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Multiple factor analysis: principal component analysis for multitable and multiblock data sets

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Abstract Multiple factor analysis (MFA, also called multiple factorial analysis) is an extension of principal component analysis (PCA) tailored to handle multiple data tables that measure sets of variables collected on the same observations, or, alternatively, (in dual‐MFA) multiple data tables where the same variables are measured on different sets of observations. MFA proceeds in two steps: First it computes a PCA of each data table and ‘normalizes’ each data table by dividing all its elements by the first singular value obtained from its PCA. Second, all the normalized data tables are aggregated into a grand data table that is analyzed via a (non‐normalized) PCA that gives a set of factor scores for the observations and loadings for the variables. In addition, MFA provides for each data table a set of partial factor scores for the observations that reflects the specific ‘view‐point’ of this data table. Interestingly, the common factor scores could be obtained by replacing the original normalized data tables by the normalized factor scores obtained from the PCA of each of these tables. In this article, we present MFA, review recent extensions, and illustrate it with a detailed example. WIREs Comput Stat 2013, 5:149–179. doi: 10.1002/wics.1246 This article is categorized under: Data: Types and Structure > Categorical Data Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

The different steps of MFA.

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MUFUBADA. Left : Discriminant factor scores for the three wine groups (regions). Right: Discriminant factor scores for the three wine groups with bootstrapped 95% confidence intervals.

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Bootstrap confidence ellipses plotted on Components 1 and 2.

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Bootstrap ratio plot for Components 1 and 2.

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Supplementary table: chemical components of the wines. Supplementary partial scores and loadings. (cf., Figure 2a).

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Partial Inertias of the tables. The sizes of the assessors' icons are proportional to their explained inertia for Components 1 and 2.

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Contributions of the tables to the compromise. The sizes of the assessors' icons are proportional to their contribution to Components 1 and 2.

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Partial factor scores and variable loadings for the first two dimensions of the compromise space. The loadings have been re‐scaled to have a variance equal the singular values of the compromise analysis.

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Compromise of the 10 tables. (a) Factor scores (wines). (b) Assessors' partial factor scores projected into the compromise as supplementary elements. Each assessor is represented by a dot, and for each wine a line connects the wine factor scores to the partial factors scores of a given assessor for this wine. (λ1 = 0.770, τ1 = 61%; λ2 = 0.123, τ2 = 10%).

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Statistical and Graphical Methods of Data Analysis > Multivariate Analysis
Data: Types and Structure > Categorical Data
Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis

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