Homework 9

#Step 1: load csv

df <- read.csv("AllCountries.csv")

#Step 2: create simple linear regresssion

linreg <- lm(LifeExpectancy ~ GDP, df)
summary(linreg)

## 
## Call:
## lm(formula = LifeExpectancy ~ GDP, data = df)
## 
## 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
##   (38 observations deleted due to missingness)
## Multiple R-squared:  0.4304, Adjusted R-squared:  0.4272 
## F-statistic: 133.7 on 1 and 177 DF,  p-value: < 2.2e-16

Interpretation:

• The intercept (68.42) represents the predicted LifeExpectancy when GDP is 0.

• The coefficient for GDP (around 0.0002476) means for every 1 dollar increase in GDP per capita, Life Expectancy increases by about 0.0002476 years.

• The R² value explains 43.04% of the variance in Life Expectancy from GDP alone, which leaves most of the variance unpredicted

#Step 3: create multiple regression model
multiple_model <- lm(LifeExpectancy ~ GDP + Health + Internet, data = df)
multiple_model

## 
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = df)
## 
## Coefficients:
## (Intercept)          GDP       Health     Internet  
##   5.908e+01    2.367e-05    2.479e-01    1.903e-01

summary(multiple_model)

## 
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = df)
## 
## 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
##   (44 observations deleted due to missingness)
## Multiple R-squared:  0.7213, Adjusted R-squared:  0.7164 
## F-statistic: 145.8 on 3 and 169 DF,  p-value: < 2.2e-16

Interpretation:

• Health has a coefficient of 0.2479, meaning every increase by 1% of government expenditures directed towards healthcare increases Life Expectancy by about 0.2479.

• The Adjusted R² of the multiple-predictor model is 0.7164, which means about 72% of Life Expectancy variance can be explained by this model. This is a large improvement over the single-predictor model, which could only predict 43.04% of variance.

#Step 4: Checking Assumptions

• To check the assumption of Homoscedasticity, we would make a Residuals vs. Fitted plot. Ideally, this plot would have a random scatter of points centered along the horizontal zero line. This would confirm that the model has a linear relationship, variance between errors is equal, and there are no major outliers.

• To check normality of residuals we would make a Q-Q plot. Ideally the points on the plot would create a straight diagonal line. An S-curve or heavy tails would indicate that the residuals are not normally distributed, which would make our p-values and confidence intervals less valid.

plot(linreg)

Reflection

• The Q-Q plot for our model has heavy tails at the bottom and top, meaning the residuals are not normally distributed.

• The Residuals vs. Fitted plot shows heteroscedasticity. It is funnel-shaped with a much wider spread of residuals at lower fitted values. This indicates that the accuracy of our model is not consistent across the data range.

#Part 5: Calculating RMSE for multiple_model

#Calculate Residuals
residuals_multiple <- resid(multiple_model)

# Calculate RMSE for multiple model
rmse_multiple <- sqrt(mean(residuals_multiple^2))
rmse_multiple

## [1] 4.056417

• On average, the model’s prediction of life expectancy is off by about 4.06 years. High residuals from countries with unusually high or low life expectancy reduce my confidence in the model because they make the distribution of the residuals less normal, negatively impacting the accuracy of confidence intervals caclulated with the model.

#Part 6: Multicollinearity in multiple regression

• The correlation of predictors Energy and Electricity would confuse the model, inflate errors, and weaken predictions.