Structural Equation Modeling (SEM): How to Run It Online and Report It in APA7
Structural equation modeling tests a theory about relationships among observed and latent (unobserved) variables. It combines a measurement model (how indicators measure latent variables) with a structural model (paths between latent variables).
What is Structural Equation Modeling (SEM)?
Structural equation modeling tests a theory about relationships among observed and latent (unobserved) variables. It combines a measurement model (how indicators measure latent variables) with a structural model (paths between latent variables).
When to use it
Use it to test theory-driven models with several outcomes, latent constructs measured by multiple items, and mediation or other path patterns tested together while accounting for measurement error.
Example question: Does the latent variable social support predict the latent variable wellbeing?
Preparing your data
One row per participant and one numeric column per observed item or scale. Name items clearly (for example sup1, sup2, sup3, well1, well2, well3).
Example dataset (simulated; first rows of 300): id, sup1 to sup3 (items of latent support), well1 to well3 (items of latent wellbeing).
| id | sup1 | sup2 | sup3 | well1 | well2 | well3 |
|---|---|---|---|---|---|---|
| 1.0 | 2.8 | 3.95 | 3.19 | 2.31 | 2.95 | 4.49 |
| 2.0 | 5.0 | 4.02 | 3.85 | 6.27 | 5.54 | 4.79 |
| 3.0 | 5.24 | 3.91 | 4.21 | 4.68 | 5.79 | 4.78 |
| 4.0 | 4.12 | 3.28 | 2.93 | 2.83 | 2.25 | 4.29 |
| 5.0 | 3.72 | 3.3 | 4.43 | 5.24 | 4.77 | 5.49 |
| 6.0 | 3.09 | 3.98 | 4.17 | 2.81 | 3.78 | 2.31 |
Download the full example CSV to practise. If the tool asks for a different layout, follow the instructions on its page.
How to run it on StatFas
- Open the Structural Equation Modeling (SEM) tool. You do not need an account to run an analysis.
- Enter your data or upload your file, laid out as described in Preparing your data.
- Select your variables. Assign each item to a latent variable, then specify the paths between latent variables (or use the default model). Choose an estimator suited to your data (for example maximum likelihood, or a robust option for non-normal data).
- Choose any options the tool offers (for example the confidence level, a post hoc test or the error bar type) and run the analysis.
- Read the results on the page. Check the assumptions first, then the test statistic, the p-value and the effect size.
- To save your results, log in or create a free account. Every analysis stays free; the account is only needed to download.
Assumptions and checks
- The model is identified: you have more known information (variances and covariances) than parameters to estimate. Each latent variable needs at least three indicators, or two with constraints.
- A sufficiently large sample. Rules of thumb are at least 200 cases, or 10 cases per estimated parameter; complex models need more.
- Continuous, approximately multivariate normal indicators for maximum likelihood; use robust or ordinal estimators for skewed or ordinal items.
- Missing data are handled sensibly (for example full-information maximum likelihood) and reported.
- Modification indices are used only with theoretical justification, because data-driven changes capitalise on chance.
Key numbers to look at
- Chi-square test of exact fit: χ²(df), p. It is often significant in large samples, so consider the other indices.
- CFI and TLI: values of about .95 or higher suggest good fit (.90 or higher is acceptable).
- RMSEA with its 90% CI: about .06 or lower is good and up to .08 is acceptable.
- SRMR: about .08 or lower is acceptable.
- Standardised loadings and path coefficients with p-values and, ideally, confidence intervals.
Effect size
Standardised path coefficients (β) are effect sizes for the structural paths. Report R2 for endogenous variables.
Benchmarks such as small, medium and large are conventions (Cohen, 1988), not rules. Interpret size in the context of your field.
Reporting Structural Equation Modeling (SEM) in APA7 style
Template
Provide the estimator, how missing data were handled, and a figure of the model with standardised coefficients.
Worked example (simulated data)
The numbers below were computed from the example CSV above, so you can reproduce them. They illustrate the format and are not real findings.
Fit indices come from the semopy package on the simulated data; they will differ in other software if estimators differ.
APA7 reminders: italicise statistical symbols (M, SD, t, F, p, r), report exact p-values to two or three decimals (use p < .001 for very small values), omit the leading zero for values that cannot exceed 1 (.45, not 0.45), and put degrees of freedom in parentheses. See the APA7 cheat sheet.
Common mistakes
- Judging fit by the chi-square test alone, or by a single index.
- Adding paths and correlated errors only to improve fit.
- Using small samples for complex models.
Related analyses and alternatives
- CFA: To test only the measurement model.
- Mediation: For simple observed-variable mediation.
- EFA: If the factor structure is still unknown.
Not sure which analysis you need? Use the test chooser.
Frequently asked questions
What fit values are acceptable?
Common guidelines are CFI and TLI ≥ .95, RMSEA ≤ .06 and SRMR ≤ .08 (Hu & Bentler, 1999). Treat them as guides rather than strict cut-offs.
Does good fit prove my model is correct?
No. Good fit shows the model is consistent with the data, but other models may fit equally well.
References
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55. https://doi.org/10.1080/10705519909540118
This guide is for learning and for preparing results for your own reports. You remain responsible for choosing an appropriate analysis, checking its assumptions and interpreting the results. Follow the formatting rules of your journal or institution where they differ from APA7.