Overview
CFA estimates a researcher-approved reflective measurement model for continuous numeric indicators and evaluates loadings, residuals, global fit, reliability and validity evidence.
Theoretical foundation
The covariance model is Σ = ΛΦΛ′ + Θ. Marker-variable identification fixes the first loading of each factor to one. Global fit compares observed and model-implied covariance structures; local diagnostics examine loadings and residuals.
Key equations
x=\Lambda\xi+\delta,\qquad \Sigma=\Lambda\Phi\Lambda^\top+\ThetaAlgorithm and computation
The service validates a deliberately small non-executable model language, requires at least three confirmed indicators per approved construct, prevents cross-loadings, and uses listwise complete data. semopy 2.3.11 fits MLW with SLSQP and marker identification. Survlytics computes standardised residuals, SRMR, a noncentral-chi-square RMSEA interval, CR, AVE and Fornell–Larcker evidence, and checks covariance conditioning, convergence and admissibility.
Required data
Continuous numeric confirmed reflective indicators, approved constructs, an identified model and at least max(50, twice the free-parameter count) complete observations.
Assumptions
Normal-theory continuous-indicator ML assumptions, independent observations and a theory-specified reflective model. Fit indices provide complementary evidence and are not automatic validity decisions.
Available settings
Approved measurement specification; correlated or orthogonal factors; listwise missing-data handling; marker identification; display precision.
Worked example
The frozen synthetic benchmark has two correlated factors with three indicators each and N = 500. Survlytics estimates the same specification as R/lavaan and compares fit and standardised loadings within stored tolerances.
Limitations and non-claims
WLSMV, ordinal thresholds, robust ML, FIML, correlated residual modification, cross-loadings and automatic modification-index model search are not implemented.
Validation notes
A frozen R 4.5.2/lavaan 0.7-2 benchmark compares chi-square, CFI, TLI, RMSEA, SRMR, AIC, BIC, loadings, factor variances and factor correlation. Fit and parameter tolerances are 0.001; information criteria tolerance is 0.01.
How Surv.lytics computes Confirmatory factor analysis
{'estimator': 'MLW', 'solver': 'SLSQP'}PASS: estimates and fit indices reconcile to the frozen lavaan normal-theory ML reference within declared tolerances. Tolerance: 0.001 for fit/parameters; 0.01 for AIC/BIC Known differences: Library objective names differ (semopy MLW versus lavaan ML); the fixture reconciles the implemented covariance convention.
References and verification status
Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39–50. https://doi.org/10.2307/3151312
VerifiedOpen DOIHu, 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
Unverified draftOpen DOI
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