The aim of principal component analysis is to explain the covariance between variables while the aim of factor analysis is to explain the variance between variables.
Factor analysis (FA) and principal component analysis (PCA) are both techniques used in data analysis, particularly in the realm of dimensionality reduction. While they share some similarities, they also have distinct differences:
- Objective:
- Factor Analysis: It aims to explain the covariance relationships among observed variables in terms of a few unobservable latent variables called factors. FA attempts to identify underlying latent variables that explain the patterns of correlations within observed variables.
- Principal Component Analysis: PCA aims to transform the original variables into a new set of orthogonal variables called principal components. These components are linear combinations of the original variables and capture as much variance in the data as possible.
- Assumption:
- Factor Analysis: It assumes that observed variables are influenced by latent factors and measurement error. FA is often used when there is a belief that the observed variables are caused by underlying latent variables.
- Principal Component Analysis: PCA does not assume the existence of latent factors. It simply aims to find orthogonal directions (components) that maximize the variance in the data.
- Output:
- Factor Analysis: The output of FA typically includes factor loadings, which indicate the strength of the relationship between observed variables and latent factors, along with eigenvalues and eigenvectors.
- Principal Component Analysis: The output of PCA includes the principal components themselves, along with the variance explained by each component, eigenvalues, and eigenvectors.
- Interpretation:
- Factor Analysis: Factors are often interpreted based on the pattern of loadings across observed variables. These factors are hypothesized to represent underlying constructs or dimensions in the data.
- Principal Component Analysis: Principal components are linear combinations of the original variables. They do not necessarily have a straightforward interpretation in terms of underlying constructs. However, they capture the maximum amount of variance in the data.
- Application:
- Factor Analysis: It is often used in psychology, sociology, and other social sciences to identify latent constructs such as intelligence, personality traits, or attitudes.
- Principal Component Analysis: PCA is widely used in various fields for data compression, visualization, noise reduction, and feature extraction.
In summary, while both factor analysis and principal component analysis are techniques used for dimensionality reduction, they differ in their assumptions, objectives, outputs, and interpretations. The choice between the two methods depends on the specific goals of the analysis and the underlying structure of the data.