Selecting and validating a GLM for a business problem

Welcome to the World of GLM Selection and Validation!

Hello there! If you’ve made it to this stage of your Exam PA journey, you already know that Generalized Linear Models (GLMs) are the bread and butter of actuarial predictive modeling. But how do we know if we’ve built a "good" model? Is it the one with the most variables? The one that fits the training data perfectly? (Spoiler: No!)

In this chapter, we are going to learn how to pick the right GLM for a specific business problem and, more importantly, how to prove that it actually works. Think of this as moving from being a "cook" who follows a recipe to a "chef" who knows exactly why they are choosing certain ingredients. Don't worry if this seems a bit heavy at first—we'll break it down step-by-step!

1. Defining the Business Problem: The "Why" Before the "How"

Before you even touch your computer, you must understand what you are trying to solve. In the context of Exam PA, the business problem dictates your choice of the target variable and the error distribution.

Step-by-Step Approach:
1. Identify the target: What are we predicting? (e.g., Number of claims, total cost, or whether a customer will churn).
2. Identify the constraints: Do we have limited data? Are there legal requirements for transparency?
3. Choose the appropriate GLM components: Specifically, the Distribution and the Link Function.

Choosing the Right Distribution

Think of the distribution as the "shape" of your data. If you pick the wrong shape, your model will never fit properly.

Common Distributions in Actuarial Work:
- Normal (Gaussian): Used for continuous data that is symmetric (rare in insurance claims, but common for height/weight).
- Poisson: The "go-to" for count data (e.g., "How many accidents happened this year?").
- Gamma: Great for claim severity (positive, continuous values that are skewed—lots of small claims, a few huge ones).
- Binomial: Used for binary outcomes (Yes/No, 1/0).
- Tweedie: A "hybrid" often used for total insurance loss because it can handle a mass of zeros (no claims) plus continuous positive values.

Quick Review: Which distribution would you use for "Number of times a machine breaks down"?
Answer: Poisson (because it is a count of events).

2. Selecting Predictors: Feature Selection

In a GLM, we don't always want to include every single variable (feature) we have. Including too many can lead to overfitting—where the model learns the "noise" in your specific data set instead of the actual pattern.

Automated Selection Methods

- Forward Selection: Start with an empty model and add variables one by one based on which improves the model the most.
- Backward Selection: Start with all variables and remove the least significant ones one by one.
- Stepwise Selection: A mix of both; it can add or remove variables at each step.

Common Mistake to Avoid: Just because a variable is "statistically significant" (low p-value) doesn't always mean it's useful for the business. Always use your common sense! If a variable says "Zip Code" and you have 5,000 zip codes, your model might explode. You might need to group them first.

3. Model Validation: The "Reality Check"

Once you have a model, you need to validate it. We do this to ensure the model performs well on unseen data (data it wasn't trained on).

The Train/Test Split

Imagine studying for a math exam. If you only practice the exact questions that will be on the test, you haven't "learned" math; you've just memorized answers.
- Training Set: The data the model "studies" to learn patterns.
- Test Set: The "final exam" to see if the model actually understood the patterns.

Cross-Validation

Sometimes, a single split isn't enough. k-fold Cross-Validation involves splitting the data into "k" pieces. We train the model on \(k-1\) pieces and test it on the remaining piece, repeating this until every piece has been the "test" set once. We then average the results.

Key Takeaway: Validation is the only way to catch Overfitting (high accuracy on training data, low accuracy on test data).

4. Evaluation Metrics: How Do We Score the Model?

We need numbers to tell us if Model A is better than Model B. Here are the big ones for Exam PA:

Deviance

In GLMs, Deviance is a measure of "goodness of fit." A lower deviance means the model fits the data better. Think of deviance as the "error" or the "gap" between your model and a perfect model.

AIC and BIC (The "Penalty" Metrics)

These metrics help us find a balance between fit and simplicity.
- AIC (Akaike Information Criterion): \(AIC = -2L + 2p\)
- BIC (Bayesian Information Criterion): \(BIC = -2L + p \ln(n)\)

Where \(L\) is the likelihood, \(p\) is the number of parameters (variables), and \(n\) is the sample size.
The Rule: Lower is better! BIC is "stricter" than AIC—it penalizes having too many variables more heavily, especially in large datasets.

Memory Aid: Think of AIC and BIC as a judge at a talent show. They give points for a good performance (Likelihood) but deduct points if you brought too many unnecessary backup dancers (Parameters/Variables).

5. Residual Analysis: Checking the "Leftovers"

Residuals are the difference between what actually happened and what your model predicted (\(Actual - Predicted\)). In a good model, the residuals should look like "white noise"—random and unpredictable.

Types of Residuals:
1. Pearson Residuals: Standardized residuals. Useful for checking constant variance.
2. Deviance Residuals: These are preferred in GLMs because their distribution is closer to being Normal, making them easier to interpret in plots.

What to look for: If you plot your residuals and see a clear pattern (like a "U" shape or a "fan" shape), your model is missing something important! It means there is still a pattern in the data that your model failed to catch.

6. Interpreting the Results for Stakeholders

The final step in a business problem is explaining your model to someone who isn't an actuary. They don't care about "Log-Likelihood"; they care about the "Bottom Line."

The Link Function Trick:
If you used a Log Link (common in Poisson and Gamma models), the coefficients act multiplicatively.
If a coefficient for "Vehicle Age" is \(0.05\), then a one-unit increase in age increases the predicted value by a factor of \(e^{0.05} \approx 1.051\), or a \(5.1\%\) increase.

Did you know? This multiplicative property is why the Log Link is so popular in insurance pricing—it's easy to say "Being a young driver increases your premium by 20%" rather than "It increases your premium by 200 dollars."

Summary Checklist for Your Exam

When you are asked to select and validate a GLM on the exam, walk through this mental checklist:
- Distribution: Does it match the nature of the target? (Count vs. Continuous)
- Link Function: Is it appropriate? (Log for positive values, Logit for binary)
- Selection: Did I use AIC/BIC or Stepwise to trim unnecessary variables?
- Validation: Did I check the performance on the Test set?
- Residuals: Did I look at the plots to ensure no patterns remain?
- Business Logic: Do the results make sense for the company?

Pro-Tip: If the exam asks you to choose between two models, and one has a slightly lower AIC but is much more complex, consider recommending the simpler model. In the real world, "simple and interpretable" often beats "complex and slightly more accurate."

You've got this! GLMs can feel like a lot of moving parts, but once you see how the distribution, link function, and validation work together, it all starts to click.