Guide to Conducting Confirmatory Factor Analysis (CFA) for Validating a Measurement Scale

Confirmatory Factor Analysis (CFA) is a structural equation modeling (SEM) technique used to test whether your data fits a predefined theoretical model of latent factors (constructs) and their relationships to observed variables (items). It's ideal for scale validation, as it confirms if your scale measures the intended factors as per theory. I'll break this down step-by-step, starting with your specific request. Note that the full process assumes you're using software like R (with lavaan package), Mplus, AMOS, or SAS. If you're new to this, I recommend starting with sample data to practice.

1. Specify the Theoretical Factor Structure Based on Prior Research

• Why based on prior research? CFA tests theory-driven models. Drawing from prior studies ensures your model is rooted in established knowledge, reducing bias and enhancing validity. For example, if validating a scale like the Big Five personality inventory, you'd reference meta-analyses or key papers (e.g., Costa & McCrae, 1992) that define factors like Neuroticism or Extraversion.


• Steps to specify the structure:

• Example: For the Perceived Stress Scale (PSS; Cohen et al., 1983), prior research specifies a single-factor structure (all items load on one "perceived stress" latent variable). If adapting it, check studies like Lee (2012) for potential two-factor models (e.g., distress vs. coping).

• Notation: Factors are denoted as η (eta) for endogenous or ξ (xi) for exogenous. Items are x1, x2, etc.


• Example structure for a hypothetical 3-factor job satisfaction scale (based on prior research like Spector, 1997):


• Factor 1: Pay Satisfaction (items: x1 = "Fair pay," x2 = "Benefits adequacy")


• Factor 2: Promotion Satisfaction (items: x3 = "Advancement opportunities," x4 = "Career growth")


• Factor 3: Supervision Satisfaction (items: x5 = "Support from boss," x6 = "Fair treatment")


• Hypothesized: Factors are correlated (e.g., r = 0.4–0.6 from literature), no cross-loadings.

d. Document assumptions: State if it's a first-order CFA (factors directly measured by items) or higher-order (e.g., a second-order "overall satisfaction" factor loading on the three sub-factors). Base on theory—e.g., if prior research supports a hierarchical model, specify it.

• Tip: If your scale is new or adapted, conduct EFA first on a pilot sample to refine, but for full validation, pivot to CFA on a separate sample. Cite 3–5 key references to justify your spec.

2. Prepare Your Data

• Sample size: Aim for 5–10 observations per parameter (e.g., 200+ for a simple model). Use representative data for your population.


• Data cleaning: Handle missing data (e.g., via FIML in software), check normality (CFA assumes multivariate normality; use robust estimators like MLR if violated), and ensure no multicollinearity (correlations < 0.9).


• Software setup:


• R: Install lavaan (install.packages("lavaan")).


• Example data: Use simulated or public datasets (e.g., from lavaan demos).

3. Specify and Estimate the Model

• Model syntax: Translate your theoretical structure into code.


• Example in R (lavaan) for the job satisfaction model above:

library(lavaan)
    model <- '
      # Latent factors defined by items
      Pay =~ x1 + x2
      Promotion =~ x3 + x4
      Supervision =~ x5 + x6
      
      # Allow factors to correlate (based on theory)
      Pay ~~ Promotion
      Pay ~~ Supervision
      Promotion ~~ Supervision
    '
    fit <- cfa(model, data = your_dataset)
    summary(fit, fit.measures = TRUE, standardized = TRUE)

• This estimates factor loadings, variances, and covariances using maximum likelihood (ML).


• Estimation checks: Fix one loading per factor to 1 for identification. Ensure the model is just-identified or over-identified.

4. Evaluate Model Fit

• Key fit indices (thresholds from Hu & Bentler, 1999):

• Interpret: If fit is poor (e.g., RMSEA > 0.10), revisit theory—don't freely modify without justification (use modification indices sparingly, e.g., MI > 10, and check Δχ²).

5. Assess Reliability and Validity

• Factor loadings (λ): > 0.7 ideal; average variance extracted (AVE) > 0.50 for convergent validity.


• Reliability: Cronbach's α > 0.70 per factor; composite reliability (CR) > 0.70.


• Discriminant validity: Fornell-Larcker criterion: AVE > squared factor correlations (φ²). Or, cross-loadings < main loadings.


• Invariance testing: If comparing groups (e.g., gender), test configural, metric, scalar invariance using multi-group CFA.


• Example output interpretation: In R, parameterEstimates(fit) shows loadings; fitMeasures(fit) gives indices.

6. Report and Iterate

• Reporting: Include path diagram, fit indices, loadings table, and references to theory. Discuss limitations (e.g., cross-sectional data can't infer causality).


• If model fails: Compare alternatives (e.g., via χ² difference test) or collect more data. Re-specify only if supported by theory.


• Common pitfalls: Over-reliance on fit indices without theory; ignoring local fit (e.g., low R² for items < 0.50, indicating poor measurement).