10. Factor Analysis
A frequently applied paradigm in analyzing data from
multivariate observations is to model the relevant information
(represented in a multivariate
variable ) as coming from a limited number of latent factors.
In a survey on household consumption, for example,
the consumption levels, , of different goods during
one month could be observed.
The variations and covariations of the components of
throughout the survey might in fact be explained by two or
three main social behavior factors of the household.
For instance, a basic desire of comfort or
the willingness to achieve a certain social level or other social latent
concepts might explain most of the consumption behavior.
These unobserved factors are
much more interesting to the social scientist than the observed quantitative
measures () themselves, because they give a better understanding of the
behavior of households. As shown in the examples below, the same kind
of factor analysis is of interest in many fields such as
psychology, marketing, economics, politic sciences, etc.
How can we provide a statistical model addressing these issues and how can
we interpret the obtained model? This is the aim of factor analysis.
As in Chapter 8 and Chapter 9,
the driving statistical theme of this chapter is to reduce the dimension
of the observed data. The perspective used, however, is different:
we assume that there is a model (it will be
called the ``Factor Model'') stating that most of the covariances between
the elements of can be explained by a limited number of latent
factors. Section 10.1 defines the basic concepts and
notations of the orthogonal factor model, stressing the non-uniqueness
of the solutions. We show how to take advantage of this non-uniqueness
to derive techniques which lead to easier interpretations.
This will involve (geometric) rotations of the factors.
Section 10.2 presents an empirical approach to factor
analysis. Various estimation procedures are proposed and an
optimal rotation procedure is defined. Many examples are used to illustrate
the method.