Multiple Linear Regression Analysis For Future Medical Expenses

Introduction:

The purposes of this analysis is to look into different features to observe their relationship, and plot a multiple linear regression based on several features of individual such as age, physical/family condition and location against their existing medical expense to be used for predicting future medical expenses of individuals that help medical insurance to make decision on charging the premium.

I have used the dataset called insurance.csv.The insurance.csv dataset contains 1338 observations (rows) and 7 features (columns). The dataset contains 4 numerical features (age, bmi, children and expenses) and 3 nominal features (sex, smoker and region) that were converted into factors with numerical value desginated for each level.

Analysys:

Importing data set and look how the structure of the insurance data set:

insurance = read.csv("insurance.csv", stringsAsFactors = T)
str(insurance)

## 'data.frame':    1338 obs. of  9 variables:
##  $ age     : int  19 18 28 33 32 31 46 37 37 60 ...
##  $ sex     : Factor w/ 2 levels "female","male": 1 2 2 2 2 1 1 1 2 1 ...
##  $ bmi     : num  27.9 33.8 33 22.7 28.9 25.7 33.4 27.7 29.8 25.8 ...
##  $ children: int  0 1 3 0 0 0 1 3 2 0 ...
##  $ smoker  : Factor w/ 2 levels "no","yes": 2 1 1 1 1 1 1 1 1 1 ...
##  $ region  : Factor w/ 4 levels "northeast","northwest",..: 4 3 3 2 2 3 3 2 1 2 ...
##  $ expenses: num  16885 1726 4449 21984 3867 ...
##  $ X       : logi  NA NA NA NA NA NA ...
##  $ X.1     : logi  NA NA NA NA NA NA ...

Let’s print the head(first 6 tuples) of the data set:

suppressMessages(library(dplyr))
head(insurance)

Now look the summary of the data set:

summary(insurance)

##       age            sex           bmi           children     smoker    
##  Min.   :18.00   female:662   Min.   :16.00   Min.   :0.000   no :1064  
##  1st Qu.:27.00   male  :676   1st Qu.:26.30   1st Qu.:0.000   yes: 274  
##  Median :39.00                Median :30.40   Median :1.000             
##  Mean   :39.21                Mean   :30.67   Mean   :1.095             
##  3rd Qu.:51.00                3rd Qu.:34.70   3rd Qu.:2.000             
##  Max.   :64.00                Max.   :53.10   Max.   :5.000             
##        region       expenses        X             X.1         
##  northeast:324   Min.   : 1122   Mode:logical   Mode:logical  
##  northwest:325   1st Qu.: 4740   NA's:1338      NA's:1338     
##  southeast:364   Median : 9382                                
##  southwest:325   Mean   :13270                                
##                  3rd Qu.:16640                                
##                  Max.   :63770

histogram of the medical expenses:

hist(insurance$expenses, main="Histogram of medical expenses",breaks = 100,xlab = "Expenses",ylab = "Frequency",col="lightblue")

A histogram of the medical expenses is plotted, and for the first 100 observations it has shown a right skewed distribution.

There are three nominal variables in the dataset.

• Sex
• Smoker
• Region

It is usefull to know the proportion distribution of the nominal features.Let’s do that.

1. Table of sex(male/female):

table_sex<-table(insurance$sex)
table_sex

## 
## female   male 
##    662    676

2. Table of smokers:

table_smoke<-table(insurance$smoker)
table_smoke

## 
##   no  yes 
## 1064  274

3. table of region:

table_region<-table(insurance$region)
table_region

## 
## northeast northwest southeast southwest 
##       324       325       364       325

Let’s visualize the above results:

par(mfrow=c(1,3))
barplot(table_sex,main="Sex",ylab="Frequency(people)",col="lightblue")
barplot(table_smoke,main="Smoke",ylab="Frequency(people)",col="lightgreen")
barplot(table_region,main="Region",ylab="Frequency(people)",col="lightgray")

Let’s make a multiple regression model to check whether there is a relationship with other variables to expenses or not, It will very useful to people for future medical expenses. That is,

• Dependent variable=expenses
• Independent variable= All the rest of variables

fit <- lm(expenses ~ (age+sex+bmi+children+smoker+region), data=insurance)
summary(fit)

## 
## Call:
## lm(formula = expenses ~ (age + sex + bmi + children + smoker + 
##     region), 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-16

A multiple linear regression is plotted by using expenses as the dependent variable, and the rest of features as indipendent variables in the regression model.

Therefore, by the analysis,

• A unit increase in age will increase the expense by 256.8 dollars.
• A person be a male will decrease the expense by 131.3 dollars.
• A unit increase in bmi will increase the expense by 339.3 dollars.
• A person make one children will increase the expense by 475.7 dollars.
• A person be a smoker will increase the expense by 23847.5 dollars.
• A person living in the southwest might decrease the expense by 1035.6 dollars.
• A person living in the northwest might decrease the expense by 352.8 dollars.

Also,

• age
• bmi
• children
• smokeyes
• region southeast
• region southwest

are significant, while the rest are not which can be dropped for model improvement. R-square ~ 0.75 indicates that about 75% of the variation in expenses is explained by the model.

Diagnostic plots provide checks for heteroscedasticity, normality, and influential observerations.

Diagnostic plots(four plots):

layout(matrix(c(1,2,3,4),2,2)) 
plot(fit)

By the above four plots we can conclude that,

+ By the **Residuals vs Fitted** graph -->> Variance is nearly constant.
+ By the **Normal QQ** graph -->>  Approximately equals to the normal assuptions.

So, we can imagine this is a multiple linear regression model(But its better if we do with imagine this is a multiple non-linear regression model) and can be applied a normal assumptions.

Conclusion:

The regression model is, expenses=-11941.6+256.8(age)+339.3(bmi)+475.7(children)+23847.5(smokeryes)-1035.6(regionsoutheast)-959.3(regionsouthwest)