Poisson regression verification record

The executed comparison fits identical observations in base R glm(), statsmodels GLM, and PHP fit_poisson(). The saved report records versions, complete outputs, maximum differences from R, warnings and retained-row metadata.

Reference cases and conventions

Six saved R fixtures cover the simulated clinic sample, a single predictor, multiple predictors, no intercept, intercept only, correlated full-rank columns and sparse counts (the main sample is the multiple-predictor case). The three-way comparison adds listwise deletion of two rows. All coefficients, covariance entries, standard errors, normal z tests and p-values, Wald confidence limits, full fitted mean counts and response residuals, likelihood, deviance, null deviance, AIC and residual df are checked. Numerical tolerance is 1e-9 + 1e-7 × abs(reference); sample accounting is exact.

R uses confint.default(fit), not profile-likelihood confint(fit). Its fitting tolerance is 1e-16 to make covariance evaluated at the preceding IRLS iterate negligible. Iteration counts can differ across implementations; they are not a reference equality requirement. Without a ones column the null baseline is mean one; the Python runner computes this convention explicitly.

Failure and integration checks

Rust/PHP tests reject insufficient observations, invalid shapes, nonfinite or negative/fractional outcomes, all-zero samples, dependent columns, and selected boundary designs. They check scaled predictor units, zero-count and near-fit deviance arithmetic, iteration exhaustion, missing-row warnings and alignment. OLS integration tests also run after changes to the shared PHP conversion code. No exhaustive finite-MLE existence test is supplied. Count overdispersion and sparse samples can invalidate conventional Wald inference. Offsets and exposure arguments are outside the first implementation. Linux and PHP-FPM deployment have not been exercised by these reports.

Reproduce

From the repository root with matching PHP development tools:

export PHP=/opt/homebrew/bin/php
export PHP_CONFIG=/opt/homebrew/bin/php-config
cargo test --workspace --locked
cargo build -p phit --locked
php -n -d "extension=$PWD/target/debug/libphit.dylib" tests/poisson.php
.context/notebooks-venv/bin/python docs/verifications/compare_poisson.py
QUARTO_PYTHON="$PWD/.context/notebooks-venv/bin/python" quarto render docs/notebooks/poisson-verification.qmd

Use .so on Linux. R fixture regeneration is an explicit development step: Rscript tests/generate-probit-poisson-fixtures.R. Ordinary Rust/PHP tests use saved fixtures without R or Python. See development for setup and large-sample benchmarks for additional evidence.