Module glm 

Diagnostics for generalized linear models beyond binary logistic — the exponential-dispersion families that share one iteratively-reweighted least-squares (IRLS) engine but differ in link, variance, and residual scale.

Where logistic is a single hand-written model, this module factors the GLM machinery into a Family trait and one generic solver GlmFit, so each family is just its link / variance / deviance formulas. Three families are provided:

• Poisson — counts, V(μ) = μ, fixed dispersion.
• NegativeBinomial — over-dispersed counts, V(μ) = μ + μ²/θ for a fixed θ.
• Gamma — positive skewed continuous responses, V(μ) = μ², with an estimated dispersion.

All three use the log link (see family for why), and each carries the same diagnostic surface:

• GlmFit — the maximum-likelihood fit via IRLS, with coefficient standard errors, Wald statistics, and p-values (normal or Student’s t depending on whether the dispersion is estimated).
• deviance_residuals / pearson_residuals — the two GLM residual scales; deviance residuals square to the residual deviance, Pearson residuals to the χ² that defines the dispersion estimate.
• GoodnessOfFit via GlmFit::goodness_of_fit — null/residual deviance, dispersion, McFadden’s pseudo-R², and AIC/BIC.
• leverage and cooks_distance — the weighted-hat-matrix influence measures.

Response conventions

As with OLS and logistic, the caller owns the design matrix, intercept column included. The response must lie in the family’s support: non-negative counts for Poisson and NegativeBinomial, strictly positive reals for Gamma. Out-of-support values are rejected with RegressionError::InvalidResponse.

Relationship to the logistic module

Binary logistic regression is itself a GLM (binomial family, logit link) and could be expressed here, but it keeps its own module: its separation diagnostics and the Hosmer–Lemeshow test are specific to the binary case, and rewriting a shipped, tested API in terms of this trait would buy nothing. This module covers the families logistic does not.

What is still out of scope

Multinomial and ordinal logistic have a vector-valued linear predictor and block covariance — a different solver shape, not another Family — and live in categorical, not here. The negative-binomial θ can be supplied fixed to NegativeBinomial or estimated jointly with fit_negative_binomial (a profile-likelihood search over θ).

Quick start

use ndarray::array;
use regression_diagnostics::glm::{GlmFit, Poisson, deviance_residuals};

// Caller supplies the intercept column (first column of ones).
let x = array![
    [1.0, 0.0],
    [1.0, 1.0],
    [1.0, 2.0],
    [1.0, 3.0],
    [1.0, 4.0],
    [1.0, 5.0],
];
let y = array![1.0, 2.0, 3.0, 5.0, 8.0, 13.0]; // roughly exponential growth
let fit = GlmFit::new(Poisson, x, y).unwrap();

let gof = fit.goodness_of_fit();
assert!(gof.residual_deviance <= gof.null_deviance + 1e-9);
let dr = deviance_residuals(&fit);
let sum_sq: f64 = dr.iter().map(|d| d * d).sum();
assert!((sum_sq - gof.residual_deviance).abs() < 1e-8);

Re-exports

pub use family::Family;

pub use family::Gamma;

pub use family::NegativeBinomial;

pub use family::Poisson;

Modules

family

Exponential-dispersion families for the generalized linear model.

Structs

GlmFit

A generalized linear model fit by iteratively reweighted least squares (Fisher scoring) for an exponential-dispersion Family.

GoodnessOfFit

Overall GLM goodness-of-fit statistics.

Functions

cooks_distance

Cook’s-distance analogue for a GLM (Pregibon):

deviance_residuals

Deviance residuals sign(yᵢ − μᵢ) · √d(yᵢ, μᵢ).

fit_negative_binomial

Fit a negative-binomial GLM estimating the dispersion θ jointly with the coefficients, by maximizing the profile log-likelihood θ ↦ ℓ(β̂(θ), θ) over θ (a one-dimensional golden-section search, refitting the coefficients at each candidate θ).

leverage

GLM leverage — the diagonal of the weighted hat matrix H = W^{1/2}X(XᵀWX)⁻¹XᵀW^{1/2}, i.e. hᵢ = wᵢ · xᵢᵀ(XᵀWX)⁻¹xᵢ with the IRLS working weight wᵢ = (dμ/dη)² / V(μᵢ).

pearson_residuals

Pearson residuals (yᵢ − μᵢ) / √V(μᵢ).