14.1 Introduction

library(corrplot)
library(psych)
library(ggfortify) # plots scores from factanal()
library(GPArotation) # to do oblimin rotation

No attempt will be made to present a comprehensive treatment of this subject. For more detail see the references mentioned in PMA6 Chapter 15.2

14.1.1 Latent Constructs

Latent variables are ones that cannot be measured directly; e.g. Depression, Anxiety, Mathematical ability. They drive how we would respond to various tasks and questions that can be measured; vocabulary, arithmetic, statistical reasoning.

How can the correlation in responses to questions help us measure these latent constructs?

Factor Analysis aims to

• Generalize of principal components analysis
• Explain interrelationships among a set of variables
• Where we select a small number of factors to convey essential information
• Can perform additional analyses to improve interpretation

14.1.2 Comparison with PCA

• Similar in that no dependent variable
• PCA:
• Select a number of components that explain as much of the total variance as possible.
• FA: Factors selected mainly to explain the interrelationships among the original variables.
• Ideally, the number of factors expected is known in advance.
• Major emphasis is placed on obtaining easily understandable factors that convey the essential information contained in the original set of variables.
• Mirror image of PCA
• Each PC is expressed as a linear combination of X’s
• Each $$X$$ is expressed as a linear combination of Factors

14.1.3 EFA vs CFA

Exploratory Factor Analysis

• Explore the possible underlying factor structure of a set of observed variables
• Does not impose a preconceived structure on the outcome.

Confirmatory Factor Analysis

• Verifies the theoretical factor structure of a set of observed variables
• Test the relationship between observed variables and theoretical underlying latent constructs
• Variable groupings are determined ahead of time.