14.2 Confirmatory factor analysis

Confirmatory Factor Analysis (CFA) is a powerful tool for testing theories about the relationships between observed variables and underlying factors. It allows researchers to validate measurement instruments and confirm hypothesized factor structures based on prior knowledge or theory.

CFA uses structural equation modeling to specify, estimate, and evaluate factor models. Researchers assess model fit using various indices, refine models as needed, and compare alternative structures. This rigorous approach helps ensure valid and reliable measurements in market research studies.

Confirmatory Factor Analysis

Exploratory vs confirmatory factor analysis

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• Exploratory Factor Analysis (EFA)
  • Investigates underlying factor structure when it is unknown or uncertain
  • Identifies the number and nature of factors that best explain the data
  • Generates hypotheses and aids in developing theories (Big Five personality traits)
  • Useful in early stages of research or when exploring a new domain (consumer behavior)
• Confirmatory Factor Analysis (CFA)
  • Tests a pre-existing theory or hypothesis about the factor structure
  • Confirms or disconfirms the proposed relationships between observed variables and latent factors
  • Validates measurement instruments (intelligence tests) and tests established theories (theory of planned behavior)
  • Appropriate when there is a strong theoretical foundation or previous empirical evidence

Confirmatory analysis with structural equations

• Model specification
  • Defines the hypothesized factor structure based on theory or previous research
  • Specifies the relationships between observed variables (survey items) and latent factors (brand loyalty)
  • Determines the number of factors and their indicators (items loading on each factor)
• Model identification
  • Ensures the model is identified, meaning unique parameter estimates can be obtained
  • Sets the scale of latent factors by fixing factor variances (to 1) or factor loadings (marker variable approach)
  • Meets the necessary conditions for identification (t-rule, three-indicator rule)
• Model estimation
  • Selects an appropriate estimation method based on data characteristics and assumptions
    1. Maximum likelihood (ML) for normally distributed data
    2. Weighted least squares (WLS) for categorical or non-normal data
  • Estimates model parameters, including factor loadings, factor variances/covariances, and error variances
• Model evaluation
  • Assesses the overall fit of the model to the observed data
  • Examines the statistical significance and practical relevance of parameter estimates
  • Compares alternative models (hierarchical models, competing theories) to select the best-fitting model

Goodness-of-fit in factor analysis

• Absolute fit indices
  • Chi-square ($\chi^2$) test assesses the discrepancy between the observed and model-implied covariance matrices (p > 0.05 indicates good fit)
  • Root Mean Square Error of Approximation (RMSEA) measures the discrepancy per degree of freedom (values $\leq$ 0.06 suggest good fit)
  • Standardized Root Mean Square Residual (SRMR) measures the average difference between observed and predicted correlations (values $\leq$ 0.08 indicate good fit)
• Incremental fit indices
  • Comparative Fit Index (CFI) compares the fit of the hypothesized model to a null model (values $\geq$ 0.95 suggest good fit)
  • Tucker-Lewis Index (TLI) adjusts the CFI for model complexity (values $\geq$ 0.95 indicate good fit)
  • Normed Fit Index (NFI) assesses the proportion of improvement in fit compared to the null model (values $\geq$ 0.95 suggest good fit)
• Parsimony fit indices
  • Akaike Information Criterion (AIC) balances model fit and complexity (lower values indicate better fit when comparing models)
  • Bayesian Information Criterion (BIC) similar to AIC but penalizes model complexity more heavily (lower values suggest better fit)
• Recommended cutoff values for good fit
  • RMSEA $\leq$ 0.06, SRMR $\leq$ 0.08, CFI and TLI $\geq$ 0.95
  • Lower AIC and BIC values are preferred when comparing competing models

Refinement of factor analysis models

• Model modification
  • Examines modification indices to identify potential sources of misfit (large values suggest areas for improvement)
  • Considers adding or removing paths based on theoretical justification and empirical evidence (significant residual covariances)
  • Avoids capitalizing on chance and overfitting the model by making too many data-driven modifications
• Model comparison
  • Tests alternative models with different factor structures (correlated vs. uncorrelated factors) or parameter constraints (equal factor loadings)
  • Uses likelihood ratio tests or information criteria (AIC, BIC) to compare nested models (significant $\chi^2$ difference or lower AIC/BIC favors the less constrained model)
  • Selects the most parsimonious model that fits the data well and aligns with theoretical expectations
• Cross-validation
  • Splits the sample into calibration and validation subsamples (random split or holdout method)
  • Develops the model using the calibration sample and tests its fit in the validation sample
  • Assesses the stability and generalizability of the model across different samples or populations (measurement invariance)