# 21 Factor Analysis

All examination, therefore, in the different sensory, school, or other specific intellectual faculties, may be regarded as so many independently obtained estimates of the one great common Intellective Function. — Charles Spearman

Factor analysis is a multivariate statistical method used to evaluate relationships among a set of observed variables. The method was developed in the early 1900s by Charles Spearman, a psychologist interested in understanding the complexities of human intelligence. Spearman found that measures of different cognitive abilities were interrelated and could be efficiently expressed in terms of a single underlying factor, “the one great common Intellective Function,” also known as general intelligence or the *g factor* (Spearman 1904).

Since Spearman’s early research on intelligence, factor analysis has been studied and applied in a variety of fields, including psychology, education, economics, and political science. The common thread in these different fields is the focus on measuring unobserved variables as potential causes of or explanations for observed behavior.

This module gives an overview of two popular applications of factor analysis, called exploratory factor analysis or EFA and confirmatory factor analysis or CFA, with details on how each is used in test development. Real data examples come from the PISA data set. The learning objectives for this module focus on understanding the basic concepts involved in EFA and CFA, recognizing how they’re appropriately applied, including common applications of each, and interpreting results.

## 21.1 Objectives

**Learning objectives**

- Compare and contrast the factor analytic model with other measurement models, including CTT and IRT, in terms of their applications in instrument development.
- Describe the differing purposes of exploratory and confirmatory factor analysis.
- Explain how an EFA is implemented, including the type of data required and steps in setting up and fitting the model.
- Interpret EFA results, including factor loadings and eigenvalues.
- Use a scree plot to visually compare factors in an EFA.
- Explain how a CFA is implemented, including the type of data required and steps in setting up and fitting the model.
- Interpret CFA results, including factor loadings and fit indices.

In this module, we will run EFA and plot results using epmr functions. We’ll also install a new package called lavaan for running CFA.

```
# R setup for this module
library("epmr")
# We're using a new package for CFA called lavaan
install.packages("lavaan")
library("lavaan")
# Functions we'll use
# fastudy() and plot() from epmr
# lavaanify() and cfa() from lavaan
```