GLM - What's the difference between linear regression and generalized linear regression?

Great question — the difference between linear regression and generalized linear regression (GLM) is fundamental in statistical modeling.

Let’s break it down clearly 👇

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🔹 1. Linear Regression (Ordinary Linear Model)

Purpose: Model a continuous response variable yyy as a linear function of predictors.

Model form:

where:

• y: continuous dependent variable

• x_i​: independent variables

• β_i: coefficients

• ϵ: normally distributed error term (ϵ∼N(0,σ²))

Assumptions:

• y is normally distributed (for given predictors).

• Variance of errors is constant (homoscedasticity).

• Relationship between predictors and y is linear.

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🔹 2. Generalized Linear Model (GLM)

Purpose: Extend linear regression to handle non-normal response variables
(e.g., binary, count, proportion, etc.)

Model structure:

GLMs have three components:

1. Random component: specifies the distribution of y
   (from the exponential family, e.g., Normal, Binomial, Poisson).

2. Systematic component: linear predictor

   
   

3. Link function: connects the mean of yyy to the linear predictor

   
   
   

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Common Examples of GLMs

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🔹 Key Difference Summary

Aspect	Linear Regression	Generalized Linear Regression
Response variable	Continuous (real-valued)	Can be binary, count, or continuous (depending on family)
Error distribution	Normal	Exponential family (Normal, Binomial, Poisson, etc.)
Link function	Identity g(μ)=μ	Any appropriate link (logit, log, etc.)
Variance	Constant	Depends on mean (varies with distribution)
Examples	Predict temperature	Predict probability or counts

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🔹 Intuition

• Linear regression assumes yyy behaves nicely (normal, continuous, linear).

• GLM is a flexible generalization that says:

  “Let’s keep the linear predictor part, but allow different response distributions and nonlinear transformations of the mean.”