Dataset

The attached 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.

Multiple R-squared: 0.2577, Adjusted R-squared: 0.2537

F-statistic: 65.26 on 1 and 188 DF

p-value: 7.714e-14

Std. Error: 7.795e-06

The p-value is less than 0.05. This model only covers about 25% of the data.

who = read.csv('who.csv')

head(who)
##               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
## 6 Antigua and Barbuda      73          0.990          0.989 0.99991 0.000142857
##        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
## 6 0.002773810     503   12543  13046
plot(who$LifeExp, who$TotExp)

who_lm = lm(who$LifeExp~who$TotExp)
summary(who_lm)
## 
## 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$TotExp  6.297e-05  7.795e-06   8.079 7.71e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared:  0.2577, Adjusted R-squared:  0.2537 
## F-statistic: 65.26 on 1 and 188 DF,  p-value: 7.714e-14
qqnorm(who_lm$residuals)
qqline(who_lm$residuals)

The p-value for the variable Totexp is 7.71e-14 which is less than 0.05. Adjusted R-squared is 0.2537

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?”

F-statistic: 507.7 on 1 and 188 DF

p-value: < 2.2e-16

Std. Error : 27518940

Multiple R-squared: 0.7298, Adjusted R-squared: 0.7283

The points follow the line on the qq plot better than the previous plot. The adjusted R squared values are higher than the previous one so this model is better.

who$LifeExp1 = who$LifeExp**4.6
who$TotExp1 = who$TotExp**0.06

plot(who$LifeExp1, who$TotExp1)

who_lm2 = lm(who$LifeExp1~who$TotExp1)
summary(who_lm2)
## 
## Call:
## lm(formula = who$LifeExp1 ~ who$TotExp1)
## 
## 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 ***
## who$TotExp1  620060216   27518940   22.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared:  0.7298, Adjusted R-squared:  0.7283 
## F-statistic: 507.7 on 1 and 188 DF,  p-value: < 2.2e-16
qqnorm(who_lm2$residuals)
qqline(who_lm2$residuals)

3.

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

slope is 620060216, y-intercept is -736527910

The life expectancy when TotExp^.06 =1.5 is 63.31153.

The life expectancy when TotExp^.06=2.5 is 86.50645.

# TotExp^.06 =1.5
# slope function
(620060216*1.5)-736527910
## [1] 193562414
#converts back to life expectancy
193562414**(1/4.6)
## [1] 63.31153
# TotExp^.06=2.5
(620060216*2.5)-736527910
## [1] 813622630
813622630**(1/4.6)
## [1] 86.50645

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

Median is close to 0. p-value is under 0.05. The residuals do not follow the line. The Multiple R-squared: 0.3574 and Adjusted R-squared: 0.3471 is not that close to 1, so there is a lot of noise. The standard errors look good.

who_multmodel = lm(who$LifeExp ~ who$PropMD + who$TotExp + (who$PropMD * who$TotExp), data=who) 

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

5.

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

We get a forecast of life expectancy of 107.6785 which is not right because the max life expectancy is 83.

# LifeExp = b0 + b1 x PropMd + b2 x TotExp + b3 x PropMD x TotExp
PropMD <- 0.03
TotExp <- 14
LifeExp_forecast<- (6.277*10) + (1.497*10^3)*PropMD + (7.233*10^-5)*TotExp - (6.026*10^-3) *PropMD*TotExp

LifeExp_forecast
## [1] 107.6785
max(who$LifeExp)
## [1] 83