How Do Principal Component Analysis (PCA) and Factor Analysis (FA) Differ?
Principal Component Analysis (PCA) and Factor Analysis (FA) are widely used multivariate statistical techniques for dimensionality reduction and extracting key information from data. Although both methods reduce data dimensionality, they differ in their objectives, underlying assumptions, and applications. The following outlines the key distinctions between PCA and FA:
Principal Component Analysis (PCA)
1. Objective
PCA aims to transform original variables into a set of uncorrelated principal components through a linear transformation, maximizing the variance explained by each component. It is designed to derive a low-dimensional representation that preserves as much of the original data variance as possible.
2. Assumptions
PCA assumes that the original variables exhibit linear relationships, allowing them to be represented as linear combinations.
3. Interpretation of Components
The importance of each principal component is determined by its explained variance ratio, which quantifies the proportion of total variance accounted for by the component.
4. Data Transformation
PCA employs orthogonal transformations to derive principal components as linear combinations of the original variables. These principal components are mutually uncorrelated.
5. Applications
PCA is commonly applied in data visualization, dimensionality reduction, and feature selection. It facilitates the identification of dominant patterns and relationships within high-dimensional datasets.
Factor Analysis (FA)
1. Objective
FA seeks to explain correlations among observed variables by identifying underlying latent factors. It assumes that observed variables are influenced by a combination of latent factors and measurement errors.
2. Assumptions
FA assumes that observed variables are driven by unobservable latent factors, which can be inferred from their shared variance patterns.
3. Interpretation of Factors
FA assesses factor importance through factor loadings, which quantify the strength of associations between latent factors and observed variables.
4. Data Transformation
FA models observed variables as linear combinations of latent factors and measurement errors. Unlike PCA, latent factors in FA may exhibit correlations with each other.
5. Applications
FA is widely used in psychometrics, social sciences, and biological research to model latent variables, explore hidden structures within data, and identify underlying factors influencing observed measurements.
While both PCA and FA reduce data dimensionality, they serve different purposes. PCA is primarily used for variance maximization and feature extraction, whereas FA aims to uncover latent structures driving observed correlations. The choice between these methods depends on the research objectives and data characteristics.
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