nishisato, analyse des correspondances du taxi, analyse d’items

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Choulakian, Vartan. Taxicab Correspondence Analysis of Ratings and Rankings. Journal de la société française de statistique, Tome 155 (2014) no. 4, pp. 1-23. http://archive.numdam.org/item/JSFS_2014__155_4_1_0/

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