Multiple Factor Analysis by Example Using R

Jérôme Pagès

• Publisher: Chapman and Hall/CRC
• Year: 2014
• Print ISBN: 9780367241032
• eBook ISBN: 9781498786690
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Multiple factor analysis (MFA) enables users to analyze tables of individuals and variables in which the variables are structured into quantitative, qualitative, or mixed groups. Written by the co-developer of this methodology, Multiple Factor Analysis by Example Using R brings together the theoretical and methodological aspects of MFA. It also inc

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Principal Component Analysis

Principal component analysis (PCA) is the most widely used factorial method. It is applied to tables in which a set of (statistical) individuals is described by a set of quantitative variables. In this chapter, we present a detailed description of this method, both in theory and in practice. This is the perfect opportunity to introduce a number of concepts used to analyse multiple tables, but also apply to simple ones. This enables the reader to see the specificities of multiple factor analysis (MFA) better.

Vocabulary: Factor Analysis or Factorial Analysis?

Both families of methods are very similar, which explains the confusion between the two names. Roughly, we can say that factor analysis is based on a model whereas factorial analysis is purely exploratory. Typically, principal component analysis is a factorial analysis. The same is true of correspondence analysis and all the methods presented in this book. In this case, why do we say multiple factor analysis? It is simply a mistranslation of a method originally introduced in French under the name Analyse factorielle Multiple. Now, multiple factor analysis is widespread and, in our opinion, changing its name to multiple factorial analysis would lead to yet more confusion.

1.1 Data, Notations

We start by studying a table with the following characteristics:

• – Each row represents a statistical individual; we denote I the number of individuals. I also designates the set of individuals. Using the same letter to designate both the set and its cardinal is not confusing as the meaning is always clear from the context.
• – Each column represents a quantitative variable; K represents the number of variables (as well as the set of variables).
• – At the intersection of row i and column k, is x_ik, the (numerical) value of individual i for variable k.

Let us add two classical notations:

• ${\overline{x}}_k$: Mean of variable k; this is not used much as the variables are generally centred, but it can sometimes be useful for the centring to appear explicitly;
• s_k: The standard deviation of variable k.

These notations are brought together in Figure 1.1

FIGURE 1.1
Data structure and notations.

An endless variety of data can be analysed using PCA. In the following paragraphs, we use an example which is both rich and easy to understand. We have the final Baccalaureate grades for 909 high school students specialising in science (I = 909), for 5 given subjects (K = 5): mathematics, physics, natural sciences, history–geography and philosophy.

1.2 Why Analyse a Table with PCA?

Let us go back to the previous example. Generally, once the means have been examined, the aim of studying such a table statistically is to investigate the students’ ‘within-subject’ and ‘between-subjects’ diversity. This diversity must first be examined by subject using indicators (mainly standard deviations) and graphs (mainly boxplots and histograms).

The choi...

Table of contents

• Cover
• Series Page
• Published Titles
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Principal Component Analysis
• 2 Multiple Correspondence Analysis
• 3 Factorial Analysis of Mixed Data
• 4 Weighting Groups of Variables
• 5 Comparing Clouds of Partial Individuals
• 6 Factors Common to Different Groups of Variables
• 7 Comparing Groups of Variables and Indscal Model
• 8 Qualitative and Mixed Data
• 9 Multiple Factor Analysis and Procrustes Analysis
• 10 Hierarchical Multiple Factor Analysis
• 11 Matrix Calculus and Euclidean Vector Space
• Bibliography
• Index