Multiple factor analysis

Table of contents

Resources

• Multiple Factor Analysis by Hervé Abdi
• Multiple Factor Analysis: main features and application to sensory data by Jérôme Pagès
• Wikipedia article

Data

Multiple factor analysis (MFA) is meant to be used when you have groups of variables. In practice, it builds a PCA on each group. It then fits a global PCA on the results of the so-called partial PCAs.

The dataset used in the following example come from this paper. In the dataset, three experts give their opinion on six different wines. Each opinion for each wine is recorded as a variable. We thus want to consider the separate opinions of each expert whilst also having a global overview of each wine. MFA is the perfect fit for this kind of situation.

import prince

dataset = prince.datasets.load_premier_league()
dataset

import pandas as pd

isinstance(dataset.columns, pd.MultiIndex)

True

Fitting

The groups are specified by the groups argument when calling fit.

groups = dataset.columns.levels[0].tolist()
groups

['2021-22', '2022-23', '2023-24']

mfa = prince.MFA(
    n_components=3,
    n_iter=3,
    copy=True,
    check_input=True,
    engine='sklearn',
    random_state=42
)
mfa = mfa.fit(
    dataset,
    groups=groups,
    supplementary_groups=None
)

There are several ways to specify the groups:

• If the columns of the dataframe are a MultiIndex:
  • By default the groups are all the columns in the first level.
  • You can also pass a list with a subset of the columns in the first level.
• You can also pass a dict that maps group names to the desired columns.

The supplementary_groups argument is expected to be a list with one more existing group names.

Eigenvalues

mfa.eigenvalues_summary

Coordinates

The MFA inherits from the PCA class, which means it provides access to the PCA methods and properties. For instance, the row_coordinates method will return the global coordinates of each wine.

mfa.row_coordinates(dataset)

There is also a partial_row_coordinates method that returns the coordinates projected onto each group.

mfa.partial_row_coordinates(dataset)

Visualization

mfa.plot(
    dataset,
    x_component=0,
    y_component=1
)

The first axis explains most of the difference between the wine ratings. This difference is actually due to the oak type of the barrels they were fermented in.

The show_partial_rows argument allows showing the global row coordinates together with the partial row coordinates. All the coordinates of each sample are connected with edges.

mfa.plot(
    dataset,
    show_partial_rows=True
)

Partial PCAs

An MFA is essentially a PCA applied to the outputs of partial PCA. Indeed, a PCA is first fitted to each group. A partial PCA can be accessed as so:

dataset['2022-23']

mfa['2022-23'].eigenvalues_summary