Life Expectancy Dataset

who_df <- read_csv("http://my-cunymsds.com/data605/data/who.csv")

EDA of Dataset

General stats of data set

summary(who_df)

##    Country             LifeExp      InfantSurvival   Under5Survival  
##  Length:190         Min.   :40.00   Min.   :0.8350   Min.   :0.7310  
##  Class :character   1st Qu.:61.25   1st Qu.:0.9433   1st Qu.:0.9253  
##  Mode  :character   Median :70.00   Median :0.9785   Median :0.9745  
##                     Mean   :67.38   Mean   :0.9624   Mean   :0.9459  
##                     3rd Qu.:75.00   3rd Qu.:0.9910   3rd Qu.:0.9900  
##                     Max.   :83.00   Max.   :0.9980   Max.   :0.9970  
##      TBFree           PropMD              PropRN             PersExp       
##  Min.   :0.9870   Min.   :0.0000196   Min.   :0.0000883   Min.   :   3.00  
##  1st Qu.:0.9969   1st Qu.:0.0002444   1st Qu.:0.0008455   1st Qu.:  36.25  
##  Median :0.9992   Median :0.0010474   Median :0.0027584   Median : 199.50  
##  Mean   :0.9980   Mean   :0.0017954   Mean   :0.0041336   Mean   : 742.00  
##  3rd Qu.:0.9998   3rd Qu.:0.0024584   3rd Qu.:0.0057164   3rd Qu.: 515.25  
##  Max.   :1.0000   Max.   :0.0351290   Max.   :0.0708387   Max.   :6350.00  
##     GovtExp             TotExp      
##  Min.   :    10.0   Min.   :    13  
##  1st Qu.:   559.5   1st Qu.:   584  
##  Median :  5385.0   Median :  5541  
##  Mean   : 40953.5   Mean   : 41696  
##  3rd Qu.: 25680.2   3rd Qu.: 26331  
##  Max.   :476420.0   Max.   :482750

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.

ggplot(who_df, aes(x=TotExp, y=LifeExp)) + 
  geom_point(alpha=0.3) +
  geom_smooth()

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

model_01 <- lm(LifeExp ~ TotExp, data=who_df)
summary(model_01)

## 
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_df)
## 
## 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

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?”

ggplot(who_df, aes(x=TotExp^0.06, y=LifeExp^4.6)) + 
  geom_point(alpha=0.3) +
  geom_smooth(method = lm)

## `geom_smooth()` using formula 'y ~ x'

model_02 <- lm(I(LifeExp^4.6) ~ I(TotExp^0.06), data=who_df)
summary(model_02)

## 
## Call:
## lm(formula = I(LifeExp^4.6) ~ I(TotExp^0.06), data = who_df)
## 
## 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 ***
## I(TotExp^0.06)  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

plot(model_02)

model_02_preds <- as_tibble(model_02$fitted.values^(1/4.6))
model_02_preds <- rename(model_02_preds, predicted= value)
model_02_preds$actual <- who_df$LifeExp

ggplot(model_02_preds, aes(x=actual, y=predicted)) + 
  geom_point(alpha=0.3) +
  geom_smooth()

3. Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.

b0 <- model_02$coefficients[1]
b1 <- model_02$coefficients[2]
Pred1 <- b0 + b1*1.5
Pred2 <- b0 + b1*2.5

# To makes sense of the prediction LifeExp^4.6, we will invert by raising the model prediction pred^(1/4.6)

print(paste0("First prediction when TotExp^.06=1.5 -> ", as.character(round(Pred1^(1/4.6),2))))

## [1] "First prediction when TotExp^.06=1.5 -> 63.31"

print(paste0("Second prediction when TotExp^.06=2.5 -> ", as.character(round(Pred2^(1/4.6),2))))

## [1] "Second prediction when TotExp^.06=2.5 -> 86.51"

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

model_03 <- lm(LifeExp ~ PropMD + TotExp + PropMD*TotExp, data=who_df)
summary(model_03)

## 
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_df)
## 
## 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

plot(model_03)

## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced

## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced

model_03_preds <- as_tibble(model_03$fitted.values)
model_03_preds <- rename(model_03_preds, predicted= value)
model_03_preds$actual <- who_df$LifeExp

ggplot(model_03_preds,aes(x=actual, y=predicted)) + 
  geom_point(alpha=0.3) +
  geom_smooth()

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

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

b0 <- model_03$coefficients[1]
b1 <- model_03$coefficients[2]
b2 <- model_03$coefficients[3]
b12 <- model_03$coefficients[4]

Pred1 <- b0 + b1*0.03 + b2*14 + b12*(0.03*14)

print(paste0("Prediction when PropMd=0.03 TotExp=14 -> ", as.character(round(Pred1,2))))

## [1] "Prediction when PropMd=0.03 TotExp=14 -> 107.7"