Problem 1
# Read the data
countries <- read.csv("AllCountries.csv")
# Keep only observations with GDP and LifeExpectancy available
countries2 <- na.omit(countries[, c("GDP", "LifeExpectancy")])
# Fit simple linear regression model
life_model <- lm(LifeExpectancy ~ GDP, data = countries2)
# Display regression results
summary(life_model)
##
## Call:
## lm(formula = LifeExpectancy ~ GDP, data = countries2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.352 -3.882 1.550 4.458 9.330
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.842e+01 5.415e-01 126.36 <2e-16 ***
## GDP 2.476e-04 2.141e-05 11.56 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.901 on 177 degrees of freedom
## Multiple R-squared: 0.4304, Adjusted R-squared: 0.4272
## F-statistic: 133.7 on 1 and 177 DF, p-value: < 2.2e-16
# Extract coefficients
coef(life_model)
## (Intercept) GDP
## 6.842208e+01 2.476441e-04
# R-squared value
summary(life_model)$r.squared
## [1] 0.4303976
# Scatterplot with regression line
plot(countries2$GDP, countries2$LifeExpectancy,
pch = 19,
col = "steelblue",
xlab = "GDP per Capita (US$)",
ylab = "Life Expectancy (Years)",
main = "Life Expectancy vs GDP")
abline(life_model, col = "red", lwd = 2)
Problem 3
# Read the data
countries <- read.csv("AllCountries.csv")
# Keep only variables needed for the model
data_mlr <- na.omit(countries[, c("LifeExpectancy", "GDP", "Health", "Internet")])
# Fit multiple linear regression model
mlr_model <- lm(LifeExpectancy ~ GDP + Health + Internet, data = data_mlr)
# Display results
summary(mlr_model)
##
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = data_mlr)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.5662 -1.8227 0.4108 2.5422 9.4161
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.908e+01 8.149e-01 72.499 < 2e-16 ***
## GDP 2.367e-05 2.287e-05 1.035 0.302025
## Health 2.479e-01 6.619e-02 3.745 0.000247 ***
## Internet 1.903e-01 1.656e-02 11.490 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.104 on 169 degrees of freedom
## Multiple R-squared: 0.7213, Adjusted R-squared: 0.7164
## F-statistic: 145.8 on 3 and 169 DF, p-value: < 2.2e-16
# Regression coefficients
coef(mlr_model)
## (Intercept) GDP Health Internet
## 5.908027e+01 2.367169e-05 2.478764e-01 1.903116e-01
# Adjusted R-squared
summary(mlr_model)$adj.r.squared
## [1] 0.7163973
# Compare with simple regression model
simple_model <- lm(LifeExpectancy ~ GDP, data = data_mlr)
summary(simple_model)$adj.r.squared
## [1] 0.4158405
summary(mlr_model)$adj.r.squared
## [1] 0.7163973
# Optional diagnostic plots
par(mfrow = c(2,2))
plot(mlr_model)
Problem 5
# Read data
countries <- read.csv("AllCountries.csv")
# Keep required variables and remove missing values
data_mlr <- na.omit(countries[, c("Country", "LifeExpectancy", "GDP", "Health", "Internet")])
# Fit multiple regression model
mlr_model <- lm(LifeExpectancy ~ GDP + Health + Internet, data = data_mlr)
# Model summary
summary(mlr_model)
##
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = data_mlr)
##
## Residuals:
## Min 1Q Median 3Q Max
## -14.5662 -1.8227 0.4108 2.5422 9.4161
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.908e+01 8.149e-01 72.499 < 2e-16 ***
## GDP 2.367e-05 2.287e-05 1.035 0.302025
## Health 2.479e-01 6.619e-02 3.745 0.000247 ***
## Internet 1.903e-01 1.656e-02 11.490 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.104 on 169 degrees of freedom
## Multiple R-squared: 0.7213, Adjusted R-squared: 0.7164
## F-statistic: 145.8 on 3 and 169 DF, p-value: < 2.2e-16
# ---------------------------
# Calculate RMSE
# ---------------------------
# Predicted values
predicted <- predict(mlr_model)
# Residuals
residuals <- data_mlr$LifeExpectancy - predicted
# RMSE
RMSE <- sqrt(mean(residuals^2))
RMSE
## [1] 4.056417
# ---------------------------
# Residual Plot
# ---------------------------
plot(predicted,
residuals,
pch = 19,
col = "steelblue",
xlab = "Predicted Life Expectancy",
ylab = "Residuals",
main = "Residuals vs Predicted Values")
abline(h = 0, col = "red", lwd = 2)
# Countries with largest residuals
data_mlr$Residual <- residuals
head(data_mlr[order(abs(data_mlr$Residual),
decreasing = TRUE), ],10)
## Country LifeExpectancy GDP Health Internet Residual
## 48 Cote d'Ivoire 54.1 1716 4.88 43.8 -14.566175
## 112 Lesotho 54.6 1324 10.90 29.8 -12.884749
## 145 Nigeria 53.9 2028 5.01 27.7 -11.741767
## 171 Sierra Leone 52.2 523 7.91 13.2 -11.365464
## 64 Eswatini 58.3 4140 15.23 30.3 -10.419868
## 178 South Africa 63.4 6340 13.32 56.2 -9.827573
## 16 Bangladesh 72.8 1698 3.38 18.0 9.416106
## 114 Libya 72.1 7235 0.00 21.8 8.699673
## 39 Chad 53.2 730 5.90 6.5 -8.597045
## 97 Italy 83.2 34318 13.47 61.3 8.302370
Probelm 6
# Read the data
countries <- read.csv("AllCountries.csv")
# Keep only variables needed and remove missing values
co2_data <- na.omit(countries[, c("CO2", "Energy", "Electricity")])
# Fit multiple regression model
co2_model <- lm(CO2 ~ Energy + Electricity, data = co2_data)
# Display model results
summary(co2_model)
##
## Call:
## lm(formula = CO2 ~ Energy + Electricity, data = co2_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.7559 -1.1406 -0.2020 0.7143 7.3751
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.998e-01 2.655e-01 3.012 0.00311 **
## Energy 3.122e-03 1.066e-04 29.290 < 2e-16 ***
## Electricity -7.044e-04 5.526e-05 -12.747 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.331 on 131 degrees of freedom
## Multiple R-squared: 0.899, Adjusted R-squared: 0.8974
## F-statistic: 582.8 on 2 and 131 DF, p-value: < 2.2e-16
# -------------------------------------------------
# Check correlation between predictors
# -------------------------------------------------
cor(co2_data$Energy,
co2_data$Electricity)
## [1] 0.7969352
# Correlation matrix
cor(co2_data[, c("Energy", "Electricity")])
## Energy Electricity
## Energy 1.0000000 0.7969352
## Electricity 0.7969352 1.0000000
# Scatterplot of Energy vs Electricity
plot(co2_data$Energy, co2_data$Electricity, pch = 19, col = "steelblue", main = "Energy vs Electricity Consumption",
xlab = "Energy (kilotons of oil equivalent)",
ylab = "Electricity (kWh per capita)")
abline(lm(Electricity ~ Energy, data = co2_data),
col = "red",
lwd = 2)
# -------------------------------------------------
# Variance Inflation Factor (VIF)
# -------------------------------------------------
library(car)
## Loading required package: carData
vif(co2_model)
## Energy Electricity
## 2.74052 2.74052
# -------------------------------------------------
# Diagnostic plots
# -------------------------------------------------
par(mfrow = c(2, 2))
plot(co2_model)
