Fitting Generalized Linear Models (GLMs)

Generalized Linear Models (GLMs) are a flexible generalization of ordinary least squares regression. They allow for response variables that have error distribution models other than a normal distribution, and for the response variable's mean to depend on the predictors through a specified link function. This makes them incredibly powerful for modeling a wide range of data types encountered in actuarial science, such as claim counts, claim amounts, and survival times.

Core Components of a GLM

A GLM is defined by three key components:

The Process of Fitting a GLM

Fitting a GLM involves estimating the model coefficients ( $\beta$ ) and assessing the model's fit. This is typically done using Maximum Likelihood Estimation (MLE).

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Step 1: Choosing the Distribution and Link Function

The choice of distribution is guided by the nature of the response variable. For example, if modeling the number of claims, a Poisson distribution is often appropriate. If modeling binary outcomes (e.g., claim occurrence), a Bernoulli distribution is used. The link function is often the 'canonical' link function associated with the chosen distribution, but others can be explored.

The canonical link function is the link function that makes the chosen distribution a member of the exponential family in its natural parameterization. It often leads to simpler calculations and desirable statistical properties.

Step 2: Estimating Coefficients (Maximum Likelihood Estimation)

Once the distribution and link function are chosen, the coefficients ( $\beta$ ) are estimated by finding the values that maximize the likelihood function. This is an iterative process, often solved using algorithms like Iteratively Reweighted Least Squares (IRLS).

The core idea behind Maximum Likelihood Estimation (MLE) for GLMs is to find the parameter values (coefficients, $\beta$ ) that make the observed data most probable. This involves defining a likelihood function, $L(\beta | \text{data})$ , which represents the probability of observing the given data for a specific set of $\beta$ values. We then maximize this function, often by maximizing its logarithm (the log-likelihood function, $\ln L(\beta | \text{data})$ ). The process is iterative because the likelihood function for GLMs is typically not analytically solvable for $\beta$ . Algorithms like Iteratively Reweighted Least Squares (IRLS) are used. IRLS approximates the log-likelihood function with a quadratic function at each step and updates the coefficient estimates based on this approximation. This process continues until the estimates converge to a stable solution.

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Step 3: Assessing Model Fit

After fitting, it's crucial to evaluate how well the model represents the data. Common methods include:

Assessment Method	Description	Interpretation
Deviance	A measure of the discrepancy between the fitted model and a saturated model (a model that perfectly fits the data).	Lower deviance generally indicates a better fit. Can be used for model comparison.
AIC/BIC	Information criteria that balance model fit with model complexity.	Lower values indicate a preferred model.
Residual Analysis	Examining different types of residuals (e.g., Pearson, deviance, response) to identify patterns or outliers.	Randomly scattered residuals suggest a good fit; systematic patterns indicate model misspecification.
Goodness-of-Fit Tests	Formal statistical tests (e.g., Chi-squared test for Poisson) to assess if the observed data is consistent with the model's predictions.	A high p-value suggests the model is a plausible fit.

Common GLMs in Actuarial Science

Several GLMs are frequently used in actuarial applications:

Key Considerations for SOA Exams

When preparing for actuarial exams, focus on understanding the underlying theory, the interpretation of model outputs (coefficients, p-values, confidence intervals), and the practical application of GLMs to insurance data. Be comfortable with selecting appropriate distributions and link functions, and interpreting model diagnostics.

What are the three fundamental components that define a Generalized Linear Model (GLM)?

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

What is the primary method used to estimate the coefficients in a GLM?

Maximum Likelihood Estimation (MLE), often implemented using Iteratively Reweighted Least Squares (IRLS).

Learning Resources

Generalized Linear Models - Society of Actuaries(paper)

A foundational monograph on GLMs specifically tailored for actuarial applications, covering theory and practical examples.

Introduction to Generalized Linear Models - StatQuest with Josh Starmer(video)

An excellent, intuitive video explanation of GLMs, breaking down the concepts with clear visuals and analogies.

An Introduction to Generalized Linear Models - University of Bristol(paper)

A comprehensive PDF document detailing the theory and application of GLMs, suitable for in-depth study.

R Documentation for glm()(documentation)

Official documentation for the `glm()` function in R, essential for understanding how to implement GLMs in practice.

Generalized Linear Models - Wikipedia(wikipedia)

A detailed overview of GLMs, covering their history, mathematical formulation, and applications across various fields.

Fitting GLMs in R - DataCamp(tutorial)

A practical tutorial demonstrating how to fit GLMs using R, with code examples and explanations.

Generalized Linear Models - UCLA Statistical Consulting(paper)

Lecture notes providing a clear and concise explanation of GLMs, focusing on their statistical underpinnings.

Understanding Generalized Linear Models - Towards Data Science(blog)

A blog post that offers a conceptual understanding of GLMs, bridging the gap between theory and application.

Actuarial Exam P/1 - Probability and Statistics(documentation)

While not solely on GLMs, this is the foundational exam for actuarial statistics, and understanding its concepts is crucial for advanced topics like GLMs.

Generalized Linear Models - Coursera (example course)(video)

An example lecture from a statistical modeling course that provides a structured explanation of GLMs, often found on platforms like Coursera.