Introduction to Data
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.
## 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. :4827501. ScatterPlot and Initial Linear Regression
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.
##
## 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-14The F-statistic is 65.26 with a p-value close to 0 gives us an indication of a significant relationship between Total Expenditure and Life Expectancy.
The \(R^2\) tells us that this model accounts for only 25.77% of the variation of the data.
Are the assumptions met?
- The residuals appear to be randomly distributed on the plot.
- Most of the residuals fall on the theoretical normal line.
- According to the histogram, the distribution of the residuals is slightly left skewed short long tails.
- The variation of the residuals appears to be constant accross the model and the residuals are independent from each other.
2. Which Model is Better?
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 re-run the simple regression model using the transformed variables.
##
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_2)
##
## 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 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
Provide and interpret the F statistics, R^2, standard error, and p-values.
- F Statistics & P-value: the value is 507.7 with p-value very close to zero, which means there is significant evidence that the current model is better than the initial model. If we recall that the original model’s
F-statisticwas 65.26, we can assume this is a better model.
- \(R^2\): around 73% of the variability of the data can be explained by the current model vs 26% with the initial one.
- Standard error: the standard error is way larger than the initial model’s. Not entirely sure what that means
Which model is “better?”
The second model is a better one. It accounts for 73% of the variability. The distribution of the residuals is nearly normal and independent.
3. Forecast
Using the results from 3, forecast life expectancy when \(TotExp^.06 =1.5\).
at_1.5 <- (-736527910 + 620060216 * 1.5) ^ (1/4.6)
paste("Life Expectancy when Tot Exp = 1.5 is: ", at_1.5)## [1] "Life Expectancy when Tot Exp = 1.5 is: 63.3115334478635"Then forecast life expectancy when \(TotExp^.06=2.5\)
at_2.5 <- (-736527910 + 620060216 * 2.5) ^ (1/4.6)
paste("Life Expectancy when Tot Exp = 2.5 is: ", at_2.5)## [1] "Life Expectancy when Tot Exp = 2.5 is: 86.5064484928337"4. Build Models.
Build the following multiple regression model and interpret the F Statistics, R^2, standard error, and p-values. How good is the model? \(LifeExp = b_0+b_1 * PropMd + b_2 * TotExp + b_3 * PropMD * TotExp\)
##
## 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- F Statistics & P-value: the value is 34.49 with p-value close to zero, the
F-statisticis lower than the original model’s which was 65.26.
- \(R^2\): around 36% of the variability of the data can be explained by the current model vs 26% with the initial one.
- Standard error: the residual standard error is 8.765, smaller than the initial model’s.
Overall, this model seems better than the initial one but not as good as the transformed model.
5. Forecast LifeExp
when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
## [1] 107.6785We saw that as the Total Expenditure increases, so does the life expectancy. That said, 107 appears to be a little high and unrealistic. Only two countries have a PropMD of 0.03 or higher and their TotExp are way higher than 14. The fact that these countries have LifeExp around 80 seem to imply that this proyection is totally unrealistic.