Confirmatory Factor Analysis Calculator

CFA Inputs

Enter numeric estimates. Mark a free parameter by adding * at the end (example: 0.72*).

Example Data Table

Illustrative 5-item, 1-factor setup. Paste into inputs to test.

Component	Example	Notes
S (5×5)	`1.00 0.62 0.55 0.48 0.50 0.62 1.10 0.58 0.45 0.46 0.55 0.58 0.95 0.44 0.43 0.48 0.45 0.44 0.90 0.52 0.50 0.46 0.43 0.52 1.05`	Symmetric covariance matrix.
Λ (5×1)	`0.82* 0.78* 0.75* 0.70* 0.73*`	One factor with five indicators.
Φ (1×1)	`1.00`	Fix factor variance for scaling.
Θ diag	`0.33* 0.49* 0.39* 0.41* 0.52*`	Positive error variances.

Formulas Used

Implied covariance: Σ = Λ Φ Λᵀ + Θ
ML fit function: F_ML = ln|Σ| + tr(SΣ⁻¹) − ln|S| − p
Chi-square: χ² = (N − 1)·F_ML
Degrees of freedom: df = p(p+1)/2 − t
RMSEA: √max((χ² − df)/(df·(N−1)), 0)
CFI: 1 − max(χ² − df,0) / max(χ²₀ − df₀, ε)
TLI: 1 − ((χ²/df) − 1) / max((χ²₀/df₀) − 1, ε)
SRMR: √mean(r²), r = (Sᵢⱼ − Σᵢⱼ)/√(SᵢᵢSⱼⱼ)

Baseline uses Σ₀ = diag(S). ε is a small stabilizer.

How to Use This Calculator

Compute your observed covariance matrix S from data.
Enter Λ to reflect your hypothesized factor structure.
Enter Φ for factor variances and correlations, if needed.
Enter Θ diagonal as positive error variances.
Use * to mark free parameters for df.
Press Calculate and review fit indices and Σ.

Model‑implied covariance as a diagnostic

Confirmatory factor analysis evaluates whether a theorized measurement structure reproduces the observed covariance matrix. This calculator builds the implied covariance Σ from your loadings Λ, factor covariance Φ, and error variances Θ. When Σ closely matches S, the measurement model is consistent with the data pattern and parameterization. Check variances for scaling.

Interpreting chi‑square with degrees of freedom

The maximum likelihood fit function produces χ² = (N−1)·FML, where larger values indicate greater discrepancy between S and Σ. Because χ² is sensitive to sample size, compare χ² to df and inspect indices. Use df computed from unique covariances minus counted free parameters. A modest χ²/df ratio can still hide localized residual problems, so inspect both.

RMSEA and confidence‑style reasoning

RMSEA summarizes approximate misfit per degree of freedom using max((χ²−df)/(df·(N−1)),0). Values near 0.05 suggest close fit, around 0.08 indicate acceptable fit, and higher values imply notable misspecification. Treat RMSEA as a model‑comparison metric, not a standalone pass/fail rule. In small samples, RMSEA can be unstable, so weigh it alongside residual diagnostics.

CFI and TLI against the independence baseline

CFI and TLI compare your model to an independence baseline that fixes all covariances to zero while retaining observed variances. These indices improve when your model reduces misfit beyond the baseline. Values at or above 0.95 are often considered strong, while 0.90 can be pragmatic in complex instruments. If baseline fit is unusually good or poor, interpret CFI/TLI with caution and context.

SRMR and residual structure

SRMR is the root mean square of standardized residual covariances, highlighting systematic localized misfit. The residual heatmap for S−Σ helps you see clusters of misfit. Concentrated large residual blocks may indicate cross‑loadings, correlated errors, or omitted factors. Target changes where theory supports modifications. When a few pairs dominate SRMR, consider revising specific items rather than broad restructuring.

Using AIC/BIC for competing specifications

AIC and BIC combine fit and parsimony using the counted free parameters. Lower values indicate a preferable balance when comparing models fit to the same dataset and variable set. Use these criteria to evaluate alternative factor structures, constraint choices, or correlated‑factor versus higher‑order specifications. Differences are most meaningful when the models use the same indicators and assumptions.

FAQs

1) Does this tool estimate CFA parameters from raw data?

No. It evaluates fit from estimates you enter, using the observed covariance matrix and your Λ, Φ, and Θ inputs to compute Σ and fit indices.

2) Should I enter a correlation matrix instead of covariances?

Use one type consistently. If you use correlations, Θ and Σ are interpreted on that scale. Covariances are preferred when indicators have different variances.

3) What does the star (*) marker change?

The star marks a parameter as free for counting. That count affects df, AIC, and BIC. It does not change numerical calculations, which use the value you enter.

4) Why do I get “non‑positive determinant” or “not invertible” errors?

Σ or S must be positive definite for ML computations. Invalid combinations of loadings, factor covariances, or error variances can make a matrix singular or indefinite.

5) How can I use the residual heatmap?

Look for large absolute residuals or clustered blocks. Those patterns often point to cross‑loadings, missing correlated errors, or an omitted factor affecting several indicators.

6) Are cutoff values like RMSEA 0.08 or CFI 0.95 universal?

No. Thresholds depend on model complexity, reliability, and sample size. Use them as guidance, then justify decisions with theory, diagnostics, and comparisons across models.