LibraryGeneralized Linear Models

Generalized Linear Models

Learn about Generalized Linear Models as part of CAS Actuarial Exams - Casualty Actuarial Society

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.

Distribution FamilyTypical Response VariableCanonical Link FunctionCommon Applications
NormalContinuous (e.g., price, temperature)Identity (E(Y)=β0+β1X1+...E(Y) = \beta_0 + \beta_1 X_1 + ...)Standard linear regression
BinomialProportion or count of successes (e.g., policy renewal, default)Logit (log(μ1μ)=β0+β1X1+...\log(\frac{\mu}{1-\mu}) = \beta_0 + \beta_1 X_1 + ...)Credit risk, insurance policy lapse rates
PoissonCount data (e.g., number of claims, accidents)Log (log(μ)=β0+β1X1+...\log(\mu) = \beta_0 + \beta_1 X_1 + ...)Frequency modeling in insurance
GammaPositive, skewed continuous data (e.g., claim severity)Inverse (1/μ=β0+β1X1+...1/\mu = \beta_0 + \beta_1 X_1 + ...)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 eβie^{\beta_i}.

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, η=Xβ\eta = X\beta, where XX are the predictor variables and β\beta are the coefficients. 3. Link Function: A function g()g(\cdot) such that g(E(Y))=ηg(E(Y)) = \eta. For example, in Poisson regression, the log link function is used: log(E(Y))=β0+β1X1+...\log(E(Y)) = \beta_0 + \beta_1 X_1 + .... 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.
What are the three fundamental components that define a Generalized Linear Model?

The random component (distribution), the systematic component (linear predictor), and the link function.

Why is the Poisson distribution often used for modeling the number of insurance claims?

Because claim counts are non-negative integers and often exhibit a mean-variance relationship consistent with the Poisson distribution.

Learning Resources

Generalized Linear Models - An Overview(paper)

A concise academic overview of GLMs, covering their theoretical underpinnings and applications, suitable for a deeper understanding.

Introduction to Generalized Linear Models (GLMs)(video)

A clear and accessible video explanation of GLMs, breaking down the core concepts and components.

Generalized Linear Models - UCLA Statistical Consulting(documentation)

Practical guide with examples using statistical software, focusing on interpretation and application of GLMs.

Generalized Linear Models (GLM) in R(blog)

A blog post demonstrating how to implement and interpret GLMs using the R programming language, with code examples.

CAS Exam 3/3L Study Notes - Generalized Linear Models(documentation)

Official study notes from the Casualty Actuarial Society specifically on Generalized Linear Models, tailored for exam preparation.

Generalized Linear Models - Wikipedia(wikipedia)

A comprehensive Wikipedia article providing a detailed theoretical background, mathematical formulation, and examples of GLMs.

Understanding Generalized Linear Models(video)

A lecture from a Coursera course that explains the intuition and practical aspects of GLMs.

An Introduction to Generalized Linear Models(paper)

A clear and concise introduction to GLMs, suitable for those with a basic understanding of statistics and regression.

GLM: Poisson Regression(video)

A focused video tutorial explaining Poisson regression, a common type of GLM used for count data.

Generalized Linear Models - A Practical Approach(documentation)

A chapter or section from a textbook or reference work that provides a practical, application-oriented view of GLMs.