Adjusted Goodness of Fit Index in Python: Formula, Worked CFA, Interpretation and Verified Results

Advanced Python implementation notes

Freeze the environment before comparing numbers. Save the Python version, semopy version, pandas version, NumPy version and optimizer information with the result. Fit indices depend on the fitted solution, and tiny optimizer or dependency changes can alter the last decimals of GFI and AGFI. A requirements file or locked environment makes a cross-software discrepancy easier to diagnose. The objective is not to force every package to print identical digits; it is to ensure that the same model and data produce the same substantive conclusion and a traceable numerical path.

Control covariance conventions. Python functions may use maximum-likelihood covariance normalization or the sample covariance with an N − 1 divisor. A custom manual GFI calculation can therefore diverge from semopy even when the model parameters appear similar. Record how the observed covariance matrix was created, and avoid mixing DataFrame.cov() output with a package objective that uses a different normalization. When the manual check begins from the package-reported GFI, the AGFI arithmetic is stable; when GFI itself is reconstructed, covariance and discrepancy conventions become part of the specification.

Inspect baseline-model degrees of freedom. semopy’s documented AGFI calculation uses the baseline model’s df divided by the target model’s df. For this conventional six-variable covariance model, that route corresponds to the 21/8 multiplier used in the verification report. More complex models can break the apparent equivalence with a simple p(p + 1)2 expression. Print DoF Baseline, DoF, GFI and AGFI together. If another library uses a different adjustment, label the implementation rather than silently replacing one result with the other.

Build a residual-pair audit table. Convert the symmetric residual correlation matrix into unique off-diagonal pairs, calculate absolute residuals, sort descending and attach interpretation flags. The first three pairs in the supplied model all involve famrel with an academic grade. This structured table is more useful than a heatmap alone because it can be filtered, exported and tested. A code assertion can verify that the matrix is symmetric, diagonal residuals are near zero and variable labels match the fitted covariance order.

Handle missing data deliberately. The worked code uses complete cases on exactly six columns. FIML or imputation would retain a different analysis sample and may change the fitted covariance structure. Never call dropna() on the full source table when unrelated variables contain missing values, because that discards cases that are complete for the CFA. Save the retained row identifiers and a missingness summary. When alternative models are compared, fit all of them to the same row set unless the analysis plan explicitly allows sample changes.

Separate model discovery from confirmation. Python makes it easy to loop over candidate cross-loadings and residual covariances, but selecting the model with the largest AGFI on the same sample inflates optimism. Use theory to define a small candidate set, reserve validation data when feasible, and report the original specification even if a revised model is explored. Automated search can be useful for diagnostics, but it should not convert a confirmatory factor analysis into an undocumented specification-mining exercise.

Check admissibility before chart generation. A pipeline should not create polished figures when the solution contains negative residual variances, nonfinite standard errors, an impossible latent correlation or failed convergence. Validate the fitted object first, then calculate statistics and visuals. Include the optimizer status in the PDF. This prevents a high AGFI from being visually promoted when the parameter solution is statistically unacceptable.

Use unit tests for the published result. Store expected quantities—N = 649, p = 6, moments = 21, parameters = 13, df = 8, GFI near 0.992635 and AGFI near 0.980667—in a small test module. Tests should allow a defensible numerical tolerance while rejecting large changes caused by wrong columns, row filtering or model edits. Reproducible analytical content benefits from the same regression-testing logic used in software engineering: when code changes, the result should either remain within tolerance or trigger a documented review.

Python validation checklist and interpretation scenarios

Scenario 1: semopy reports a high AGFI and the optimizer succeeded. Continue to inspect parameter admissibility. Successful numerical convergence means the optimizer found a solution, not that every variance, covariance and loading is scientifically acceptable. Build checks for negative residual variances, nonfinite estimates and factor correlations beyond plausible bounds before exporting charts or an APA sentence.

Scenario 2: AGFI is high but the residual matrix has a visible cluster. The current famrel–grade residuals illustrate this case. Use a long-form residual table and group pairs by measurement block. A cluster across factors can indicate an omitted cross-loading, a direct relationship, a weak factor definition or a sample-specific pattern. The software identifies the location; theory determines the response.

Scenario 3: Python and R differ in the seventh decimal. Compare full-precision GFI, target df and baseline df before investigating deeper causes. The supplied results differ by roughly two ten-millionths and round identically. Such a difference is not a substantive conflict. A discrepancy in the third decimal, however, would justify checking data rows, covariance normalization, model constraints and package definitions.

Scenario 4: adding a residual covariance improves GFI. Recalculate q and df automatically. The revised model’s AGFI should be computed from its own fit and complexity, not by reusing the original df. Store both models in a comparison DataFrame and include the theoretical reason for the added covariance. A loop that frees paths solely by largest modification evidence is exploratory model search.

Scenario 5: columns are read as strings. pandas can import numeric-looking fields as object type when files contain stray characters. Coerce the six analysis variables explicitly, count conversion failures and stop if unexpected missing values are created. A covariance model built from silently altered columns can still run and produce persuasive output, making type validation an essential pre-fit step.

Scenario 6: feature scaling is changed. Covariance-based fit can respond to scale conventions, while standardized residual interpretation depends on correct standardization. Do not standardize some indicators and leave others raw unless that is the planned model. If all indicators are standardized for a correlation-model analysis, label the method and do not compare its GFI directly with a raw-covariance result.

Scenario 7: FIML or imputation is introduced. Save an analysis flag describing the missing-data path and ensure the PDF states it. Imputed datasets may require pooled parameter and fit treatment rather than one completed file treated as observed truth. The complete-case result in this post is simple to reproduce because exactly six columns define row inclusion.

Scenario 8: a dashboard displays AGFI as 98.1%. Correct the label. Multiplying by 100 can be visually convenient, but it encourages the false claim that the model explains 98.1% of variance. Keep the axis and card labeled “AGFI” and display .981. Explain that it is an adjusted covariance-reproduction index, not a percentage of outcome variance.

Scenario 9: the result is deployed in a web calculator. Require users to provide either p and df with a validated GFI or p and q with GFI. Validate df > 0, retain full precision, show all intermediate steps and warn that the calculator does not estimate a CFA. The calculation engine should be separate from the explanatory content and should reproduce the worked result exactly.

Scenario 10: the pipeline is rerun after code changes. Unit tests should confirm row count, variable order, moment count, df, GFI, AGFI and the identity of the largest residual pair. Snapshot tests can verify that tables contain the expected labels. This converts the statistical report into a maintainable analytical product rather than a one-time notebook.