HW9
Phillip Alexander
2025-11-25
1. Upload the data set as a csv
all_countries <- read.csv("AllCountries.csv")
head(all_countries)## Country Code LandArea Population Density GDP Rural CO2 PumpPrice
## 1 Afghanistan AFG 652.86 37.172 56.9 521 74.5 0.29 0.70
## 2 Albania ALB 27.40 2.866 104.6 5254 39.7 1.98 1.36
## 3 Algeria DZA 2381.74 42.228 17.7 4279 27.4 3.74 0.28
## 4 American Samoa ASM 0.20 0.055 277.3 NA 12.8 NA NA
## 5 Andorra AND 0.47 0.077 163.8 42030 11.9 5.83 NA
## 6 Angola AGO 1246.70 30.810 24.7 3432 34.5 1.29 0.97
## Military Health ArmedForces Internet Cell HIV Hunger Diabetes BirthRate
## 1 3.72 2.01 323 11.4 67.4 NA 30.3 9.6 32.5
## 2 4.08 9.51 9 71.8 123.7 0.1 5.5 10.1 11.7
## 3 13.81 10.73 317 47.7 111.0 0.1 4.7 6.7 22.3
## 4 NA NA NA NA NA NA NA NA NA
## 5 NA 14.02 NA 98.9 104.4 NA NA 8.0 NA
## 6 9.40 5.43 117 14.3 44.7 1.9 23.9 3.9 41.3
## DeathRate ElderlyPop LifeExpectancy FemaleLabor Unemployment Energy
## 1 6.6 2.6 64.0 50.3 1.5 NA
## 2 7.5 13.6 78.5 55.9 13.9 808
## 3 4.8 6.4 76.3 16.4 12.1 1328
## 4 NA NA NA NA NA NA
## 5 NA NA NA NA NA NA
## 6 8.4 2.5 61.8 76.4 7.3 545
## Electricity Developed
## 1 NA NA
## 2 2309 1
## 3 1363 1
## 4 NA NA
## 5 NA NA
## 6 312 1dim(all_countries)## [1] 217 26colSums(is.na(all_countries))## Country Code LandArea Population Density
## 0 0 8 1 8
## GDP Rural CO2 PumpPrice Military
## 30 3 13 50 67
## Health ArmedForces Internet Cell HIV
## 29 49 13 15 81
## Hunger Diabetes BirthRate DeathRate ElderlyPop
## 52 10 15 15 24
## LifeExpectancy FemaleLabor Unemployment Energy Electricity
## 18 30 30 82 76
## Developed
## 752. Simple Linear Regression (Fitting and Interpretation): Using the AllCountries dataset, fit a simple linear regression model to predict LifeExpectancy (average life expectancy in years) based on GDP (gross domestic product per capita in $US). Report the intercept and slope coefficients and interpret their meaning in the context of the dataset. What does the R² value tell you about how well GDP explains variation in life expectancy across countries?
#summary(all_countries)
simple_model <- lm(LifeExpectancy ~ GDP,data = all_countries)
summary(simple_model)##
## Call:
## lm(formula = LifeExpectancy ~ GDP, data = all_countries)
##
## 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-16The Interpretation
- The intercept is about 6.842e+01 which represents the predicted life
expectancy when GPD is 0. This is not practically meaningful, but
mathematically its the y-intercept.
- The coefficient is about 2.476e-4 which means for every 1 point
increase in GDP, life expectancy increases by 2.476e-4 years.
- Both p-values (2e-16) are < 0.05 which indicates statistical
significance.
- The adjusted r-squared is around 0.4272 explains about 42.72% of the
variance in life expectancy from GDP alone. Decent but room for
improvement.
3. Multiple Linear Regression (Fitting and Interpretation) Fit a
multiple linear regression model to predict LifeExpectancy using GDP,
Health (percentage of government expenditures on healthcare), and
Internet (percentage of population with internet access) as predictors.
Interpret the coefficient for Health, explaining what it means in terms
of life expectancy while controlling for GDP and Internet. How does the
adjusted R² compare to the simple regression model from Question 1, and
what does this suggest about the additional predictors?
multiple_model <- lm(LifeExpectancy ~ GDP + Health + Internet, data = all_countries)
summary(multiple_model)##
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = all_countries)
##
## 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-16The Interpretations
- Holding GDP and Internet constant, the Health coefficient is about
2.479e-1 which means for every 1 point increase in health, health
expectancy increases by 2.479e-1 years while holding GDP and internet
constant.
- The adjusted r-squared is 0.7164 which compared to the linear
regressions resulted adjusted r-squared of 0.4272 which suggests a big
improvement to the adjusted r-squared with the additional
predictors
4. Checking Assumptions (Homoscedasticity and Normality) For the
simple linear regression model from Question 1 (LifeExpectancy ~ GDP),
describe how you would check the assumptions of homoscedasticity and
normality of residuals. For each assumption, explain what an ideal
outcome would look like and what a violation might indicate about the
model’s reliability for predicting life expectancy. Afterwords, code
your answer and reflect if it matched the ideal outcome
# Visual Linearity Check
plot(all_countries$LifeExpectancy, all_countries$GDP,
xlab="Life Expectancy", ylab="GDP", main="Life Expectancy vs GDP")
abline(simple_model, col=1, lwd=2)
# Core Diagnostics -
par(mfrow=c(2,2)); plot(simple_model); par(mfrow=c(1,1))
Answer:
- For the simple linear regression model from Question 1 (Life
Expectancy ~ GDP), to check assumptions of homescedasticity is to use
Residuals vs Fitted Plot and for normality is to use a Q-Q Plot.
- For the Residuals vs Fitted Plot, the ideal outcome are residuals
scattered randomly around 0 and spread of points are consistent across
the line. A violation would indicate
- For the Q-Q plot, the ideal outcome are residuals should follow the
straight diagonal line. A violation would indicate
- Result of the Residuals vs Fitted Plot is not a flat cloud, the
residuals are not scattered randomly around 0, and the spread of points
are not consistent across the line. Therefore it does not match the
ideal outcome.
- The result of the Q-Q plot is the tails are not quite on the line but
do not deviate too far from the line. So normality is checked. Therefore
it does match the ideal outcome.
5. Diagnosing Model Fit (RMSE and Residuals) For the multiple regression model from Question 2 (LifeExpectancy ~ GDP + Health + Internet), calculate the RMSE and explain what it represents in the context of predicting life expectancy. How would large residuals for certain countries (e.g., those with unusually high or low life expectancy) affect your confidence in the model’s predictions, and what might you investigate further?
# calculate residuals
residuals_multiple <- resid(multiple_model)
# calculate RMSE
rmse_multiple <- sqrt(mean(residuals_multiple^2))
rmse_multiple## [1] 4.056417
- The RMSE for the multiple regression model from Question 2
(LifeExpectancy ~ GDP + Health + Internet) is 3.411562 which means that
the prediction misses by ~3.41 years on average.
- Large residuals for certain countries affect the confidence in the
model’s prediction because the data is skewed because of them and also
because there is a possiblity and there is missing data present that is
not accounted for. This would suggest to look further into the data
set’s variables and data for each column.
6. Hypothetical Example (Multicollinearity in Multiple Regression) Suppose you are analyzing the AllCountries dataset and fit a multiple linear regression model to predict CO2 emissions (metric tons per capita) using Energy (kilotons of oil equivalent) and Electricity (kWh per capita) as predictors. You notice that Energy and Electricity are highly correlated. Explain how this multicollinearity might affect the interpretation of the regression coefficients and the reliability of the model.s
- This multicollinearity might affect the interpretation of the
regression coefficients and reliability of the model because it makes it
hard to tell which predictor really matters and the coefficients can
become unstable and p-values unreliable.