Predicting Medical Expenses
Step 1-collecting data —-
For this analysis, we will use a simulated dataset containing hypothetical medical expenses for patients in the United States. This data was created for this book using demographic statistics from the US Census Bureau, and thus, approximately reflect real-world conditions.
Step 2: Exploring and preparing the data —-
insurance <- read.csv("insurance.csv", stringsAsFactors = TRUE)cannot open file 'insurance.csv': No such file or directoryError in file(file, "rt") : cannot open the connectionsummarize the charges variable
summary(insurance$expenses) Min. 1st Qu. Median Mean 3rd Qu. Max.
1122 4740 9382 13270 16640 63770 histogram of insurance charges
hist(insurance$expenses)
table of region
table(insurance$region)
northeast northwest southeast southwest
324 325 364 325 exploring relationships among features: correlation matrix
cor(insurance[c("age", "bmi", "children", "expenses")]) age bmi children expenses
age 1.0000000 0.10934101 0.04246900 0.29900819
bmi 0.1093410 1.00000000 0.01264471 0.19857626
children 0.0424690 0.01264471 1.00000000 0.06799823
expenses 0.2990082 0.19857626 0.06799823 1.00000000visualing relationships among features: scatterplot matrix paris creates scatter plot matrix
pairs(insurance[c("age", "bmi", "children", "expenses")])
The relationship between age and expenses displays several relatively straight lines, while the bmi versus expenses plot has two distinct groups of points.
more informative scatterplot matrix
library(psych)
pairs.panels(insurance[c("age", "bmi", "children", "expenses")])
The stretched ellipse represents a strong coreelation and Vice Versa. Therefore, there is week correlation between bmi and children.
Step 3: Training a model on the data —-
ins_model <- lm(expenses ~ age + children + bmi + sex + smoker + region,
data = insurance)
ins_model <- lm(expenses ~ ., data = insurance) # this is equivalent to abovesee the estimated beta coefficients
ins_model
Call:
lm(formula = expenses ~ ., data = insurance)
Coefficients:
(Intercept) age sexmale
-11941.6 256.8 -131.4
bmi children smokeryes
339.3 475.7 23847.5
regionnorthwest regionsoutheast regionsouthwest
-352.8 -1035.6 -959.3 Interpretation: for each one year increase in the age, the expense increase by 256.8; A male has 131.4 less expense each year than a female.
Step 4: Evaluating model performance —-
see more detail about the estimated beta coefficients
summary(ins_model)
Call:
lm(formula = expenses ~ ., data = insurance)
Residuals:
Min 1Q Median 3Q Max
-11302.7 -2850.9 -979.6 1383.9 29981.7
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -11941.6 987.8 -12.089 < 2e-16 ***
age 256.8 11.9 21.586 < 2e-16 ***
sexmale -131.3 332.9 -0.395 0.693255
bmi 339.3 28.6 11.864 < 2e-16 ***
children 475.7 137.8 3.452 0.000574 ***
smokeryes 23847.5 413.1 57.723 < 2e-16 ***
regionnorthwest -352.8 476.3 -0.741 0.458976
regionsoutheast -1035.6 478.7 -2.163 0.030685 *
regionsouthwest -959.3 477.9 -2.007 0.044921 *
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6062 on 1329 degrees of freedom
Multiple R-squared: 0.7509, Adjusted R-squared: 0.7494
F-statistic: 500.9 on 8 and 1329 DF, p-value: < 2.2e-16Step 5: Improving model performance —-
add a higher-order “age” term : we assume that there is a linear relation between age and expenditure which might not be true.Therefore,adding a square term to verify.
insurance$age2 <- insurance$age^2Add an indicator for BMI>=30: The estimated beta for this binary feature would then indicate the average net impact on medical expenses for individuals with BMI of 30 or above, relative to those with BMI less than 30.
insurance$bmi30 <- ifelse(insurance$bmi >= 30, 1, 0)create final model
ins_model2 <- lm(expenses ~ age + age2 + children + bmi + sex +
bmi30*smoker + region, data = insurance)
summary(ins_model2)
Call:
lm(formula = expenses ~ age + age2 + children + bmi + sex + bmi30 *
smoker + region, data = insurance)
Residuals:
Min 1Q Median 3Q Max
-17297.1 -1656.0 -1262.7 -727.8 24161.6
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 139.0053 1363.1359 0.102 0.918792
age -32.6181 59.8250 -0.545 0.585690
age2 3.7307 0.7463 4.999 6.54e-07 ***
children 678.6017 105.8855 6.409 2.03e-10 ***
bmi 119.7715 34.2796 3.494 0.000492 ***
sexmale -496.7690 244.3713 -2.033 0.042267 *
bmi30 -997.9355 422.9607 -2.359 0.018449 *
smokeryes 13404.5952 439.9591 30.468 < 2e-16 ***
regionnorthwest -279.1661 349.2826 -0.799 0.424285
regionsoutheast -828.0345 351.6484 -2.355 0.018682 *
regionsouthwest -1222.1619 350.5314 -3.487 0.000505 ***
bmi30:smokeryes 19810.1534 604.6769 32.762 < 2e-16 ***
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 4445 on 1326 degrees of freedom
Multiple R-squared: 0.8664, Adjusted R-squared: 0.8653
F-statistic: 781.7 on 11 and 1326 DF, p-value: < 2.2e-16Interpretation: 50% of the predict values is above th true values in the range 1656 and 727.8.The model fit statistics help to determine whether our changes improved the performance of the regression model. Relative to our first model, the R-squared value has improved from 0.75 to about 0.87. Similarly, the adjusted R-squared value, which takes into account the fact that the model grew in complexity, also improved from 0.75 to 0.87. Our model is now explaining 87 percent of the variation in medical treatment costs. Additionally, our theories about the model’s functional form seem to be validated. The higher-order age2 term is statistically significant, as is the obesity indicator, bmi30. The interaction between obesity and smoking suggests a massive effect; in addition to the increased costs of over $13,404 for smoking alone, obese smokers spend another $19,810 per year. This may suggest that smoking exacerbates diseases associated with obesity.