Estimation walkthroughs

Each walkthrough derives one estimator step by step on a small data set: the objective, the matrices, the solution, the uncertainty and, where it applies, the likelihood. Each shows the intermediate quantities, and the comparison notebooks fit the same models in R, statsmodels and phit. The pages show saved outputs, so reading them needs no Python or R.

OLS

OLS mathematics: derives the estimator and evaluates each intermediate quantity on the PHP example data.
OLS verification: estimates identical models in R, Python statsmodels and the PHP extension, displays results and checks agreement.

Binary logistic regression

Math walkthrough: likelihood, matrix calculations, Newton/IRLS and Wald inference.
Executed comparison: identical models in R, statsmodels and PHP.
Verification record: fixtures, edge cases and tolerances.
Scale and performance: standard release benchmarks.

Probit regression

Math walkthrough: likelihood, scoring, matrices and inference.
Executed comparison: identical fits in R, statsmodels and PHP.
Verification record: conventions and failure cases.
Scale and performance: standard release benchmarks.

Poisson regression

Math walkthrough: likelihood, scoring, matrices and inference.
Executed comparison: identical fits in R, statsmodels and PHP.
Verification record: conventions and failure cases.
Scale and performance: standard release benchmarks.

Gaussian random-intercept mixed model

Math walkthrough: covariance, marginal likelihood and shrinkage.
Synthetic reference comparison: lme4 ML, statsmodels and PHP.
Published dataset comparison: four studies, lme4, nlme, statsmodels and PHP.
Verification record: conventions, boundary fits and limits.
Scale and performance: release API benchmarks and charts.

Expanded mixed designs and R compatibility

Random slopes, penalized least squares and REML: observed matrices, covariance dot products, QR elimination, ML/REML determinant terms and plug-in fixed-effect covariance on published sleepstudy data.
Expanded R and optimizer audit: weighted/offset fits, grouped binomial trials, both native optimizers and all four actual lme4 wrappers.

Binomial random-intercept likelihoods

Likelihood and integration: observed cattle counts, log-odds dot products, posterior modes, Laplace/AGQ and joint information.
R and independent Python verification: normalized likelihoods, zero-variance reduction and measured approximation error.