What CFA does
Confirmatory factor analysis tests whether a set of observed items measures the latent constructs you specified in advance. Each item loads on one construct, the constructs are allowed to correlate, and the model is estimated by maximum likelihood. If the model fits and the items load strongly, your questionnaire measures what it is meant to measure, which is the foundation for any structural model.
CFA or EFA?
Use exploratory factor analysis when you do not yet know how items group, for example when developing a new scale. Use CFA when the structure comes from theory or from established scales. Running both on the same sample is circular; if you develop a scale, split the sample or collect new data for the CFA.
Build the model
In CB-SEM Studio, load your data and click Create constructs: items named with a shared prefix, such as SAT1, SAT2 and SAT3, become one construct. Do not draw any paths: a CFA has only correlations between constructs, which the app adds automatically when you leave the structural part empty and choose Correlate all constructs. In syntax:
SAT =~ SAT1 + SAT2 + SAT3 + SAT4
TRUST =~ TR1 + TR2 + TR3
LOY =~ LOY1 + LOY2 + LOY3Check model fit
Report χ² with its degrees of freedom and p value, plus CFI, TLI, RMSEA with its 90% confidence interval, and SRMR. Good fit is commonly read as CFI and TLI ≥ 0.95, RMSEA ≤ 0.06 and SRMR ≤ 0.08 (Hu and Bentler, 1999); CFI ≥ 0.90 and RMSEA ≤ 0.08 are often called acceptable. The chi-square test is nearly always significant in large samples, so do not reject a model on that alone. See the guide to fit indices.
Inspect the loadings
Standardized loadings should be significant and ideally 0.70 or higher, meaning the construct explains about half of the item’s variance. Items between 0.50 and 0.70 can stay if reliability and AVE are adequate; items below 0.50 are candidates for removal, but check whether dropping them changes the meaning of the construct.
Reliability and convergent validity
| Measure | Formula | Threshold |
|---|---|---|
| Composite reliability (CR) | (Σλ)² / [(Σλ)² + Σθ] | ≥ 0.70 |
| Average variance extracted (AVE) | Σλ² / k | ≥ 0.50 |
| Cronbach’s α | From item variances and covariances | ≥ 0.70 |
λ are standardized loadings, θ the standardized error variances (1 − λ²) and k the number of items (Fornell and Larcker, 1981).
Discriminant validity
- HTMT: the heterotrait–monotrait ratio of correlations should be below 0.85, or 0.90 for conceptually close constructs (Henseler, Ringle and Sarstedt, 2015). It is the most sensitive test.
- Fornell–Larcker: the square root of each construct’s AVE should exceed its correlations with the other constructs.
- MSV: the maximum squared correlation with any other construct should be smaller than the AVE.
Common problems and fixes
- Poor fit. Inspect modification indices. A large MI for a residual covariance between two items of the same construct with similar wording can justify freeing it; a large cross-loading suggests an ambiguous item.
- Heywood case (a negative error variance or a standardized loading above 1). Often caused by a construct with only two items, a small sample or an outlier; consider adding items or constraining the parameter.
- Low AVE. Drop the weakest items if content validity allows.
- High HTMT. The two constructs may not be distinct; consider merging them or a second-order model.
Reporting checklist
- Estimator, software and how latent variables were scaled.
- Fit indices with cut-offs, and any modifications with their justification.
- A table of standardized loadings, CR, AVE and α per construct.
- An HTMT or Fornell–Larcker table for discriminant validity.
References
- Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd ed.). Guilford.
- Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39–50.
- Henseler, J., Ringle, C. M., & Sarstedt, M. (2015). A new criterion for assessing discriminant validity in variance-based structural equation modeling. Journal of the Academy of Marketing Science, 43(1), 115–135.
- Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis. Structural Equation Modeling, 6(1), 1–55.