Generalized Linear Regression

https://www.dataspoof.info/post/generalized-linear-regression-in-python/

Generalized Linear Regression (GLR) is a statistical framework that extends the concept of linear regression to handle various types of response variables beyond the continuous numeric outcomes in traditional linear regression. GLR is particularly useful when dealing with non-normal and non-continuous data types, such as binary, count, and categorical data.

In regular linear regression, the response variable (dependent variable) is assumed to be continuous and normally distributed, and the goal is to model the relationship between the response variable and one or more predictor variables (independent variables) through a linear equation. The equation can be represented as:

$�={�}_0+{�}_1{�}_1+{�}_2{�}_2+\dots +{�}_{�}{�}_{�}+�$

Where:

• $�$ is the response variable
• ${�}_0$ is the intercept
• ${�}_1,{�}_2,\dots ,{�}_{�}$ are the coefficients corresponding to predictor variables ${�}_1,{�}_2,\dots ,{�}_{�}$
• ${�}_1,{�}_2,\dots ,{�}_{�}$ are the predictor variables
• $�$ is the error term assumed to be normally distributed with mean zero.

Generalized Linear Regression, on the other hand, allows us to model different types of response variables by introducing a link function and a probability distribution from the exponential family.

The key components of a Generalized Linear Regression model are:

1. Link Function: The link function relates the mean of the response variable to the linear combination of the predictors. It is denoted by $�(�)$, where $�$ is the mean of the response variable. It ensures that the predicted values lie within the appropriate range for the type of response variable. Common link functions include the identity function (for Gaussian distribution), logit function (for binary data), log function (for Poisson distribution), etc.

2. Probability Distribution: GLR assumes that the response variable follows a probability distribution from the exponential family, which includes various distributions like Gaussian (normal), binomial (binary), Poisson (count data), gamma (continuous positive data), etc.

The generalized linear model equation can be written as:

$�(�)={�}_0+{�}_1{�}_1+{�}_2{�}_2+\dots +{�}_{�}{�}_{�}$

where $�(�)$ is the link function applied to the mean of the response variable.

Examples of GLR applications include:

1. Logistic Regression: Used for binary classification problems, where the response variable is binary (0 or 1).

2. Poisson Regression: Suitable for modeling count data, such as the number of events occurring in a fixed interval.

3. Gamma Regression: Applied to continuous positive outcomes with right-skewed distributions.

4. Ordinal Regression: Utilized for ordinal categorical response variables.

5. Multinomial Regression: Used when the response variable has multiple categories.

Generalized Linear Regression is a powerful and flexible tool for handling a wide range of data types and is widely used in various fields such as economics, biology, medicine, social sciences, and more.