Generalized Linear Models: Linear Regression Beyond the Normal Error

A straight line fitted by least squares predicts minus 2.4 bike rentals on a cold hour, and a pass chance of 106 percent after ten hours of study. This lesson shows what ordinary linear regression quietly assumes, and how generalized linear models keep the straight line while changing the distribution and adding a link function: Poisson regression for counts, logistic regression for yes or no outcomes, one fitting method for both, and a check for when the noise is wrong. It is for students who know simple linear regression, the idea of maximum likelihood, logarithms and a first derivative. Chapters 0:00 A straight line that predicts minus 2.4 bikes 1:31 What least squares quietly assumes 2:54 Counts and the Poisson distribution 3:39 The log link: effects multiply 4:44 A whole distribution, not just a curve 5:30 Yes or no: logistic regression 7:07 One recipe: the definition of a GLM 8:11 Fitting: the score equation 9:22 Iteratively reweighted least squares 10:26 When the noise is wrong: overdispersion 11:53 The answer, and what comes next Further reading P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd edition (Chapman and Hall, 1989) A. Agresti, Foundations of Linear and Generalized Linear Models (Wiley, 2015) A. J. Dobson and A. G. Barnett, An Introduction to Generalized Linear Models, 4th edition (CRC Press, 2018) The likelihood idea is in the channel's Maximum Likelihood Estimation video. A later episode may look at generalized linear mixed models, where every day gets its own random level. Narration uses a synthetic voice (IndexTTS2, the channel's usual voice); the script, mathematics and animations are written and checked for this channel.