Confirmatory Factor Analysis (CFA)

Confirmatory Factor Analysis (CFA) is an advanced statistical method that belongs to the family of structural equation modeling techniques. Unlike its exploratory counterpart (exploratory factor analysis), CFA begins with a theory-driven model that specifies how measured variables relate to unobserved constructs.

Core Principles

Model Specification

• Researchers must define a priori:
  • Number of factors (latent variables)
  • Which observed variables load onto which factors
  • Whether factors are correlated
  • Error term relationships

Mathematical Foundation

The basic CFA model can be expressed as:

x = Λξ + δ

Where:

• x represents observed variables
• Λ (lambda) is the matrix of factor loadings
• ξ (xi) represents latent factors
• δ (delta) represents measurement errors

Applications

CFA serves several critical functions in research:

1. Construct Validation

   • Testing theoretical measurement models
   • Assessing measurement invariance
   • Validating psychometric instruments

2. Scale Development

   • Confirming factor structures
   • Evaluating item performance
   • Refining measurement tools

3. Theory Testing

   • Examining hypothesized relationships
   • Comparing competing models
   • Supporting theoretical framework development

Model Evaluation

Fit Indices

Researchers typically examine multiple fit indices:

• Chi-square test of model fit
• Comparative Fit Index (CFI)
• Root Mean Square Error of Approximation (RMSEA)
• Standardized Root Mean Square Residual (SRMR)

Model Modification

When model fit is inadequate:

• Modification indices guide potential changes
• model specification adjustments
• Cross-validation with new data

Assumptions and Requirements

1. Statistical Assumptions

   • multivariate normality
   • Adequate sample size
   • Continuous variables (or appropriate estimators)

2. Practical Considerations

   • Theoretical justification
   • Sufficient indicators per factor
   • identification constraints

Software Implementation

Common statistical packages for CFA include:

• LISREL
• Mplus
• R (lavaan package)
• AMOS
• EQS

Limitations and Considerations

• Requires strong theoretical foundation
• Sensitive to sample size and missing data
• Model modifications should be theoretically justified
• Results may not generalize across populations

Best Practices

1. Pre-Analysis

   • Clear theoretical framework
   • Adequate sample size planning
   • proper variable screening

2. Analysis

   • Multiple fit indices examination
   • Parameter estimate evaluation
   • reliability assessment

3. Reporting

   • Complete model specification
   • Fit indices with confidence intervals
   • Standardized and unstandardized results

CFA continues to evolve with advances in statistical computing and methodology, remaining a crucial tool in social sciences, psychology, and other fields requiring latent variable modeling.