What CB-SEM is
Covariance-based structural equation modeling (CB-SEM) estimates a set of relationships among latent variables, such as trust, satisfaction or loyalty, each measured by several observed indicators. It combines a measurement model, a confirmatory factor analysis linking each construct to its items, with a structural model of hypothesised paths between constructs.
The parameters are chosen so that the covariance matrix implied by the model is as close as possible to the observed covariance matrix, usually by maximum likelihood (ML). Because the model reproduces the data, CB-SEM can test how well the whole theory fits, through the chi-square test and fit indices such as CFI and RMSEA. That makes it the method of choice for confirming theory, whereas PLS-SEM is preferred for prediction and exploratory models.
Prepare the data
- Sample size. 200 cases is a common minimum for moderately complex models; models with many parameters, low loadings or non-normal data need more. A rule of thumb is at least 10 cases per estimated parameter.
- Missing data and outliers. Check the amount and pattern of missing data with Little’s MCAR test, and screen for multivariate outliers with Mahalanobis distance.
- Normality. ML assumes multivariate normality. Report skewness, kurtosis and Mardia’s coefficient; CB-SEM Studio computes all three. Absolute skewness below 2 and kurtosis below 7 are usually considered tolerable.
- Reverse-coded items. Reverse negatively worded items before analysis, for example with the free data cleaning tool.
Specify the model
Draw one construct per latent variable and assign its items, then draw an arrow for each hypothesis. Exogenous constructs, the ones with no incoming arrows, are allowed to correlate by default. In lavaan syntax, which CB-SEM Studio writes for you, a model looks like this:
QUAL =~ Q1 + Q2 + Q3 + Q4
TRUST =~ T1 + T2 + T3
SAT =~ S1 + S2 + S3
LOY =~ L1 + L2 + L3
SAT ~ QUAL + TRUST
LOY ~ SAT + TRUSTIdentification
A model is identified when there is enough information to estimate every parameter. Each latent variable needs a scale, either by fixing its first loading to 1 (the marker method, the default) or by fixing its variance to 1. Each construct should have at least three indicators, or two if it correlates with other constructs. The model’s degrees of freedom, the number of observed variances and covariances minus the number of free parameters, must be zero or positive. CB-SEM Studio checks this before it runs and handles single-item constructs by fixing the item’s error variance to zero.
Step 1: assess the measurement model
Anderson and Gerbing’s (1988) two-step approach first tests the measurement model on its own as a CFA, with all constructs correlated, and only then the structural model. See the CFA tutorial for details.
| Criterion | What to report | Common threshold |
|---|---|---|
| Indicator reliability | Standardized loadings | ≥ 0.70; 0.50–0.70 acceptable if CR and AVE hold |
| Internal consistency | Composite reliability (CR), Cronbach’s α | ≥ 0.70 |
| Convergent validity | Average variance extracted (AVE) | ≥ 0.50 |
| Discriminant validity | HTMT, Fornell–Larcker, MSV | HTMT < 0.85; √AVE above correlations; MSV < AVE |
| Model fit | χ²/df, CFI, TLI, RMSEA, SRMR | See the fit indices guide |
Step 2: assess the structural model
- Fit. Report the same fit indices for the structural model. With a saturated structural part, fit equals the CFA; constraining paths makes it worse.
- Path coefficients. Report each standardized coefficient β with its unstandardized estimate, standard error, z value and p value. A hypothesis is supported when p < .05 and the sign matches the prediction.
- Explained variance. Report R² for every endogenous construct.
Mediation
With paths X → M, M → Y and X → Y, the indirect effect is the product of the X → M and M → Y paths. Its sampling distribution is not normal, so test it with a Monte Carlo or bootstrap confidence interval rather than the Sobel test (Preacher and Selig, 2012). CB-SEM Studio finds every indirect route, reports its estimate, delta-method standard error and a 95% Monte Carlo interval, and classifies the mediation following Zhao, Lynch and Chen (2010).
Modification indices
If fit is poor, modification indices show which fixed parameters would most reduce the chi-square if freed. The usual candidates are residual covariances between items of the same construct that share wording, or a cross-loading that signals an ambiguous item. Free parameters one at a time, only with a theoretical reason, and report every change; data-driven modifications capitalise on chance (MacCallum, Roznowski and Necowitz, 1992).
Report the results
Report the software, the estimator (maximum likelihood), how latent variables were scaled, how missing data were handled, all fit indices with their cut-offs, the measurement model table, the structural paths with R², the indirect effects with their interval method, and any modifications made.
CB-SEM Studio exports every table to Word, Excel or LaTeX and drafts a methods paragraph that covers these points.
References
- Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103(3), 411–423.
- Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate data analysis (8th ed.). Cengage.
- Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis. Structural Equation Modeling, 6(1), 1–55.
- Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford.
- MacCallum, R. C., Roznowski, M., & Necowitz, L. B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111(3), 490–504.
- Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6(2), 77–98.
- Zhao, X., Lynch, J. G., & Chen, Q. (2010). Reconsidering Baron and Kenny: Myths and truths about mediation analysis. Journal of Consumer Research, 37(2), 197–206.