14.1 Introduction

This set of notes uses functions from several new packages. See the links in the Additional Resources section for more information

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.