# ~~~~~~~~~~~~~~~~~~~~~~~~~~
# ~ CRP 245 Homework 1 ~
# ~ Problem 1 ~
# ~~~~~~~~~~~~~~~~~~~~~~~~~~
# These data were used to examine the effect of participant characteristics
# on medical costs billed by health insurance and are comprised of a sample
# of 1138 beneficiaries currently enrolled in a specific insurance plan along
# with multiple beneficiary characteristics and total medical expenses charged
# to the plan for the calendar year. One set of analyses examined the
# association between charges and age.
# Data Set: insurance
# Data Dictionary:
# (1) age This is an integer indicating the age of the primary
# beneficiary (excluding those above 64 years, since
# they are generally covered by the government).
# (2) sex This is the policy holder's sex at birth, either male or female.
# (3) bmi This is the body mass index (BMI), which provides a sense of
# how over or under-weight a person is relative to their height.
# BMI is equal to weight (in kilograms) divided by height
# (in meters) squared. Normal BMI is usually considered within
# the range of 18.5 to 24.9.
# (4) children This is an integer indicating the number of children/dependents
# covered by the insurance plan.
# (5) smoker This is yes or no depending on whether the insured regularly
# smokes tobacco.
# (6) region This is the beneficiary's place of residence in the U.S.,
# divided into four geographic regions: northeast, southeast,
# southwest, or northwest.
# (7) charges Individual medical costs billed by health insurance
## Download and load the data file = insurance
load(url("http://www.duke.edu/~sgrambow/crp241data/insurance.RData"))
# examine structure of data
str(insurance)
## 'data.frame': 1338 obs. of 7 variables:
## $ age : int 19 18 28 33 32 31 46 37 37 60 ...
## $ sex : chr "female" "male" "male" "male" ...
## $ bmi : num 27.9 33.8 33 22.7 28.9 ...
## $ children: int 0 1 3 0 0 0 1 3 2 0 ...
## $ smoker : chr "yes" "no" "no" "no" ...
## $ region : chr "southwest" "southeast" "southeast" "northwest" ...
## $ charges : num 16885 1726 4449 21984 3867 ...
# summarize data
summary(insurance)
## age sex bmi children
## Min. :18.00 Length:1338 Min. :15.96 Min. :0.000
## 1st Qu.:27.00 Class :character 1st Qu.:26.30 1st Qu.:0.000
## Median :39.00 Mode :character Median :30.40 Median :1.000
## Mean :39.21 Mean :30.66 Mean :1.095
## 3rd Qu.:51.00 3rd Qu.:34.69 3rd Qu.:2.000
## Max. :64.00 Max. :53.13 Max. :5.000
## smoker region charges
## Length:1338 Length:1338 Min. : 1122
## Class :character Class :character 1st Qu.: 4740
## Mode :character Mode :character Median : 9382
## Mean :13270
## 3rd Qu.:16640
## Max. :63770
# visualize data using ggplot package
library(ggplot2)
# Scatter Plot with overlaid fitted regression line
# and 95% confidence bands
ggplot(insurance, aes(x=age, y=charges))+
geom_point()+
geom_smooth(method=lm, se=TRUE)
## `geom_smooth()` using formula 'y ~ x'
# fitting regression model
fit.age <- lm(charges ~ age, data=insurance)
summary(fit.age)
##
## Call:
## lm(formula = charges ~ age, data = insurance)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8059 -6671 -5939 5440 47829
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3165.9 937.1 3.378 0.000751 ***
## age 257.7 22.5 11.453 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11560 on 1336 degrees of freedom
## Multiple R-squared: 0.08941, Adjusted R-squared: 0.08872
## F-statistic: 131.2 on 1 and 1336 DF, p-value: < 2.2e-16
# Key Output
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 3165.9 937.1 3.378 0.000751 ***
# age 257.7 22.5 11.453 < 2e-16 ***
#
# Slope estimate is key inference target
# Interpretation of Slope
# For each 1 unit increase in age
# the mean charges increase by $257.7
#
confint(fit.age)
## 2.5 % 97.5 %
## (Intercept) 1327.4403 5004.3297
## age 213.5788 301.8665
# 2.5 % 97.5 %
# (Intercept) 1327.4403 5004.3297
# age 213.5788 301.8665
# standard 95% confidence interval is (213.58, 301.87)
#
# if we want to rescale the slope and report
# a point estimate and 95% confidence interval
# for each 5 year increment in age we can multiply
# the slope estimate and confidence limits by 5
fit.age$coefficients*5
## (Intercept) age
## 15829.425 1288.613
# yields
#
# (Intercept) age
# 15829.425 1288.613
#
# estimate = 1288.61, meaning that
# For each 5 unit increase in age
# the mean charges increase by 1288.61
# and for 95% CI
confint(fit.age)*5
## 2.5 % 97.5 %
## (Intercept) 6637.201 25021.649
## age 1067.894 1509.332
# 2.5 % 97.5 %
# (Intercept) 6637.201 25021.649
# age 1067.894 1509.332
#
# so point estimate and 95% CI is
# 1288.61 with 95% CI: 1067.89, 1509.33
#
# Determine the estimated mean annual charges for 28 year old beneficiaries
# and associated confidence interval
# We can do this by hand in R or use the predicted function as
# shown below. Note that we need to provide the desired value of the
# X variable using the data.frame function and we specify the type
# of interval we want for that estimate -- we want a confidence interval
# for the mean estimated annual charges for 28 year old beneficiaries
#
predict(fit.age,data.frame(age=28),interval = "confidence")
## fit lwr upr
## 1 10382.12 9588.939 11175.3
# fit lwr upr
# 1 10382.12 9588.94 11756.30
# Estimated mean annual charges are 10382.12 with 95% CI from 9588.94 to 11756.30
#
# END OF PROGRAM