# Confirmatory Factor Analysis

**Confirmatory Factor Analysis (CFA)** is a statistical technique used in the field of psychometrics to verify the factor structure of a set of observed variables. Unlike exploratory factor analysis, which allows the data itself to reveal the underlying structure, CFA tests a pre-specified factor model to see how well it fits the observed data.

## History

The development of Confirmatory Factor Analysis is closely tied to the evolution of Structural Equation Modeling (SEM). Jöreskog was a seminal figure in formalizing the method as a way to validate factor structures suggested by exploratory factor analysis or theoretical constructs.

## Key Concepts

Factors: Factors are latent variables that explain the correlations or covariances among observed variables.

- Factor Loadings: Factor loadings are the coefficients that indicate the relationship between each observed variable and the underlying factor.
- Error Terms: Error terms account for the variability in observed variables not explained by the factors.
- Model Specification: In CFA, researchers must a priori specify which observed variables are related to which factors. This is typically based on theoretical considerations or previous research.
- Model Estimation: Various methods can be used for estimating the model parameters, such as maximum likelihood, generalized least squares, or weighted least squares.

## Model Evaluation

- Goodness-of-Fit: Several goodness-of-fit indices exist to evaluate how well the model fits the data, including Chi-square, Root Mean Square Error of Approximation (RMSEA), Comparative Fit Index (CFI), and Tucker-Lewis Index (TLI).
- Modification Indices: Modification indices can be inspected to identify paths or constraints that, if changed, would improve the model fit.

## Applications

- Psychology: CFA is widely used in psychology to confirm the validity of constructs like intelligence, self-esteem, and many others.
- Business: In business, CFA can be used to validate the structure of customer satisfaction surveys, organizational climate scales, and more.
- Healthcare: In healthcare, it is often employed to validate measurement instruments such as health-related quality of life scales.
- Software Tools: Several statistical software packages can perform CFA, including SPSS Amos, Mplus, and R packages like lavaan.

## Limitations

- Requires large sample sizes for stable and reliable estimates.
- It is confirmatory in nature, meaning it doesn't allow for the discovery of unexpected or novel structures in the data.
- Assumes linear relationships between factors and observed variables.