Description: The Generalized Linear Model (GLM) is an extension of linear regression that allows modeling response variables that do not follow a normal distribution. Unlike ordinary linear regression, which assumes that errors are normally distributed and that the relationship between variables is linear, GLMs allow the dependent variable to have more general probability distributions, such as binomial, Poisson, or gamma. This is achieved by introducing a link function that relates the mean of the dependent variable to a linear combination of the independent variables. This flexibility makes GLMs powerful tools in statistics, as they can adapt to a wide variety of situations and types of data. Additionally, GLMs are particularly useful in various contexts where the assumptions of linear regression are not met, allowing researchers and analysts to obtain more accurate and meaningful inferences. Their ability to handle different error distributions and their focus on the functional relationship between variables make them a fundamental pillar in modern statistical analysis.
History: The concept of Generalized Linear Models was introduced by John Nelder and Robert Wedderburn in 1972. Their work revolutionized the way statisticians approached data modeling, allowing for greater flexibility in the choice of error distributions and link functions. Since then, GLMs have evolved and been integrated into numerous statistical packages, facilitating their use across various disciplines.
Uses: Generalized Linear Models are used in a variety of fields, including biology, economics, medicine, and social sciences. They are particularly useful in situations where data do not meet normality assumptions, such as in the analysis of event counts, success rates, or binary data. They are also applied in survival studies and in modeling longitudinal data.
Examples: A practical example of a GLM is the use of a logistic model to predict the probability of a patient developing a disease based on risk factors such as age and body mass index. Another example is the use of a Poisson model to analyze the number of calls received at a customer service center over a given period.
