Structural equation modeling

CB-SEM: Structural Equation Modeling Online

Estimate structural equation models with latent variables by maximum likelihood, the method used by AMOS and lavaan. Write the model in lavaan syntax and get fit indices, loadings, structural paths, R², indirect effects with Monte Carlo confidence intervals, and reliability and validity.

Free · No sign-upReplaces AMOS · lavaan · MplusWorks with your data

Updated 13 October 2026 · By Bibin Xavier

Data 180 × 12

Paste from Excel, or drop a CSV, TSV or Excel file here.Upload file

Example data 180 rows, 12 variables. Data stays in this browser tab.

Respondenttext · 180 levels	TR1numeric · coded	TR2numeric · coded	TR3numeric · coded	TR4numeric · coded	SAT1numeric · coded	SAT2numeric · coded	SAT3numeric · coded	SAT4numeric · coded	LOY1numeric · coded	LOY2numeric · coded	LOY3numeric · coded
R001	2	2	1	2	3	1	3	3	4	2	3
R002	5	4	5	4	3	4	3	4	4	3	4
R003	4	4	3	4	1	2	2	4	2	3	1
R004	1	2	3	2	4	4	4	4	3	4	4
R005	3	4	2	3	3	3	3	4	3	2	2
R006	5	4	5	4	5	5	4	4	4	3	4
R007	3	4	4	4	3	3	2	4	1	3	3
R008	3	3	4	4	5	4	5	5	5	5	4

Variables and options

Model syntax

TR =~ TR1 + TR2 + TR3 + TR4
SAT =~ SAT1 + SAT2 + SAT3 + SAT4
LOY =~ LOY1 + LOY2 + LOY3

SAT ~ a*TR
LOY ~ b*SAT + c*TR

indirect := a*b
total := c + a*b

lavaan syntax: F =~ x1 + x2 measurement, Y ~ a*X regression with label, x1 ~~ x2 covariance, ind := a*b defined effect.

Identify latent variables by fixing their variance to 1 (instead of the first loading)

Results update automatically when you change an option.

Results

180 rows · example data

χ²(41)50.45p = .148

CFI.989

TLI.985

RMSEA.03690% CI [.000, .065]

SRMR.037

Table 1

Model fit

Index	Value	Threshold	Verdict
χ² / df	1.23	≤ 3 (≤ 5 acceptable)	Good
CFI	.989	≥ .95 (≥ .90 acceptable)	Good
TLI	.985	≥ .95 (≥ .90 acceptable)	Good
RMSEA	.036 [.000, .065]	≤ .06 (≤ .08 acceptable)	Good
SRMR	.037	≤ .08	Good
p-close (RMSEA ≤ .05)	.755	> .05	Good
GFI	.952	≥ .90	Good
AIC / BIC	4986.9 / 5066.7	lower is better (compare models)	—

Note. Thresholds follow Hu and Bentler (1999). With small samples or many indicators, judge fit on several indices together.

Structural model (standardized estimates)

Table 2

Measurement model (factor loadings)

Construct	Indicator	Estimate	SE	z	p	Standardized (λ)
TR	TR1	1.000^a	—	—	—	.730
TR	TR2	1.014	0.111	9.17	< .001	.754
TR	TR3	1.071	0.117	9.12	< .001	.749
TR	TR4	1.092	0.117	9.35	< .001	.772
SAT	SAT1	1.000^a	—	—	—	.723
SAT	SAT2	1.065	0.116	9.20	< .001	.771
SAT	SAT3	1.054	0.114	9.21	< .001	.772
SAT	SAT4	0.862	0.111	7.74	< .001	.636
LOY	LOY1	1.000^a	—	—	—	.795
LOY	LOY2	1.023	0.097	10.55	< .001	.786
LOY	LOY3	1.098	0.100	11.02	< .001	.830

Note. ^a Fixed to 1 to set the scale of the construct. Standardized loadings of .70 or more are ideal; below .50 suggests the item measures the construct poorly (Hair et al., 2019).

Table 3

Structural paths (regressions)

Path	Label	Estimate	SE	z	p	95% CI	Standardized (β)
TR → SAT	a	0.591	0.101	5.85	< .001	[0.393, 0.789]	.576
SAT → LOY	b	0.579	0.117	4.95	< .001	[0.350, 0.808]	.539
TR → LOY	c	0.227	0.108	2.10	.035	[0.016, 0.437]	.206

Table 4

Explained variance (R²)

Variable	R²
TR1	.532
TR2	.569
TR3	.562
TR4	.595
SAT1	.523
SAT2	.594
SAT3	.595
SAT4	.405
LOY1	.632
LOY2	.617
LOY3	.689
SAT	.332
LOY	.460

Table 5

Variances and residual variances

Variable	Estimate	SE	z	p	Standardized
TR1	0.485	0.064	7.53	< .001	.468
TR2	0.430	0.060	7.21	< .001	.431
TR3	0.494	0.068	7.28	< .001	.438
TR4	0.448	0.064	6.94	< .001	.405
SAT1	0.530	0.070	7.54	< .001	.477
SAT2	0.451	0.066	6.89	< .001	.406
SAT3	0.440	0.064	6.87	< .001	.405
SAT4	0.636	0.077	8.30	< .001	.595
LOY1	0.390	0.058	6.67	< .001	.368
LOY2	0.436	0.064	6.86	< .001	.383
LOY3	0.365	0.062	5.87	< .001	.311
TR	0.552	0.104	5.29	< .001	1.000
SAT	0.389	0.080	4.84	< .001	.668
LOY	0.362	0.069	5.23	< .001	.540

Table 6

Defined effects (e.g. indirect effects)

Effect	Definition	Estimate	SE	z	p	95% CI (delta)	95% CI (Monte Carlo)
indirect	`a*b`	0.3419	0.0826	4.14	< .001	[0.180, 0.504]	[0.189, 0.520]
total	`c + a*b`	0.5685	0.1023	5.56	< .001	[0.368, 0.769]	[0.360, 0.773]

Note. Monte Carlo confidence intervals (10,000 draws; Preacher and Selig, 2012) do not assume the indirect effect is normally distributed and are preferred for products of coefficients.

Table 7

Construct reliability and convergent validity

Construct	Items	Cronbach’s α	CR (ρ_c)	McDonald’s ω	AVE	√AVE	MSV	Verdict
TR	4	.838	.838	.838	.565	.751	.332	Supported
SAT	4	.816	.817	.750	.529	.727	.432	Supported
LOY	3	.846	.846	.748	.646	.804	.432	Supported

Note. Thresholds (Hair et al., 2019; Fornell and Larcker, 1981): α, CR and ω ≥ .70 (≤ .95); AVE ≥ .50; MSV below AVE for discriminant validity. CR and AVE use standardized loadings; ω uses the unstandardized solution.

Table 8

Fornell–Larcker criterion

	TR	SAT	LOY
TR	.751
SAT	.576	.727
LOY	.516	.657	.804

Note. Diagonal: √AVE. Off-diagonal: latent correlations. Discriminant validity holds when each √AVE exceeds that construct’s correlations.

Table 9

Heterotrait–monotrait ratio (HTMT)

	TR	SAT	LOY
TR	—
SAT	.584	—
LOY	.510	.648	—

Note. HTMT below .85 (strict) or .90 (liberal) supports discriminant validity (Henseler, Ringle and Sarstedt, 2015). Computed from absolute item correlations.

Interpretation

The model fits the data well (CFI = .989, RMSEA = .036, SRMR = .037). Significant paths: TR → SAT (β = 0.58); SAT → LOY (β = 0.54); TR → LOY (β = 0.21). The indirect effect is 0.342, 95% Monte Carlo CI [0.189, 0.520].

The structural model showed acceptable fit, χ²(41) = 50.45, p = .148, CFI = .989, TLI = .985, RMSEA = .036, 90% CI [.000, .065], SRMR = .037.

How to cite this tool

Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.

APA 7

Xavier, B. (2026). CB-SEM: Structural Equation Modeling Online [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/cb-sem-structural-equation-modeling/

BibTeX

@software{xavier_cbsem_2026,
  author = {Xavier, Bibin},
  title = {CB-SEM: Structural Equation Modeling Online},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/cb-sem-structural-equation-modeling/},
  note = {Web application}
}

Suggested sentence for your methods section: “The analysis was carried out with the free CB-SEM: Structural Equation Modeling Online tool on FreeStatisticalTools.com (Xavier, 2026).”

What is CB-SEM?

Covariance-based structural equation modeling (CB-SEM) estimates relationships among latent constructs measured by several indicators, and tests how well the whole model reproduces the observed covariances. This tool uses maximum likelihood estimation, the default in AMOS and lavaan, and returns the fit indices, loadings, paths and indirect effects most journals require.

When to use CB-SEM

Use CB-SEM to test a theory that links latent constructs measured by several indicators, when you care about overall model fit. It is the standard approach in psychology, education and much of management research; PLS-SEM is the alternative for prediction-oriented or formative models.

Assumptions

Multivariate normality of the indicators (maximum likelihood).
A sample size of roughly 200 or more, or at least 10 cases per free parameter.
Every latent variable needs at least three indicators, or two with constraints, and one fixed loading or variance.

How to set up your data

One row per respondent and one numeric column per indicator. Items named with a shared prefix and a number (TR1, TR2, TR3) can be turned into a model automatically.

How to use this tool

Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
Choose the variables. Select the model syntax.
Set the options. Fill in identify latent variables by fixing their variance to 1 (instead of the first loading).
Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.

How to report CB-SEM in APA style

Report the estimator, χ²(df), CFI, TLI, RMSEA with its 90% CI and SRMR, then standardized loadings, structural coefficients with p-values and R², and indirect effects with confidence intervals.

Frequently asked questions

What is the difference between CB-SEM and PLS-SEM?

CB-SEM estimates common factors and tests overall model fit, so it suits theory testing. PLS-SEM builds composites and maximises explained variance, so it suits prediction, complex models and formative constructs.

Which fit indices should I report?

Report χ² with its degrees of freedom and p-value, CFI, TLI, RMSEA with its 90% confidence interval, and SRMR. Common cut-offs are CFI and TLI ≥ .95, RMSEA ≤ .06 and SRMR ≤ .08 (Hu and Bentler, 1999).

How do I test an indirect effect in SEM?

Label the paths, for example SAT ~ a*TR and LOY ~ b*SAT, then add a line ind := a*b. The tool reports the indirect effect with delta-method and Monte Carlo confidence intervals.

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