The who.csv dataset contains real-world data from 2008. The variables included follow.
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.
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.
who <- read.csv("who.csv")

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  
## 
head(who,5)
##       Country LifeExp InfantSurvival Under5Survival  TBFree      PropMD
## 1 Afghanistan      42          0.835          0.743 0.99769 0.000228841
## 2     Albania      71          0.985          0.983 0.99974 0.001143127
## 3     Algeria      71          0.967          0.962 0.99944 0.001060478
## 4     Andorra      82          0.997          0.996 0.99983 0.003297297
## 5      Angola      41          0.846          0.740 0.99656 0.000070400
##        PropRN PersExp GovtExp TotExp
## 1 0.000572294      20      92    112
## 2 0.004614439     169    3128   3297
## 3 0.002091362     108    5184   5292
## 4 0.003500000    2589  169725 172314
## 5 0.001146162      36    1620   1656
who_lm1 <- lm(who$LifeExp ~ who$TotExp)

plot(LifeExp~TotExp, data=who, xlab="Total Expenditures", ylab="Life Expectancy")
abline(who_lm1, col='blue')

summary(who_lm1)
## 
## Call:
## lm(formula = who$LifeExp ~ who$TotExp)
## 
## 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 ***
## who$TotExp  6.297e-05  7.795e-06   8.079 7.71e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared:  0.2577, Adjusted R-squared:  0.2537 
## F-statistic: 65.26 on 1 and 188 DF,  p-value: 7.714e-14
# Residuals Analysis
plot(who_lm1$fitted.values, who_lm1$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0, col="red")

qqnorm(who_lm1$residuals)
qqline(who_lm1$residuals)

#plot(who_lm) #more extensive residual analysis plots
Interpretation1
R_squared is considerable low (0.2577), which means the model can only explain 25% of data variability. The F statistic is not very representative as it is a one-factor model. The Std Error of the coefficient TotExp is about 8.5 times smaller than the coefficient which is good in terms of variability size of the slope. The p-value of the coefficient TotExp is very low, which means it is very relevant to the prediction of the target variable in the model.
# Residuals plot shows a very disperse variability with a clear pattern (not centered around zero). # Q-Q plot shows the residuals clearly deviating from the theoretical straight line
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?”
LifeExp_powered <- who$LifeExp^4.6
TotExp_powered <- who$TotExp^0.06

who_lm2 <- lm(LifeExp_powered ~ TotExp_powered)

plot(LifeExp_powered ~ TotExp_powered, data=who, xlab="Total Expenditures (power 0.06)", ylab="Life Expectancy(power 4.6)")
abline(who_lm2, col='blue')

summary(who_lm2)
## 
## Call:
## lm(formula = LifeExp_powered ~ TotExp_powered)
## 
## 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 ***
## TotExp_powered  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 Analysis
plot(who_lm2$fitted.values, who_lm2$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0, col="red")

qqnorm(who_lm2$residuals)
qqline(who_lm2$residuals)

#plot(who_lm) #more extensive residual analysis plots
Interpretation2
R_squared went up considerably from (0.2577) to (0.7298), which means the model can only explain almost 73% of data variability. The F statistic is not very representative as it is a one-factor model. The Std Error of the coefficient TotExp is about 22 times smaller than the coefficient which is good in terms of variability size of the slope. The p-value of the coefficient TotExp is much smaller (2e-16), still very relevant to the prediction of the target variable in the model.
Residuals plot shows a relatively constant variability with no apparent patterns. Q-Q plot shows the residuals relatively following the theoretical straight line (except on the ends), which denotes a normal distribution
This model performs much better than the first degree one
3. Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
test_data <- data.frame(TotExp_powered=c(1.5, 2.5))
predicted.dat <- predict(who_lm2, newdata=test_data, interval='confidence')^(1/4.6)

predicted.dat
##        fit      lwr      upr
## 1 63.31153 62.10686 64.43893
## 2 86.50645 85.41583 87.54970
For TotExp = 1.5, LifeExp = 63.3 with a Confidence Interval (62.1, 64.4) at 95% Confidence level
For TotExp = 2.5, LifeExp = 86.5 with a Confidence Interval (85.4, 87.5) at 95% Confidence level
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
who_lm3 <- lm(LifeExp ~ PropMD + TotExp + PropMD*TotExp, data=who)

summary(who_lm3)
## 
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, 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 Analysis
plot(who_lm3$fitted.values, who_lm3$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0, col="red")

qqnorm(who_lm3$residuals)
qqline(who_lm3$residuals)

Interpretation3
R_squared went down considerably from (0.7298) to (0.3574), which means the model can only explain almost 36% of data variability. The F statistic is 34.5 by considering 3 variables. The Std Error of the first 2 coefficients are several times smaller than the coefficients except for the last combined feature. The p-values of the coefficients very small so all variables are very relevant to the prediction of the target variable in the model.
# Residuals plot shows a very disperse variability with a clear pattern (not centered around zero). # Q-Q plot shows the residuals clearly deviating from the theoretical straight line
This model is not very good, very similar to the first model with 1 variable of order 1
5. Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
test_data <- data.frame(PropMD=0.03, TotExp=14)
predicted.dat <- predict(who_lm3, newdata=test_data, interval='confidence')

predicted.dat
##       fit      lwr      upr
## 1 107.696 91.85996 123.5321
For PropMD = 0.03 & TotExp = 14, LifeExp = 107.6 with a Confidence Interval (91.8, 123.5) at 95% Confidence level
The prediction and Confidence Interval seem unrealistically high, primarily because the LifeExp caps up to 83 and the TotExp of 14 is very low compared to the training set numbers (min of 13)