Generalized Linear Models (GLMs)
Generalized Linear Models (GLMs) extend the familiar linear regression framework to accommodate response variables that have error distribution models other than a normal distribution, and for which the mean function is related to the predictors via a link function. This makes them incredibly versatile for modeling a wide range of data types encountered in actuarial science, such as claim counts, claim severities, and binary outcomes.
Core Components of a GLM
A GLM is defined by three key components:
Why GLMs are Essential for Actuaries
Actuarial data often violates the assumptions of ordinary least squares (OLS) regression. For instance, claim counts are non-negative integers and often follow a Poisson distribution, while claim severities are positive and can be skewed, often modeled by a Gamma or log-normal distribution. GLMs provide a robust framework to handle these situations appropriately.
Think of the link function as a 'translator' that allows us to use a linear model on a transformed version of the expected outcome, which then maps back to the original scale of the outcome.
Common GLM Families and Link Functions
Distribution Family | Typical Response Variable | Canonical Link Function | Common Applications |
---|---|---|---|
Normal | Continuous (e.g., price, temperature) | Identity () | Standard linear regression |
Binomial | Proportion or count of successes (e.g., policy renewal, default) | Logit () | Credit risk, insurance policy lapse rates |
Poisson | Count data (e.g., number of claims, accidents) | Log () | Frequency modeling in insurance |
Gamma | Positive, skewed continuous data (e.g., claim severity) | Inverse () | Severity modeling in insurance |
Model Fitting and Interpretation
GLMs are typically fitted using Maximum Likelihood Estimation (MLE). The interpretation of coefficients depends on the link function. For example, with a log link function in a Poisson regression, a one-unit increase in a predictor corresponds to a multiplicative change in the expected count by a factor of .
A Generalized Linear Model (GLM) connects a response variable's probability distribution to a linear combination of predictor variables through a link function. The three core components are: 1. Random Component: The probability distribution of the response variable (e.g., Poisson for counts, Binomial for proportions). 2. Systematic Component: The linear predictor, , where are the predictor variables and are the coefficients. 3. Link Function: A function such that . For example, in Poisson regression, the log link function is used: . This allows modeling non-normally distributed data and non-linear relationships between the mean of the response and the predictors.
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Key Considerations for Actuarial Applications
When applying GLMs in actuarial exams, pay close attention to:
- Model Selection: Choosing the appropriate distribution family and link function based on the nature of the response variable and domain knowledge.
- Overdispersion: For count data (Poisson), the variance often exceeds the mean. This requires using a quasi-Poisson model or a Negative Binomial distribution.
- Interpretation of Coefficients: Understanding how coefficients relate to the expected value of the response on the original scale, considering the link function.
- Model Diagnostics: Assessing model fit using residual analysis and goodness-of-fit tests.
The random component (distribution), the systematic component (linear predictor), and the link function.
Because claim counts are non-negative integers and often exhibit a mean-variance relationship consistent with the Poisson distribution.
Learning Resources
A concise academic overview of GLMs, covering their theoretical underpinnings and applications, suitable for a deeper understanding.
A clear and accessible video explanation of GLMs, breaking down the core concepts and components.
Practical guide with examples using statistical software, focusing on interpretation and application of GLMs.
A blog post demonstrating how to implement and interpret GLMs using the R programming language, with code examples.
Official study notes from the Casualty Actuarial Society specifically on Generalized Linear Models, tailored for exam preparation.
A comprehensive Wikipedia article providing a detailed theoretical background, mathematical formulation, and examples of GLMs.
A lecture from a Coursera course that explains the intuition and practical aspects of GLMs.
A clear and concise introduction to GLMs, suitable for those with a basic understanding of statistics and regression.
A focused video tutorial explaining Poisson regression, a common type of GLM used for count data.
A chapter or section from a textbook or reference work that provides a practical, application-oriented view of GLMs.