Data 605 - Assignment 12
Hazal Gunduz
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.
Loading the Data
who <- read.csv("~/Downloads/who.csv")
summary(who)## 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. :482750head(who, 10)## Country LifeExp InfantSurvival Under5Survival TBFree
## 1 Afghanistan 42 0.835 0.743 0.99769
## 2 Albania 71 0.985 0.983 0.99974
## 3 Algeria 71 0.967 0.962 0.99944
## 4 Andorra 82 0.997 0.996 0.99983
## 5 Angola 41 0.846 0.740 0.99656
## 6 Antigua and Barbuda 73 0.990 0.989 0.99991
## 7 Argentina 75 0.986 0.983 0.99952
## 8 Armenia 69 0.979 0.976 0.99920
## 9 Australia 82 0.995 0.994 0.99993
## 10 Austria 80 0.996 0.996 0.99990
## PropMD PropRN PersExp GovtExp TotExp
## 1 0.000228841 0.000572294 20 92 112
## 2 0.001143127 0.004614439 169 3128 3297
## 3 0.001060478 0.002091362 108 5184 5292
## 4 0.003297297 0.003500000 2589 169725 172314
## 5 0.000070400 0.001146162 36 1620 1656
## 6 0.000142857 0.002773810 503 12543 13046
## 7 0.002780191 0.000741044 484 19170 19654
## 8 0.003698671 0.004918937 88 1856 1944
## 9 0.002331953 0.009149391 3181 187616 190797
## 10 0.003610904 0.006458749 3788 189354 193142Scatterplot of the LifeExp ~ TotExp and linear regression.
#Simple linear regression of LifeExp ~ TotExp
who_lm <- lm(LifeExp ~ TotExp, data = who)
summary(who_lm)##
## 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-14R-squared of 0.2577 shows that 25.77% of the model data variation. We see that assumptions of simple linear regression aren’t met. The variable that we used shows for 25.77% of the variance.
#Scatterplot of the LifeExp ~ TotExp
plot(LifeExp ~ TotExp, data = who,
main = "Life Expectancy vs Total Expenditures",
xlab = "Total Expenditures", ylab = "Life Expectancy")
abline(who_lm, col = "blue")
plot(who_lm$fitted.values, who_lm$residuals,
main = "Residuals Plot",
xlab = "Fitted Values", ylab = "Residuals")
abline(h = 0, col = "green")
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?”
#Life Expectancy ^ 4.6
who_LifeExp <- who$LifeExp ^ (4.6)
#Total Expenditures ^ 0.06
who_TotExp <- who$TotExp ^ (0.06)#Scatterplot transformed variables
plot(who_TotExp ~ who_LifeExp,
main = "Total Expenditure vs Life Expectancy",
xlab = "TotalExpenditure", ylab = "LifeExpentancy")
#Simple regression LifeExp ~ TotExp
who_lm2 <- lm(who_LifeExp ~ who_TotExp, data = who)
summary(who_lm2)##
## Call:
## lm(formula = who_LifeExp ~ who_TotExp, data = who)
##
## 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_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-16This model is better. The plot is more linear than first linear. R-squared value has increased.
3. Using the results from 3, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.
new_data <- data.frame(who_TotExp = 1.5)
predict_LifeExp <- predict(who_lm2, newdata = new_data) ^ (1/4.6)
predict_LifeExp## 1
## 63.31153new_data2 <- data.frame(who_TotExp = 2.5)
predict_LifeExp2 <- predict(who_lm2, newdata = new_data2) ^ (1/4.6)
predict_LifeExp2## 1
## 86.50645The model predicts a life expectancy of ~63 and ~87.
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
#multiple regression model
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-16F-statistic: 34.49 on 3 and 186 DF.The 34.49 F statistic means that the model is significant at the 0.01 level, as the corresponding p-value (2.2e-16) is less than 0.01.
R-squared: 0.3471. This means that 34.71% of the change in life expectancy is explained by the sum of total expenditures in the model.
Standard error: 8.765 on 186 degrees of freedom. The standard error of 8.765 means that the average difference between the predicted and actual values of the dependent variable is 8.765.
P-values: 2.2e-16, PropMD:2.32e-07, TotExp:9.39e-14, PropMD:TotExp:6.35e-05. These p values for the total expenditure variable are significant at the 0.01. This means that the total expenditure, Proportion of MD are all associated with life expectancy variable in the model.
5. Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?
predict(who_lm3, data.frame(PropMD = 0.03, TotExp = 14))## 1
## 107.696The forecast doesn’t seem very realistic as humans don’t live that long.