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Multiple Factor Analysis by Example Using R

Book Description

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 includes examples of applications and details of how to implement MFA using an R package (FactoMineR).

The first two chapters cover the basic factorial analysis methods of principal component analysis (PCA) and multiple correspondence analysis (MCA). The next chapter discusses factor analysis for mixed data (FAMD), a little-known method for simultaneously analyzing quantitative and qualitative variables without group distinction. Focusing on MFA, subsequent chapters examine the key points of MFA in the context of quantitative variables as well as qualitative and mixed data. The author also compares MFA and Procrustes analysis and presents a natural extension of MFA: hierarchical MFA (HMFA). The final chapter explores several elements of matrix calculation and metric spaces used in the book.

Table of Contents

  1. Preliminaries
  2. Preface
  3. Chapter 1 Principal Component Analysis
        1. Vocabulary: Factor Analysis or Factorial Analysis?
    1. 1.1 Data, Notations
    2. 1.2 Why Analyse a Table with PCA?
    3. 1.3 Clouds of Individuals and Variables
        1. Cloud of Individuals <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N<span class="cSubscript">I</span></span>
        2. Cloud of Variables <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N<span class="cSubscript">K</span></span>
    4. 1.4 Centring and Reducing
    5. 1.5 Fitting Clouds NI and NK
      1. 1.5.1 General Principles and Formalising Criteria
        1. Fitting <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N<span class="cSubscript">I</span></span> in &#8477; in ℝ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic"><span class="cSuperscript">K</span></span>
        2. Fitting <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N<span class="cSubscript">K</span></span> in &#8477; in ℝ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic"><span class="cSuperscript">I</span></span>
      2. 1.5.2 Interpreting Criteria
      3. 1.5.3 Solution
        1. In the Individuals’ Space
        2. In the Variables’ Space
      4. 1.5.4 Relationships Between the Analyses of the Two Clouds
      5. 1.5.5 Representing the Variables
      6. 1.5.6 Number of Axes
      7. 1.5.7 Vocabulary: Axes and Factors
    6. 1.6 Interpretation Aids
      1. 1.6.1 Percentage of Inertia Associated with an Axis
      2. 1.6.2 Contribution of One Point to the Inertia of an Axis
        1. Case of an Individual
        2. Case of a Variable
      3. 1.6.3 Quality of Representation of a Point by an Axis
    7. 1.7 First Example: 909 Baccalaureate Candidates
      1. 1.7.1 Projected Inertia (Eigenvalues)
      2. 1.7.2 Interpreting the Axes
      3. 1.7.3 Methodological Remarks
        1. Interpretability and Percentage of Inertia
        2. Two Readings of the Correlation Circle
        3. Validating the Interpretation
        4. PCA and Synthetic Visualisation
    8. 1.8 Supplementary Elements
        1. Remark
    9. 1.9 Qualitative Variables in PCA
        1. Principle
        2. Baccalaureate Grades Example
    10. 1.10 Second Example: Six Orange Juices
    11. 1.11 PCA in FactoMineR
        1. Drop-Down Menu in R Commander
          1. Main Menu (See Figure 1.11)
          2. Graphical Options (See Figure 1.12)
        2. Examples of Commands
        3. Remark
        4. Script for Analysing the Orange Juice Data
    1. Figure 1.1
    2. Figure 1.2
    3. Figure 1.3
    4. Figure 1.4
    5. Figure 1.5
    6. Figure 1.6
    7. Figure 1.7
    8. Figure 1.8
    9. Figure 1.9
    10. Figure 1.10
    11. Figure 1.11
    12. Figure 1.12
    1. Table 1.1
    2. Table 1.2
    3. Table 1.3
    4. Table 1.4
    5. Table 1.5
    6. Table 1.6
  4. Chapter 2 Multiple Correspondence Analysis
    1. 2.1 Data
    2. 2.2 Complete Disjunctive Table
    3. 2.3 Questioning
    4. 2.4 Clouds of Individuals and Variables
      1. 2.4.1 Cloud of Individuals
        1. Distance Between an Individual i and the Centre of Gravity of NI
        2. Total Inertia of NI (With Respect to GI)
        3. Distance Between Two Individuals i and l
        4. Remark
      2. 2.4.2 Cloud of Categories
      3. 2.4.3 Qualitative Variables
    5. 2.5 Fitting Clouds NI and NK
      1. 2.5.1 Cloud of Individuals
      2. 2.5.2 Cloud of Categories
      3. 2.5.3 Relationships Between the Two Analyses
    6. 2.6 Representing Individuals, Categories and Variables
        1. Representing Individuals and Categories
        2. Representing the Variables
        3. Number of Axes
    7. 2.7 Interpretation Aids
        1. Percentage of Inertia Associated with an Axis
        2. Contributions
        3. Supplementary Elements
    8. 2.8 Example: Five Educational Tools Evaluated by 25 Students
      1. 2.8.1 Data
      2. 2.8.2 Analyses and Representations
        1. Representing Individuals
        2. Representing Categories
      3. 2.8.3 MCA/PCA Comparison for Ordinal Variables
    9. 2.9 MCA in FactoMineR
        1. Drop-Down Menu in R Commander
        2. FactoMineR offers three types of graph for MCA (see Figure 2.7).
        3. Examples of Commands
    1. Figure 2.1
    2. Figure 2.2
    3. Figure 2.3
    4. Figure 2.4
    5. Figure 2.5
    6. Figure 2.6
    7. Figure 2.7
    1. Table 2.1
    2. Table 2.2
  5. Chapter 3 Factorial Analysis of Mixed Data
    1. 3.1 Data, Notations
    2. 3.2 Representing Variables
    3. 3.3 Representing Individuals
    4. 3.4 Transition Relations
        1. Relationships from ℝK Toward ℝ1
        2. Relationship from ℝI Toward ℝK
        3. Remark
    5. 3.5 Implementation
    6. 3.6 Example: Biometry of Six Individuals
    7. 3.7 FAMD in FactoMineR
        1. Drop-Down Menu in R Commander
          1. Main Menu (see Figure 3.5)
        2. Graphical Options (see Figure 3.6)
        3. Examples of Commands
    1. Figure 3.1
    2. Figure 3.2
    3. Figure 3.3
    4. Figure 3.4
    5. Figure 3.5
    6. Figure 3.6
    1. Table 3.1
    2. Table 3.2
    3. Table 3.3
  6. Chapter 4 Weighting Groups of Variables
    1. 4.1 Objectives
    2. 4.2 Introductory Numerical Example
    3. 4.3 Weighting Variables in MFA
    4. 4.4 Application to the Six Orange Juices
    5. 4.5 Relationships with Separate Analyses
    6. 4.6 Conclusion
    7. 4.7 MFA in FactoMineR (First Results)
        1. Drop-Down Menu in R Commander
        2. Defining the Groups (See Figure 4.9)
        3. Graphical Options (See Figure 4.10)
        4. Command Examples
        5. Table 4.5
        6. Simplified Output
    1. Figure 4.1
    2. Figure 4.2
    3. Figure 4.3
    4. Figure 4.4
    5. Figure 4.5
    6. Figure 4.6
    7. Figure 4.7
    8. Figure 4.8
    9. Figure 4.9
    10. Figure 4.10
    1. Table 4.1
    2. Table 4.2
    3. Table 4.3
    4. Table 4.4
    5. Table 4.5
    6. Table 4.6
    7. Table 4.7
  7. Chapter 5 Comparing Clouds of Partial Individuals
    1. 5.1 Objectives
    2. 5.2 Method
    3. 5.3 Application to the Six Orange Juices
    4. 5.4 Interpretation Aids
        1. Remark
    5. 5.5 Distortions in Superimposed Representations
      1. 5.5.1 Example (Trapeziums Data)
      2. 5.5.2 Geometric Interpretation
        1. Remark
      3. 5.5.3 Algebra Approach
        1. Notations (Reminders and Additions)
        2. Reconstitution of <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">j</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsub">I</span>
        3. Numerical Example
    6. 5.6 Superimposed Representation: Conclusion
    7. 5.7 MFA Partial Clouds in FactoMineR
        1. The Drop-Down Menu
        2. Command Lines
    1. Figure 5.1
    2. Figure 5.2
    3. Figure 5.3
    4. Figure 5.4
    5. Figure 5.5
    6. Figure 5.6
    7. Figure 5.7
    8. Figure 5.8
    9. Figure 5.9
    10. Figure 5.10
    1. Table 5.1
    2. Table 5.2
    3. Table 5.3
    4. Table 5.4
    5. Table 5.5
  8. Chapter 6 Factors Common to Different Groups of Variables
    1. 6.1 Objectives
      1. 6.1.1 Measuring the Relationship between a Variable and a Group
      2. 6.1.2 Factors Common to Several Groups of Variables
      3. 6.1.3 Back to the Six Orange Juices
      4. 6.1.4 Canonical Analysis
    2. 6.2 Relationship Between a Variable and Groups of Variables
    3. 6.3 Searching for Common Factors
    4. 6.4 Searching for Canonical Variables
        1. Remark
    5. 6.5 Interpretation Aids
      1. 6.5.1 Lg Relationship Measurement
      2. 6.5.2 Canonical Correlation Coefficients
    1. Figure 6.1
    1. Table 6.1
  9. Chapter 7 Comparing Groups of Variables and Indscal Model
    1. 7.1 Cloud NJ of Groups of Variables
        1. Remark
    2. 7.2 Scalar Product and Relationship Between Groups of Variables
            1. Lg and RV
    3. 7.3 Norm in the Groups’ Space
    4. 7.4 Representation of Cloud NJ
      1. 7.4.1 Principle
        1. Remark
      2. 7.4.2 Criterion
    5. 7.5 Interpretation Aids
    6. 7.6 The Indscal Model
      1. 7.6.1 Model
      2. 7.6.2 Estimating Parameters and Properties
      3. 7.6.3 Example of an Indscal model via MFA (cards)
      4. 7.6.4 Ten Touraine White Wines
        1. Data (Wines)
        2. Results
        3. Comparing the Two Estimations: Quality of the Fit
        4. Nature of the Factors
        5. Conclusion
    7. 7.7 MFA in FactoMineR (groups)
    1. Figure 7.1
    2. Figure 7.2
    3. Figure 7.3
    4. Figure 7.4
    5. Figure 7.5
    6. Figure 7.6
    7. Figure 7.7
    8. Figure 7.8
    9. Figure 7.9
    1. Table 7.1
    2. Table 7.2
    3. Table 7.3
    4. Table 7.4
    5. Table 7.5
  10. Chapter 8 Qualitative and Mixed Data
    1. 8.1 Weighted MCA
      1. 8.1.1 Cloud of Categories in Weighted MCA
      2. 8.1.2 Transition Relations in Weighted MCA
    2. 8.2 MFA of Qualitative Variables
      1. 8.2.1 From the Perspective of Factorial Analysis
        1. Principle of Weighting Groups of Variables
        2. MFA of Qualitative Variables Is Based on a Weighted MCA
        3. Remark
      2. 8.2.2 From the Perspective of Multicanonical Analysis
        1. Lg Measurement for Qualitative Variables
        2. Remark
        3. Looking for General Variables
      3. 8.2.3 Representing Partial Individuals
        1. Remark
      4. 8.2.4 Representing Partial Categories
        1. Remark
      5. 8.2.5 Analysing in Space of Groups of Variables (ℝ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">I</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">2</span>))
        1. Cloud of Groups of Variables
        2. Interpreting the Scalar Product Between Two Groups
    3. 8.3 Mixed Data
      1. 8.3.1 Weighting the Variables
      2. 8.3.2 Properties
        1. Representing the Variables
        2. Representing Clouds of Partial Individuals
        3. Multicanonical Analysis
        4. Analysis in ℝ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">I</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">2</span>
    4. 8.4 Application (Biometry2)
      1. 8.4.1 Separate Analyses
        1. Inertias (Table 8.4)
        2. Correlations Between the Factors of the Separate Analyses (Table 8.5)
      2. 8.4.2 Inertias in the Overall Analysis
      3. 8.4.3 Coordinates of the Factors of the Separate Analyses
      4. 8.4.4 First Factor
        1. Individuals and Variables (See Figure 8.2)
        2. Partial Individuals (See Figure 8.3)
        3. Transition Relations
        4. Partial Categories (see Figure 8.3)
        5. Conclusion
      5. 8.4.5 Second Factor
      6. 8.4.6 Third Factor
      7. 8.4.7 Representing Groups of Variables
      8. 8.4.8 Conclusion
    5. 8.5 MFA of Mixed Data in FactoMineR
        1. Table 8.5 (Correlations Between Partial Axes)
        2. Table 8.7 (Contributions of the Cells to Partial Individuals)
        3. Table 8.8 (Qualities of Representation in ℝ<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">I</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">2</span>))
        4. Table 8.9 (Lg an RV Relationship Indicators)
    1. Figure 8.1
    2. Figure 8.2
    3. Figure 8.3
    4. Figure 8.4
    1. Table 8.1
    2. Table 8.2
    3. Table 8.3
    4. Table 8.4
    5. Table 8.5
    6. Table 8.6
    7. Table 8.7
    8. Table 8.8
    9. Table 8.9
  11. Chapter 9 Multiple Factor Analysis and Procrustes Analysis
    1. 9.1 Procrustes Analysis
      1. 9.1.1 Data, Notations
      2. 9.1.2 Objectives
      3. 9.1.3 Methods and Variations
        1. Depending on the Number of Clouds
        2. Remark
        3. Depending on the Number of Dimensions
        4. Influence of the Dimensions on the Objectives
    2. 9.2 Comparing MFA and GPA
      1. 9.2.1 Representing <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">j</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsub">I</span>
        1. In MFA
        2. In GPA
      2. 9.2.2 Mean Cloud
        1. In MFA
        2. In GPA
      3. 9.2.3 Objective, Criterion, Algorithm
        1. In MFA
        2. In GPA
        3. Criterion
        4. Remark
      4. 9.2.4 Properties of the Representations of <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">j</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsub">I</span>
        1. In GPA
        2. In MFA
      5. 9.2.5 A First Appraisal
      6. 9.2.6 Harmonising the Inertia of <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cItalic">N</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsup">j</span><span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="cIsub">I</span>
      7. 9.2.7 Relationships Between Homologous Factors
        1. Remark
      8. 9.2.8 Representing Individuals
      9. 9.2.9 Interpretation Aids
            1. Decomposition by Dimension
            2. Decomposition by Group
            3. Decomposition by Individual
            4. Summary
      10. 9.2.10 Representing the Variables
    3. 9.3 Application (Data 23−1)
      1. 9.3.1 Data 23−1
      2. 9.3.2 Results of the MFA
        1. Projected Inertias of the Mean Cloud
        2. Relationship Measurements Between Factors and Groups
        3. Inertias of Individuals in the Mean Cloud
        4. Superimposed Representation
      3. 9.3.3 Results of the GPA
        1. Projected Inertias of the Mean Cloud
        2. Representing Individuals
        3. Indicators of the Discrepancy to the Procrustes Model
        4. Conclusion
    4. 9.4 Application to the Ten Touraine Wines
    5. 9.5 Conclusion
    6. 9.6 GPA in FactoMineR
        1. Procrustes MFA (PMFA; Figure 9.8)
    1. Figure 9.1
    2. Figure 9.2
    3. Figure 9.3
    4. Figure 9.4
    5. Figure 9.5
    6. Figure 9.6
    7. Figure 9.7
    8. Figure 9.8
    1. Table 9.1
    2. Table 9.2
    3. Table 9.3
    4. Table 9.4
    5. Table 9.5
  12. Chapter 10 Hierarchical Multiple Factor Analysis
    1. 10.1 Data, Examples
    2. 10.2 Hierarchy and Partitions
    3. 10.3 Weighting the Variables
        1. Remark
    4. 10.4 Representing Partial Individuals
      1. 10.4.1 Method
      2. 10.4.2 Application to the Six Orange Juices
    5. 10.5 Canonical Correlation Coefficients
    6. 10.6 Representing the Nodes
    7. 10.7 Application to Mixed Data: Sorted Napping®
      1. 10.7.1 Data and Methodology
      2. 10.7.2 Intermediary Analysis: MFA on a Sorted Nappe
      3. 10.7.3 Decompositions of Inertia
      4. 10.7.4 Representing Partial and Mean Individuals
        1. Representation of Groups of Variables (see Figure 10.13)
    8. 10.8 HMFA in FactoMineR
        1. Remark
    1. Figure 10.1
    2. Figure 10.2
    3. Figure 10.3
    4. Figure 10.4
    5. Figure 10.5
    6. Figure 10.6
    7. Figure 10.7
    8. Figure 10.8
    9. Figure 10.9
    10. Figure 10.10
    11. Figure 10.11
    12. Figure 10.12
    13. Figure 10.13
    14. Figure 10.14
    1. Table 10.1
    2. Table 10.2
    3. Table 10.3
    4. Table 10.4
    5. Table 10.5
    6. Table 10.6
    7. Table 10.7
  13. Chapter 11 Matrix Calculus and Euclidean Vector Space
    1. 11.1 Matrix Calculus
        1. Definitions
        2. Transposition
        3. Matrix Multiplication
        4. Trace of a Square Matrix
        5. Matrix and Function, Orthogonal Matrix, Diagonalisation
    2. 11.2 Euclidean Vector Space
      1. 11.2.1 Vector Space Endowed with the Usual Distance
        1. Two-Dimensional Space
        2. Vector Space with n Dimensions
      2. 11.2.2 Euclidean Space Endowed with a Diagonal Metric
      3. 11.2.3 Visualising a Cloud in a Space Endowed with a Metric Different from the Identity
    1. Figure 11.1
    2. Figure 11.2
    1. Table 11.1
    2. Table 11.2
  14. Bibliography