When we look at medical insurance premiums, we often wonder if moving to a different part of the country could change our medical premiums (bills).

Our Investigation: Today we are using a dataset of 1,338 insurance customers to see if there is a statistically significant relationship between Region and Medical Charges.

The Roadmap:

• Visualize the “Spread of Costs”
• Define our statistical “Ground Rules”
• Testing for significant differences using ANOVA
• Understanding how health factors (BMI/AGE) complicate the story

Before diving into the analysis, we verified the data for 1,338 individuals to ensure there were no illogical extremes, like an age of 200.

##       age             bmi           charges     
##  Min.   :18.00   Min.   :15.96   Min.   : 1122  
##  1st Qu.:27.00   1st Qu.:26.30   1st Qu.: 4740  
##  Median :39.00   Median :30.40   Median : 9382  
##  Mean   :39.21   Mean   :30.66   Mean   :13270  
##  3rd Qu.:51.00   3rd Qu.:34.69   3rd Qu.:16640  
##  Max.   :64.00   Max.   :53.13   Max.   :63770

A boxplot is the best way to see the “spread” and “outliers” of costs for each region.
By removing the outliers in the second graph, we can focus on charges that are under $20,000.
We can see that the median costs are actually quite similar, though the Northeast trends slightly higher with the Southeast right behind it.

This violin plot shows us where the “density” of people lies.
Despite the southeast having a lesser amount of people, the taller point shows that they have a higher density of expensive claims.

To prove that region matters, we can’t just look at averages; we have to look at Variance. We use a test called ANOVA.

The F-statistic is calculated as:

\[F = \frac{\text{Variance between regions}}{\text{Variance within regions}}\]
Where:
Variance between regions: is how much the regional averages differ from each other.
Variance within regions: is how much the individuals vary within their own region.

If the variance between regions is much larger than the variance within them, we conclude that the location has a significant impact on charges.

For our conclusions to be reliable, our data must follow three main rules:

1. Independence: One person’s bill does not affect anothers.
2. Normality: The variation in our data follows a bell curve: \[\epsilon \sim N(0, \sigma^2)\]
3. Equal Variance: The “spread” of costs is roughly similar across all regions: \[\sigma^2_{NE} \approx \sigma^2_{NW} \approx \sigma^2_{SE} \approx \sigma^2_{SW}\]

If these hold, our \(p\text{-value}\) will accurately tell us if Region is a significant factor.

Now we calculate our p-value. If the p-value is less than 0.05, we reject the idea that all regions are equal.
We do that with this code:

region_model <- aov(charges ~ region, data = ds)
anova_summary <- summary(region_model)

p_value <- anova_summary[[1]]["region", "Pr(>F)"]

cat("ANOVA Result\n")

ANOVA Result

cat(sprintf("**p-value = %.4f**\n\n", p_value))

p-value = 0.0309

## - **Statistically significant difference between regions**

There is a positive linear trend between BMI and medical charges. Most importantly, the trend lines for all four regions are nearly parallel. This tells us that while a region might be slightly more expensive overall, the higher BMI is a better indicator for increased charges.

Here we have the code for an interactive model that maps * ‘Age’ on the x-axis * ‘BMI’ on the y-axis * ‘Chagres ($)’ on the z-axis.

plot_ly(ds, x = ~age, y = ~bmi, z = ~charges, 
        color = ~region, colors = "Set1",
        type = "scatter3d", mode = "markers", marker = list(size = 3)) %>%
  layout(title = "The Complexity of Costs",
         scene = list(xaxis = list(title = 'Age'),
                      yaxis = list(title = 'BMI'),
                      zaxis = list(title = 'Charges ($)')))

Costs aren’t just about region; they are driven by health. Here we see how Age and BMI drive Charges, colored by Region.

The Findings:

Our ANOVA test (Slide 5) typically shoes a low p-value, meaning region does influence cost.
The Southeast region often has the highest average charges.

The ‘But’:
While geography matters, our 3D and BMI plot shows that Age & BMI create the largest cost differences.

Conclusion: Where you live matters for your insurance bill, but who you are and your health lifestyle matter more.