Multiple factor analysis

Multiple Factor Analysis (MFA) new article content ... Introduction The Multiple Factor Analysis is a factorial method devoted to the study of tables in which a group of individuals is described by a set of variables (quantitative and / or qualitative) structured in groups. It may be seen as an extension of: • the Principal component analysis (PCA) when variables are quantitative, • the Multiple correspondence analysis (MCA) when variables are qualitative, • the Factorial Analysis of Mixed Data (FAMD) when the active variables belong to the two types.

Introductory Example

Why introduce several groups of variables active in the same factorial analysis?

Data

Let us consider the case of quantitative variables, that is to say, within the framework of the PCA. An example of data from ecological research provides a useful illustration. There are, for 72 stations, two types of measurements.

1. The abundance-dominance coefficient of 50 plant species (coefficient ranging from 0 = the plant is absent, to 9 = the species covers more than three-quarters of the surface). The whole set of the 50 coefficients defines the floristic profile of a station.
2. Eleven pedological measurements (Pedology = soil science): particle size, physical, chemistry, etc. The set of these eleven measures defines the pedological profile of a station.

Three possible analyses

PCA of flora (pedology as supplementary) This analysis focuses on the variability of the floristic profiles. Two stations are close one another if they have similar floristic profiles. In a second step, the main dimensions of this variability (i.e. the principal components) are related to the pedological variables introduced as supplementary.

PCA of pedology (flora as supplementary) This analysis focuses on the variability of soil profiles. Two stations are close if they have the same soil profile. The main dimensions of this variability (i.e. the principal components) are then related to the abundance of plants.

PCA of the two groups of variables as active One may want to study the variability of stations from both the point of view of flora and soil. In this approach, two stations should be close if they have both similar flora 'and' similar soils.

Balance between groups of variables

Methodology

The third analysis of the introductory example implicitly assumes a balance between flora and soil. However, in this example, the mere fact that the flora is represented by 50 variables and the soil by 11 variables implies that the PCA with 61 active variables will be influenced mainly by the flora at least on the first axis). This is not desirable: there is no reason to wish one group play a more important role in the analysis. The core of MFA is based on a factorial analysis (PCA in the case of quantitative variables, MCA in the case of qualitative variables) in which the variables are weighted. These weights are identical for the variables of the same group (and vary from one group to another). They are such that the maximum axial inertia of a group is equal to 1: in other words, by applying the PCA (or, where applicable, the MCA) to one group with this weighting, we obtain a first eigenvalue equal to 1. To get this property, MFA assigns to each variable of group ${\displaystyle j}$  a weight equal to the inverse of the first eigenvalue of the analysis (PCA or MCA according to the type of variable) of the group ${\displaystyle j}$ . Formally, noting ${\displaystyle \lambda _{1}^{j}}$  the first eigenvalue of the factorial analysis of one group ${\displaystyle j}$ , the MFA assigns weight ${\displaystyle 1/\lambda _{1}^{j}}$  for each variable of the group ${\displaystyle j}$ .

Balancing maximum axial inertia rather than the total inertia (= the number of variables in standard PCA) gives the MFA several important properties for the user. More directly, its interest appears in the following example.

Example

Let two groups of variables defined on the same set of individuals.

1. The group 1 is composed of two uncorrelated variables A and B.
2. The group 2 is composed of two variables {C1, C2} identical to the same variable C uncorrelated with the first two.

This example is not completely unrealistic. It is often necessary to simultaneously analyse multi-dimensional and (quite) one-dimensional groups.

Each group having the same number of variables has the same total inertia.

In this example the first axis of the PCA is almost coincident with C. Indeed, in the space of variables, there are two variables in the direction of C: group 2, with all its inertia concentrated in one direction, influences predominantly the first axis. For its part, group 1, consisting of two orthogonal variables (= uncorrelated), has its inertia uniformly distributed in a plane (the plane generated by the two variables) and hardly weighs on the first axis.

Numerical Example width=100% border="0" |- | width=50% |

Table 1. MFA. Test data. A et B (group 1) are uncorrelated. C1 and C2 (group 2) are identical.

	${\displaystyle A}$	${\displaystyle B}$	${\displaystyle C_{1}}$	${\displaystyle C_{2}}$
${\displaystyle 1}$	1	1	1	1
${\displaystyle 2}$	2	3	4	4
${\displaystyle 3}$	3	5	2	2
${\displaystyle 4}$	4	5	2	2
${\displaystyle 5}$	5	3	4	4
${\displaystyle 6}$	6	1	2	2

| width=50% |

Table 2. Test data. Decomposition of the inertia in the PCA and in the MFA applied to data in Table 1.

	${\displaystyle F_{1}}$	${\displaystyle F_{2}}$
PCA
Inertia	2.14 (100%)	1
group 1	0.24(11%)	1
group 2	1.91(89%)	0
MFA
Inertia	1.28(100%)	1
group 1	0.64(50%)	1
group 2	0.64(50%)	0

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Table 2 summarizes the inertia of the first two axes of the PCA and of the MFA applied to Table 1.

Group 2 variables contribute to 88.95 % of the inertia of the axis 1 of the PCA. The first axis (${\displaystyle F_{1}}$ ) is almost coincident with C: the correlation between C and ${\displaystyle F_{1}}$  is .976;

The first axis of the MFA (on Table 1 data) shows the balance between the two groups of variables: the contribution of each group to the inertia of this axis is strictly equal to 50%.

The second axis, meanwhile, depends only on the group 1. This is natural since this group is two-dimensional while the second group, being one-dimensional, can be highly related to only one axis (here the first axis).

Conclusion about the balance between groups

Introducing several active groups of variables in a factorial analysis implicitly assumes a balance between these groups.

This balance must take into account that a multidimensional group influences naturally more axes than a one-dimensional group does (which may not be closely related to one axis).

The weighting of the MFA, which makes equal to 1 the maximum axial inertia of each group, plays this role.

Overview on few application areas

Survey Questionnaires are always structured according to different themes. Each theme is a group of variables, for example, questions about opinions and questions about behaviour. Thus, in this example, we may want to perform a factorial analysis in which two individuals are close if they have expressed both the same opinions and the same behaviour.

Sensory analysis A same set of products has been evaluated by a panel of experts and a panel of consumers. For its evaluation, each jury uses a list of descriptors (sour, bitter, etc.). Each judge scores each descriptor for each product on a scale of intensity ranging for example from 0 = null or very low to 10 = very strong. In the table associated with a jury, at the intersection of the row ${\displaystyle i}$  and column ${\displaystyle k}$ , is the average score assigned to product ${\displaystyle i}$  for descriptor ${\displaystyle k}$ .

Individuals are the products. Each jury is a group of variables. We want to achieve a factorial analysis in which two products are similar if they were evaluated in the same way and that by both juries.

Multidimensional time series ${\displaystyle K}$  variables are measured on ${\displaystyle I}$  individuals. These measurements are made at ${\displaystyle J}$  dates. There are many ways to analyse such data set. One of them suggested by the MFA, is to consider each day as a group of variables in the analysis of the tables (each table corresponds to one date) juxtaposed row-wise (table analysed thus has ${\displaystyle I}$  rows and ${\displaystyle J}$ x${\displaystyle K}$  columns).

Conclusion about these examples: in practice, the variables are organized into groups very often.

Graphics from MFA

Beyond the weighting of variables, the interest of the MFA lies in a series of graphics and indicators valuable in the analysis of a table whose columns are organized into groups.

Graphics common to all the simple factorial analyses (PCA, MCA)

The core of the MFA is a weighted factorial analysis: MFA firstly provides the classical results of the factorial analyses.

1. Representations of individuals in which two individuals are much closer than they have similar values for all variables in all groups; in practice the user especially studies the first factorial plane. 2.Representations of quantitative variables as in PCA (correlation circle).

Figure1. MFA. Test data. Representation of individuals on the first plane.

Figure2. MFA. Test data. Representation of variables on the first plane.

In the example:

• The first axis mainly opposes individuals 1 and 5 (Figure 1).
• The four variables have a positive coordinate (Figure 2): the first axis is a size effect. Thus, the individual 1 has low values for all the variables and individual 5 has high values for all the variables.

3. Indicators aiding interpretation: projected inertia, contributions and quality of representation. In the example, the contribution of individuals 1 and 5 to the inertia of the first axis is 45.7% + 31.5% = 77.2% which justifies the interpretation focussed on these two points.

4. Representations of categories of qualitative variables as in MCA (a category lies at the centroid of the individuals who possess it). No qualitative variables in the example.

Graphics specific to this kind of multiple table

5. Superimposed representations of individuals « seen » by each group. An individual considered from the point of view of a single group is called « partial individual »' (in parallel, an individual considered from the point of view of all variables is said “mean individual because it lies at the center of gravity of its partial points). Partial cloud ${\displaystyle N_{i}^{j}}$  gathers the ${\displaystyle I}$  individuals from the perspective of the single group ${\displaystyle j</math>(ie<math>{i^{j},j=1,J}}$ ): that is the cloud analysed in the separate factorial analysis (PCA or MCA) of the group ${\displaystyle j}$ . The superimposed representation of the ${\displaystyle N_{i}^{j}}$  provided by the MFA is similar in its purpose to that provided by the Procrustes analysis.

 

Figure 3. MFA. Test data. Superimposed representation of mean and partial clouds.

In the example (figure 3), individual 1 is characterized by a small size (i.e. small values) both in terms of group 1 and group 2 (partial points of the individual 1 have a negative coordinate and are close one another). On the contrary, the individual 5 is more characterized by high values for the variables of group 2 than for the variables of group 1 (for the individual 5, group 2 partial point lies further from the origin than group 1 partial point). This reading of the graph can be checked directly in the data.

References

External links

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