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Generalized Linear Model Notes

This repo is a collection of notes for the generalized models. Some notes are mathematical. Some notes are in code.

GLM Exponential Forms.qmd

This document states all formulas necessary to get pdfs and pmfs into exponential form and is largely based on Generalized Linear Models and Extensions by Hardin and Hilbe.

gamma dispersion.R

Both R and Stata’s glm implementation use the method of moments estimator of dispersion instead of the maximum likelihood estimator. This R script explores the numerical differences and some of the statistical consequences of different estimators.

lm and glm comparisons.R

Many statistical properties of the linear model are applicable to the generalized linear model. The linear models’ SSE is the generalized linear model’s deviance. Linear model’s SSTO is the generalized linear model’s null deviance. etc. Unfortunately, many connections are obfuscated by different names.

This R script explores connections by performing many calculations with two models. One trained with lm() and the other with glm(family = gaussian()).

glm diagnostics.R

The statistical community has not reached a consensus on how to analyze a generalized linear model’s fit.

  • Some textbooks recommend one approach. Other textbooks recommend a different approach.
  • Some approaches are implemented in R and some are not.
  • Some assessment techniques are applicable to the binomial family, but not the gamma family and vice versa.
  • Some textbooks make a distinction between deviance and scaled deviance. Other textbooks don’t.
  • Sometimes deviance and scaled deviance are the same so the distinction is irrelevant (Poisson and binomial).

These details make it nearly impossible to asses a model’s fit. This script takes a different strategy. It uses R’s influence.measures on a linear model. The calculations are checked against examples in Linear Regressions Analysis by Montgomery, Peck and Vining.

Then a glm model with gaussian family is trained. Equality between R’s influence.measure’s calculations is confirmed. Influence.measure() can be used across all glm families so I have a simple, consistent, and easy way of assessing fit across all types of generalized linear models.