Analyze World Health Organization Data

Import data

Data is real-world World Health Organization data from 2008. It includes r nrow(who) observations for r ncol(who) variables. Data dictionary:

  • Country: name of the country
  • LifeExp: average life expectancy for the country in years
  • InfantSurvival: proportion of those surviving to one year or more
  • Under5Survival: proportion of those surviving to five years or more
  • TBFree: proportion of the population without TB
  • PropMD: proportion of the population who are MDs
  • PropRN: proportion of the population who are RNs
  • PersExp: mean personal expenditures on healthcare in US dollars at average exchange rate
  • GovtExp: mean government expenditures per capita on healthcare, US dollars at average exchange rate
  • TotExp: sum of personal and government expenditures
who <- read.csv('who.csv')
head(who)
##               Country LifeExp InfantSurvival Under5Survival  TBFree
## 1         Afghanistan      42          0.835          0.743 0.99769
## 2             Albania      71          0.985          0.983 0.99974
## 3             Algeria      71          0.967          0.962 0.99944
## 4             Andorra      82          0.997          0.996 0.99983
## 5              Angola      41          0.846          0.740 0.99656
## 6 Antigua and Barbuda      73          0.990          0.989 0.99991
##        PropMD      PropRN PersExp GovtExp TotExp
## 1 0.000228841 0.000572294      20      92    112
## 2 0.001143127 0.004614439     169    3128   3297
## 3 0.001060478 0.002091362     108    5184   5292
## 4 0.003297297 0.003500000    2589  169725 172314
## 5 0.000070400 0.001146162      36    1620   1656
## 6 0.000142857 0.002773810     503   12543  13046

Data Exploration

summary(who)
##                 Country       LifeExp      InfantSurvival  
##  Afghanistan        :  1   Min.   :40.00   Min.   :0.8350  
##  Albania            :  1   1st Qu.:61.25   1st Qu.:0.9433  
##  Algeria            :  1   Median :70.00   Median :0.9785  
##  Andorra            :  1   Mean   :67.38   Mean   :0.9624  
##  Angola             :  1   3rd Qu.:75.00   3rd Qu.:0.9910  
##  Antigua and Barbuda:  1   Max.   :83.00   Max.   :0.9980  
##  (Other)            :184                                   
##  Under5Survival       TBFree           PropMD              PropRN         
##  Min.   :0.7310   Min.   :0.9870   Min.   :0.0000196   Min.   :0.0000883  
##  1st Qu.:0.9253   1st Qu.:0.9969   1st Qu.:0.0002444   1st Qu.:0.0008455  
##  Median :0.9745   Median :0.9992   Median :0.0010474   Median :0.0027584  
##  Mean   :0.9459   Mean   :0.9980   Mean   :0.0017954   Mean   :0.0041336  
##  3rd Qu.:0.9900   3rd Qu.:0.9998   3rd Qu.:0.0024584   3rd Qu.:0.0057164  
##  Max.   :0.9970   Max.   :1.0000   Max.   :0.0351290   Max.   :0.0708387  
##                                                                           
##     PersExp           GovtExp             TotExp      
##  Min.   :   3.00   Min.   :    10.0   Min.   :    13  
##  1st Qu.:  36.25   1st Qu.:   559.5   1st Qu.:   584  
##  Median : 199.50   Median :  5385.0   Median :  5541  
##  Mean   : 742.00   Mean   : 40953.5   Mean   : 41696  
##  3rd Qu.: 515.25   3rd Qu.: 25680.2   3rd Qu.: 26331  
##  Max.   :6350.00   Max.   :476420.0   Max.   :482750  
##

The above provides a glimpse of what the data is showing for each column and category.

Question 1

Provide a scatterplot of LifeExp~TotExp, and run simple linear regression. Do not transform the variables. Provide and interpret the F statistics, R^2, standard error,and p-values only. Discuss whether the assumptions of simple linear regression met.

# Linear regression model build
lm <- lm(LifeExp ~ TotExp, data=who)

# Scatterplot of dependent and independent variables
plot(LifeExp~TotExp, data=who, 
     xlab="Total Expenditures", ylab="Life Expectancy",
     main="Life Expectancy vs Total Expenditures")
abline(lm)

# Linear regression model summary
summary(lm)
## 
## Call:
## lm(formula = LifeExp ~ TotExp, data = who)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -24.764  -4.778   3.154   7.116  13.292 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 6.475e+01  7.535e-01  85.933  < 2e-16 ***
## TotExp      6.297e-05  7.795e-06   8.079 7.71e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared:  0.2577, Adjusted R-squared:  0.2537 
## F-statistic: 65.26 on 1 and 188 DF,  p-value: 7.714e-14
# Residuals variability plot
plot(lm$fitted.values, lm$residuals, 
     xlab="Fitted Values", ylab="Residuals",
     main="Residuals Plot for Linear Model")
abline(h=0)

## Residuals Q-Q plot
qqnorm(lm$residuals)
qqline(lm$residuals)

Results

Three items are providing insight into the shape of the data.

  • \(R^2\) ~ 0.25
  • Non-random shape of residuals
  • Major deviation at the tails for the QQ plot

Looking a the above it is clear that the relationship is not linear and that there is a differnt underlying relationship in the data.

Question 2

Raise life expectancy to the 4.6 power (i.e., LifeExp^4.6). Raise total expenditures to the 0.06 power (nearly a log transform, TotExp^.06). Plot LifeExp^4.6 as a function of TotExp^.06, and r re-run the simple regression model using the transformed variables. Provide and interpret the F statistics, R^2, standard error, and p-values. Which model is “better?”

Transforming the variables as noted in Question 2.

# Transformation
LifeExpNew <- who$LifeExp^4.6
TotExpNew <- who$TotExp^0.06

# Linear regression model build
lmNew <- lm(LifeExpNew ~ TotExpNew)

# Scatterplot of dependent and independent variables
plot(LifeExpNew~TotExpNew, 
     xlab="Total Expenditures", ylab="Life Expectancy",
     main="Life Expectancy vs Total Expenditures (Modified)")
abline(lmNew)

# Linear regression model summary
summary(lmNew)
## 
## Call:
## lm(formula = LifeExpNew ~ TotExpNew)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -308616089  -53978977   13697187   59139231  211951764 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -736527910   46817945  -15.73   <2e-16 ***
## TotExpNew    620060216   27518940   22.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared:  0.7298, Adjusted R-squared:  0.7283 
## F-statistic: 507.7 on 1 and 188 DF,  p-value: < 2.2e-16
# Residuals variability plot
plot(lmNew$fitted.values, lmNew$residuals, 
     xlab="Fitted Values", ylab="Residuals",
     main="Residuals Plot New")
abline(h=0)

# Residuals Q-Q plot
qqnorm(lmNew$residuals)
qqline(lmNew$residuals)

Results

Three items are providing insight into the shape of the data.

  • R^2 ~ 0.75
  • Random shape of residuals
  • Minimal deviation at the tails for the QQ plot

Looking at the above it is clear that the relationship is better described by the power functions. The QQ plot shows minimal deviations at the tail indicating that there is no skewness based upon the new linear model developed. The random residuals around the 0 line also indicate that the linear model

Question 3

predictdata <- data.frame(TotExpNew=c(1.5,2.5))
predict(lmNew, predictdata,interval="predict")^(1/4.6)
##        fit      lwr      upr
## 1 63.31153 35.93545 73.00793
## 2 86.50645 81.80643 90.43414

Predicting the values at 1.5 adn 2.5 provides the following results.

The prediction at 1.5 is 63 years with a CI(35.93545, 73.00793).

The prediction at 2.5 is 87 year with a CI(81.80643, 90.43414).

Question 4

Build the following multiple regression model and interpret the F Statistics, R^2, standard error, and p-values. How good is the model?

LifeExp = b0+b1 x PropMd + b2 x TotExp + b3 x PropMD x TotExp

# Multiple linear regression model build
lm4 <- lm(LifeExp ~ PropMD + TotExp + TotExp:PropMD, data=who)

# Linear regression model summary
summary(lm4)
## 
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + TotExp:PropMD, data = who)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.320  -4.132   2.098   6.540  13.074 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    6.277e+01  7.956e-01  78.899  < 2e-16 ***
## PropMD         1.497e+03  2.788e+02   5.371 2.32e-07 ***
## TotExp         7.233e-05  8.982e-06   8.053 9.39e-14 ***
## PropMD:TotExp -6.026e-03  1.472e-03  -4.093 6.35e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.765 on 186 degrees of freedom
## Multiple R-squared:  0.3574, Adjusted R-squared:  0.3471 
## F-statistic: 34.49 on 3 and 186 DF,  p-value: < 2.2e-16
# Residuals variability plot
plot(lm4$fitted.values, lm4$residuals, 
     xlab="Fitted Values", ylab="Residuals",
     main="Residuals Plot Cross Variable")
abline(h=0)

# Residuals Q-Q plot
qqnorm(lm4$residuals)
qqline(lm4$residuals)

Results

Three items are providing insight into the shape of the data.

  • \(R^2\) ~ 0.34
  • Non-random shape of residuals
  • Major deviation at the tails for the QQ plot

Looking a the above it is clear that the relationship is not linear and that there is a differnt underlying relationship in the data. In this case, the interaction between Total Expenditure and MD Population didn’t generate a new variable that was a better indicator of the data.

Question 5

Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?

newdata <- data.frame(PropMD=0.03, TotExp=14)
predict(lm4, newdata,interval="predict")
##       fit      lwr      upr
## 1 107.696 84.24791 131.1441

Predicting the values at PropMD=0.03, TotExp=14 provides the following results.

The prediction is 108 years with a CI(84.24791, 131.1441).

The data maxes out about the 90-100 range. Seeing a prediction of 108 becomes unrealistic when the CI also shows 132 years.

The model does what it is supposed to which is predict but it’s up the data scientist to also interpret the results of the model.