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.
Load data from CSV file
| Country | LifeExp | InfantSurvival | Under5Survival | TBFree | PropMD | PropRN | PersExp | GovtExp | TotExp |
|---|---|---|---|---|---|---|---|---|---|
| Afghanistan | 42 | 0.835 | 0.743 | 0.99769 | 0.0002288 | 0.0005723 | 20 | 92 | 112 |
| Albania | 71 | 0.985 | 0.983 | 0.99974 | 0.0011431 | 0.0046144 | 169 | 3128 | 3297 |
| Algeria | 71 | 0.967 | 0.962 | 0.99944 | 0.0010605 | 0.0020914 | 108 | 5184 | 5292 |
| Andorra | 82 | 0.997 | 0.996 | 0.99983 | 0.0032973 | 0.0035000 | 2589 | 169725 | 172314 |
| Angola | 41 | 0.846 | 0.740 | 0.99656 | 0.0000704 | 0.0011462 | 36 | 1620 | 1656 |
| Antigua and Barbuda | 73 | 0.990 | 0.989 | 0.99991 | 0.0001429 | 0.0027738 | 503 | 12543 | 13046 |
## Country LifeExp InfantSurvival Under5Survival
## Afghanistan : 1 Min. :40.00 Min. :0.8350 Min. :0.7310
## Albania : 1 1st Qu.:61.25 1st Qu.:0.9433 1st Qu.:0.9253
## Algeria : 1 Median :70.00 Median :0.9785 Median :0.9745
## Andorra : 1 Mean :67.38 Mean :0.9624 Mean :0.9459
## Angola : 1 3rd Qu.:75.00 3rd Qu.:0.9910 3rd Qu.:0.9900
## Antigua and Barbuda: 1 Max. :83.00 Max. :0.9980 Max. :0.9970
## (Other) :184
## 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.
Scatterplot of LifeExp~TotExp
plot(who_data$LifeExp ~ who_data$TotExp, main = "LifeExp vs TotExp", xlab = "Sum of Personal and government expenditures", ylab = "Average life expectancy In Years")
abline(lm(who_data$LifeExp ~ who_data$TotExp), col="blue")
Perform Linear Regression
##
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_data)
##
## 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-14F-statistic is 65.26 and p-value is close to 0 (7.714e-14). It is likely that the model is explaining the data well, Due to the \(R^2\) value 0.2577, we can conclude that only 25% of the variation can be explained by our data. Standard error is very low. The assumptions of of simple linear regression are met.
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?”
Adding Column LifeExp_4.6
Adding Column TotExp_0.06
## 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 LifeExp_4.6 TotExp_0.06
## 1 0.000572294 20 92 112 29305338 1.327251
## 2 0.004614439 169 3128 3297 327935478 1.625875
## 3 0.002091362 108 5184 5292 327935478 1.672697
## 4 0.003500000 2589 169725 172314 636126841 2.061481
## 5 0.001146162 36 1620 1656 26230450 1.560068
## 6 0.002773810 503 12543 13046 372636298 1.765748Scatterplot of LifeExp_4.6~TotExp_0.06
plot(who_data$TotExp_0.06 , who_data$LifeExp_4.6, main = "LifeExp^{4.6} vs TotExp^{.06}", xlab = "Average life expectancy In Years^0.06", ylab = "Sum of Personal and government expenditures^4.6")
abline(lm( who_data$LifeExp_4.6 ~ who_data$TotExp_0.06), col="blue")
Perform Linear Regression
##
## Call:
## lm(formula = LifeExp_4.6 ~ TotExp_0.06, data = who_data)
##
## 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_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-16From this above, the linear relationship looks very clear between these 2 variables.
Also as we see the R squared values are high to assume that this is a good fit for the data.
We see that the new model fits better as compared to the model we built earlier.
3.Using the results from 2, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
predict_LifeExp <- function(TotExp_0.06_val){
return ((predict(transformed.lm, newdata = data.frame(TotExp_0.06 = TotExp_0.06_val))) ^(1/4.6))
}Forecast life expectancy when \(TotExp^{.06}\)=1.5. Then forecast life expectancy when \(TotExp^{.06}\)=2.5
## 1
## 63.31153## 1
## 86.506454. 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
LifeExp.lm <- lm(LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_data)
summary(lm(LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_data))##
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_data)
##
## 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\(R^2\) value is low so the model in not a great fit.
5. Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
## 1
## 107.696The forecast does not seem to be realistic as it is a very high value of life expectancy which is not very very rare.