Multiple Taxicab Correspondence Analysis of a Survey Related to Health Services

Vartan Choulakian,Biagio Simonetti,Jacques Allard

doi:10.6339/jds.2013.11(2).1113

Abstract

We present an analysis of a health survey data by multiple cor respondence analysis (MCA) and multiple taxicab correspondence analysis (MTCA), MTCA being a robust L1 variant of MCA. The survey has one passive item, gender, and 22 active substantive items representing health services offered by municipal authorities; each active item has four answer categories: this service is used, never tried, tried with no access, non re sponse. We show that the first principal MTCA factor is perfectly charac terized by the sum score of the category this service is used over all service items. Further, we prove that such a sum score characterization always exists for any survey data.

Highlights

The data, that will be discussed in this paper, represent a survey of 3530 individuals residing in downtown eastside Vancouver with high incidence of AIDS/HIV related diseases
In a series of papers Choulakian (2003; 2005; 2006a; 2006b) developed principle component analysis (PCA) based on matrix norms, generalizing the classical PCA, or equivalently generalizing the well known singular value decomposition (SVD). This led to the development of taxicab principal component analysis (TPCA) based on the most robust matrix norm named taxicab matrix norm, and on which taxicab correspondence analysis (TCA) is based
To see that TPCA is similar to and has the same mathematical framework of classical PCA, we start with an overview of classical PCA, which can be described in many ways, see Jolliffe (2002) for a comprehensive account

Summary

Introduction

The data, that will be discussed in this paper, represent a survey of 3530 individuals residing in downtown eastside Vancouver with high incidence of AIDS/HIV related diseases. In a series of papers Choulakian (2003; 2005; 2006a; 2006b) developed principle component analysis (PCA) based on matrix norms, generalizing the classical PCA, or equivalently generalizing the well known singular value decomposition (SVD). This led to the development of taxicab principal component analysis (TPCA) based on the most robust matrix norm named taxicab matrix norm, and on which taxicab correspondence analysis (TCA) is based. The classical principal component analysis (PCA) consists of successive maximization of the variance or the square of the L2 -norm of the linear combination of the variables of the matrix T subject to a quadratic constraint; that is, it is based on the following optimization problem max ||T u||2 subject to ||u||2 = 1;

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