Factor analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. Essentially, it models observed variables as linear combinations of potential factors plus "error" terms. The technique is used widely across various fields, including psychology, sociology, marketing, business, and medicine, to identify underlying relationships between measured variables.

Overview

Factor analysis aims to find independent latent variables, known as factors, that can explain the patterns of correlations within a set of observed variables. It is often used in data reduction to identify a small number of factors from a large set of variables, making it easier to interpret the data. There are two main types of factor analysis: exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). EFA is used to uncover the underlying structure of a relatively large set of variables, while CFA tests the hypothesis that a relationship between observed variables and their underlying latent constructs exists.

Mathematical Formulation

The mathematical model for factor analysis can be expressed as:

\[X = \mu + \Lambda F + \epsilon\]

where \(X\) is the vector of observed variables, \(\mu\) is the vector of means, \(\Lambda\) is the matrix of factor loadings, \(F\) is the vector of unobserved latent factors, and \(\epsilon\) is the vector of error terms. The factor loadings represent the correlations between the factors and the observed variables.

Applications

Factor analysis is applied in various domains to uncover the underlying structure in datasets:

- In psychology, it is used to identify factors that influence mental processes and behaviors. - In marketing, it helps in identifying underlying dimensions of consumer preferences and attitudes. - In sociology, it is utilized to explore social attitudes and values. - In medicine, factor analysis can reveal patterns in symptoms, aiding in the diagnosis and understanding of diseases.

Limitations

While factor analysis is a powerful tool, it has limitations. The interpretation of factors can be subjective, and the method relies heavily on the assumption of linearity between variables and factors. Additionally, the choice of the number of factors to retain can be somewhat arbitrary and influence the results.

See Also

• Principal component analysis
• Cluster analysis
• Multivariate statistics

References

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