# Lesson 12: Factor Analysis

### Introduction

**Factor Analysis** is a method for modeling observed variables, and their covariance structure, in terms of a smaller number of underlying unobservable (latent) “factors.” The factors typically are viewed as broad concepts or ideas that may describe an observed phenomenon. For example, a basic desire of obtaining a certain social level might explain most consumption behavior. These unobserved factors are more interesting to the social scientist than the observed quantitative measurements.

Factor analysis is generally an exploratory/descriptive method that requires many subjective judgments. It is a widely used tool and often controversial because the models, methods, and subjectivity are so flexible that debates about interpretations can occur.

The method is similar to principal components although, as the textbook points out, factor analysis is more elaborate. In one sense, factor analysis is an inversion of principal components. In factor analysis we model the observed variables as linear functions of the “factors.” In principal components, we create new variables that are linear combinations of the observed variables. In both PCA and FA, the dimension of the data is reduced. Recall that in PCA, the interpretation of the principal components is often not very clean. A particular variable may, on occasion, contribute significantly to more than one of the components. Ideally we like each variable to contribute significantly to only one component. A technique called factor rotation is employed towards that goal. Examples of fields where factor analysis is involved include physiology, health, intelligence, sociology, and sometimes ecology among others.

### Learning objectives & outcomes

Upon completion of this lesson, you should be able to do the following:

- Understand the terminology of factor analysis, including the interpretation of factor loadings, specific variances, and communalities;
- Understand how to apply both principal component and maximum likelihood methods for estimating the parameters of a factor model;
- Understand factor rotation, and interpret rotated factor loadings.