Predicting Insurance Costs with Multiple Regression: Age and BMI Analysis

Introduction And Lesson Overview

Welcome to the first lesson of PredictHealth's Multi-Factor Cost Models course. In this lesson, you will learn how to analyze and predict insurance costs using combined factor analysis . Combined factor analysis means looking at more than one variable at a time to understand how they work together to affect insurance costs. This approach is at the heart of PredictHealth's cost modeling, where we use real-world data to make better predictions and decisions. By the end of this lesson, you will know how to build a simple multiple regression model using two important factors — age and BMI — to predict insurance charges . You will also learn how to interpret the results, evaluate the model's performance, and visualize the predictions. This lesson will give you a strong foundation for more advanced cost modeling techniques later in the course.

Understanding The Dataset And Feature Selection

Let's start by understanding the insurance dataset we will use. The insurance dataset contains information about individuals, including their age, sex, BMI (Body Mass Index), number of children, smoking status, region, and the insurance charges they paid. Here is a quick look at what the data might look like: age sex bmi children smoker region charges 19 female 27.9 0 yes southwest 16884.92 18 male 33.8 1 no southeast 1725.55 28 male 33.0 3 no southeast 4449.46 In this lesson, we will focus on two numerical features: age and bmi . These are important because age often affects health risks, and BMI is a common measure of body fat, which can also impact health and insurance costs. While there are other features in the dataset, starting with these two helps us build a clear and simple model.

age	sex	bmi	children	smoker	region	charges
19	female	27.9	0	yes	southwest	16884.92
18	male	33.8	1	no	southeast	1725.55
28	male	33.0	3	no	southeast	4449.46

Preparing The Data For Multiple Regression

Before we can build a model, we need to prepare the data. First, let's import all the necessary libraries: Python import pandas as pd import numpy as np import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from sklearn.linear_model import LinearRegression from sklearn.metrics import mean_squared_error, r2_score Next, we need to separate the features (the variables we use to predict) from the target (the value we want to predict, which is charges). We also need to split the data into a training set and a testing set. The training set is used to build the model, and the testing set is used to see how well the model works on new data. Here is how you can do this in Python: Python # Separating features and target X = insurance_data.drop('charges', axis=1) y = insurance_data['charges'] # First, let's build a model with just age and BMI X_simple = insurance_data[['age', 'bmi']] X_train, X_test, y_train, y_test = train_test_split(X_simple, y, test_size=0.2, random_state=42) While age (typically 18-65 years) and BMI (typically 15-50) are on similar scales in our dataset, it's important to know when standardization is needed. Standardization (scaling variables to have mean 0 and standard deviation 1) is recommended when features have very different scales (e.g., age vs. income in dollars), when using regularized models (Ridge, Lasso), when using distance-based algorithms, or when coefficients need to be directly compared for importance. In our case, since both features are on comparable scales, standardization is optional, but it's good practice to consider it for more robust modeling.

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Evaluating Model Performance And Business Interpretation

After building the model, it's important to check how well it works. We use several metrics for this:

Mean Squared Error (MSE): Measures the average squared difference between actual and predicted values.
Root Mean Squared Error (RMSE): The square root of MSE, which brings the error back to the original units (dollars).
R-squared (R²): Shows how much of the variation in insurance charges is explained by the model.

Let's look at the code and the output:

Python

# Model evaluation
mse = mean_squared_error(y_test, y_pred)
rmse = np.sqrt(mse)
r2 = r2_score(y_test, y_pred)

# Printing the model results
print("Multiple Regression Model (Age + BMI):")
print(f"Age coefficient: {multi_model.coef_[0]:.2f}")
print(f"BMI coefficient: {multi_model.coef_[1]:.2f}")
print(f"Intercept: {multi_model.intercept_:.2f}")
print("\nModel Equation:")
print(f"Insurance Cost = {multi_model.intercept_:.2f} + {multi_model.coef_[0]:.2f} × Age + {multi_model.coef_[1]:.2f} × BMI")
print("\nModel Evaluation:")
print(f"Mean Squared Error (MSE): {mse:.2f}")
print(f"Root Mean Squared Error (RMSE): {rmse:.2f}")
print(f"R-squared (R²): {r2:.2f}")

# Model evaluation
mse = mean_squared_error(y_test, y_pred)
rmse = np.sqrt(mse)
r2 = r2_score(y_test, y_pred)

# Printing the model results
print("Multiple Regression Model (Age + BMI):")
print(f"Age coefficient: {multi_model.coef_[0]:.2f}")
print(f"BMI coefficient: {multi_model.coef_[1]:.2f}")
print(f"Intercept: {multi_model.intercept_:.2f}")
print("\nModel Equation:")
print(f"Insurance Cost = {multi_model.intercept_:.2f} + {multi_model.coef_[0]:.2f} × Age + {multi_model.coef_[1]:.2f} × BMI")
print("\nModel Evaluation:")
print(f"Mean Squared Error (MSE): {mse:.2f}")
print(f"Root Mean Squared Error (RMSE): {rmse:.2f}")
print(f"R-squared (R²): {r2:.2f}")

Example output might look like this:

plaintext

Multiple Regression Model (Age + BMI):
Age coefficient: 223.80
BMI coefficient: 330.79
Intercept: -5571.08

Model Equation:
Insurance Cost = -5571.08 + 223.80 × Age + 330.79 × BMI

Model Evaluation:
Mean Squared Error (MSE): 131440262.76
Root Mean Squared Error (RMSE): 11464.74
R-squared (R²): 0.15

Multiple Regression Model (Age + BMI):
Age coefficient: 223.80
BMI coefficient: 330.79
Intercept: -5571.08

Model Equation:
Insurance Cost = -5571.08 + 223.80 × Age + 330.79 × BMI

Model Evaluation:
Mean Squared Error (MSE): 131440262.76
Root Mean Squared Error (RMSE): 11464.74
R-squared (R²): 0.15

The coefficients tell us how much the insurance cost changes for each unit increase in age or bmi. For example, if the age coefficient is 223.80, then for each additional year of age, the insurance cost increases by about $223.80, holding BMI constant. Similarly, for each unit increase in BMI, the cost increases by $330.79.

This kind of interpretation is very useful for business decisions. It helps insurance companies understand how much each factor contributes to the overall cost.

Introduction And Lesson Overview

Understanding The Dataset And Feature Selection

Preparing The Data For Multiple Regression

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Key Assumptions Of Multiple Regression

Building The Combined Factor Regression Model

Evaluating Model Performance And Business Interpretation

Visualizing Predictions Vs. Actual Values

Summary And Next Steps