HW 9
Ayan Elmi
2025-12-01
Setting working directory
setwd("~/Desktop/Data 101")
countries <- read.csv("AllCountries.csv")Checking the structure of the dataset
str(countries)## 'data.frame': 217 obs. of 26 variables:
## $ Country : chr "Afghanistan" "Albania" "Algeria" "American Samoa" ...
## $ Code : chr "AFG" "ALB" "DZA" "ASM" ...
## $ LandArea : num 652.86 27.4 2381.74 0.2 0.47 ...
## $ Population : num 37.172 2.866 42.228 0.055 0.077 ...
## $ Density : num 56.9 104.6 17.7 277.3 163.8 ...
## $ GDP : int 521 5254 4279 NA 42030 3432 16864 11653 4212 NA ...
## $ Rural : num 74.5 39.7 27.4 12.8 11.9 34.5 75.4 8.1 36.9 56.6 ...
## $ CO2 : num 0.29 1.98 3.74 NA 5.83 1.29 5.74 4.78 1.9 8.41 ...
## $ PumpPrice : num 0.7 1.36 0.28 NA NA 0.97 NA 1.1 0.77 NA ...
## $ Military : num 3.72 4.08 13.81 NA NA ...
## $ Health : num 2.01 9.51 10.73 NA 14.02 ...
## $ ArmedForces : int 323 9 317 NA NA 117 0 105 49 NA ...
## $ Internet : num 11.4 71.8 47.7 NA 98.9 14.3 76 75.8 69.7 97.2 ...
## $ Cell : num 67.4 123.7 111 NA 104.4 ...
## $ HIV : num NA 0.1 0.1 NA NA 1.9 NA 0.4 0.2 NA ...
## $ Hunger : num 30.3 5.5 4.7 NA NA 23.9 NA 3.8 4.3 NA ...
## $ Diabetes : num 9.6 10.1 6.7 NA 8 3.9 13.2 5.5 7.1 11.6 ...
## $ BirthRate : num 32.5 11.7 22.3 NA NA 41.3 16.1 17 13.1 11 ...
## $ DeathRate : num 6.6 7.5 4.8 NA NA 8.4 5.8 7.6 9.7 8.9 ...
## $ ElderlyPop : num 2.6 13.6 6.4 NA NA 2.5 7.2 11.3 11.4 13.6 ...
## $ LifeExpectancy: num 64 78.5 76.3 NA NA 61.8 76.5 76.7 74.8 76 ...
## $ FemaleLabor : num 50.3 55.9 16.4 NA NA 76.4 NA 57.1 55.8 NA ...
## $ Unemployment : num 1.5 13.9 12.1 NA NA 7.3 NA 9.5 17.7 NA ...
## $ Energy : int NA 808 1328 NA NA 545 NA 2030 1016 NA ...
## $ Electricity : int NA 2309 1363 NA NA 312 NA 3075 1962 NA ...
## $ Developed : int NA 1 1 NA NA 1 NA 2 1 NA ...Summary statistics to explore the data
summary(countries)## Country Code LandArea Population
## Length:217 Length:217 Min. : 0.01 Min. : 0.0120
## Class :character Class :character 1st Qu.: 10.83 1st Qu.: 0.7728
## Mode :character Mode :character Median : 94.28 Median : 6.5725
## Mean : 608.38 Mean : 35.0335
## 3rd Qu.: 446.30 3rd Qu.: 25.0113
## Max. :16376.87 Max. :1392.7300
## NA's :8 NA's :1
## Density GDP Rural CO2
## Min. : 0.1 Min. : 275 Min. : 0.00 Min. : 0.0400
## 1st Qu.: 37.5 1st Qu.: 2032 1st Qu.:19.62 1st Qu.: 0.8575
## Median : 92.1 Median : 5950 Median :38.15 Median : 2.7550
## Mean : 361.4 Mean : 14733 Mean :39.10 Mean : 4.9780
## 3rd Qu.: 219.8 3rd Qu.: 17298 3rd Qu.:57.83 3rd Qu.: 6.2525
## Max. :20777.5 Max. :114340 Max. :87.00 Max. :43.8600
## NA's :8 NA's :30 NA's :3 NA's :13
## PumpPrice Military Health ArmedForces
## Min. :0.1100 Min. : 0.000 Min. : 0.000 Min. : 0.0
## 1st Qu.:0.7450 1st Qu.: 3.015 1st Qu.: 6.157 1st Qu.: 12.0
## Median :0.9800 Median : 4.650 Median : 9.605 Median : 31.5
## Mean :0.9851 Mean : 6.178 Mean :10.597 Mean : 162.1
## 3rd Qu.:1.1800 3rd Qu.: 8.445 3rd Qu.:13.713 3rd Qu.: 146.5
## Max. :2.0000 Max. :31.900 Max. :39.460 Max. :3031.0
## NA's :50 NA's :67 NA's :29 NA's :49
## Internet Cell HIV Hunger
## Min. : 1.30 Min. : 13.70 Min. : 0.100 Min. : 2.50
## 1st Qu.:29.18 1st Qu.: 83.83 1st Qu.: 0.175 1st Qu.: 2.50
## Median :58.35 Median :110.00 Median : 0.400 Median : 6.50
## Mean :54.47 Mean :107.05 Mean : 1.941 Mean :11.25
## 3rd Qu.:78.92 3rd Qu.:127.50 3rd Qu.: 1.400 3rd Qu.:14.80
## Max. :98.90 Max. :328.80 Max. :27.400 Max. :61.80
## NA's :13 NA's :15 NA's :81 NA's :52
## Diabetes BirthRate DeathRate ElderlyPop
## Min. : 1.000 Min. : 7.00 Min. : 1.600 Min. : 1.200
## 1st Qu.: 5.350 1st Qu.:11.40 1st Qu.: 5.800 1st Qu.: 3.600
## Median : 7.200 Median :17.85 Median : 7.250 Median : 6.600
## Mean : 8.542 Mean :20.11 Mean : 7.683 Mean : 8.953
## 3rd Qu.:10.750 3rd Qu.:27.65 3rd Qu.: 9.350 3rd Qu.:14.500
## Max. :30.500 Max. :47.80 Max. :15.500 Max. :27.500
## NA's :10 NA's :15 NA's :15 NA's :24
## LifeExpectancy FemaleLabor Unemployment Energy
## Min. :52.20 Min. : 6.20 Min. : 0.100 Min. : 66
## 1st Qu.:66.90 1st Qu.:50.15 1st Qu.: 3.400 1st Qu.: 738
## Median :74.30 Median :60.60 Median : 5.600 Median : 1574
## Mean :72.46 Mean :57.95 Mean : 7.255 Mean : 2664
## 3rd Qu.:77.70 3rd Qu.:69.25 3rd Qu.: 9.400 3rd Qu.: 3060
## Max. :84.70 Max. :85.80 Max. :30.200 Max. :17923
## NA's :18 NA's :30 NA's :30 NA's :82
## Electricity Developed
## Min. : 39 Min. :1.00
## 1st Qu.: 904 1st Qu.:1.00
## Median : 2620 Median :2.00
## Mean : 4270 Mean :1.81
## 3rd Qu.: 5600 3rd Qu.:3.00
## Max. :53832 Max. :3.00
## NA's :76 NA's :75Checking for NA’s
colSums(is.na(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
## 75Simple linear regression
# Fit simple linear regression: mpg ~ weight
simple_model <- lm(LifeExpectancy ~ GDP, data = countries)
# View the model summary
summary(simple_model)##
## Call:
## lm(formula = LifeExpectancy ~ GDP, data = 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-16Interpretation:
The intercept (around 68.42) represents the predicted life expectancy for a country with a GDP of $0. This is not not practically meaningful, but mathematically it’s the y-intercept.
The coefficient for weight (around 0.0002476) means for every $1 increase in GDP, life expectancy increases by about 0.0002476 years.
Look at the p-values: Both are < 0.05, indicating statistical significance.
R² (around 0.43) explains about 43% of the variance in life expectancy is from GDP which is low.
Multiple linear regression
multiple_model <- lm(LifeExpectancy ~ GDP+ Health + Internet, data = countries)
# View the model summary
summary(multiple_model)##
## Call:
## lm(formula = LifeExpectancy ~ GDP + Health + Internet, data = 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-16Interpretation:
Now life expectancy is explained by GDP, Health and Internet.
Coefficients (holding the other variables constant)
GDP: +0.0000237 years per $1 GDP (p = 0.302, not significant). After accounting for health spending and internet access, GDP doesn’t add clear independent information. Roughly +0.024 years per +$1000 GDP.
Health: +0.2479 years per 1% increase in health spending (p = 0.000247). Countries that devote a higher share of government spending to healthcare tend to live longer—about +2.48 years per +10 percentage points.
Internet: +0.1903 years per +1% internet access (p < 2e-16). Greater internet access strongly predicts longer life expectancy, even after controlling for GDP and health spending.
par(mfrow=c(2,2))
plot(simple_model)
par(mfrow=c(1,1))Checking for assumptions in Homoscedasticity
Homoscedasticity (Residuals vs Fitted Plot)
How to check: Look at a Residuals vs Fitted plot. Then compare all of the residuals to the Residual vs fitted plot. Thus, that way you can check for consistency between the residuals.
Ideal outcome: Residuals should be evenly scattered around 0 with no funnel shape or pattern and should be pretty equal.
Violation would look like: Patterns or widening/narrowing spread indicate non-constant variance, making predictions less reliable across GDP levels. If variance isn’t constant, the model may give incorrect p-values and confidence intervals, so your conclusions about significance might be wrong.
Normality (Q–Q Plot)
How to check for assumption for Normality in residuals: Examine a Q–Q plot of the residuals to see if they follow a normal distribution.
Ideal outcome: Not many outliers, no S curves, most points should be close to the line.Points fall roughly on the 45° line, showing residuals are approximately normal.
Violation meaning: Strong deviations or curve shapes indicate skewness or outliers, making p-values and confidence intervals less trustworthy.
Comparing to outcome Homoscedascity Analysis:
Homoscedasticity Analysis:
Overall, the assumption seems mostly reasonable. The spread of the residuals stays fairly steady, though a few outliers do pop up. They’re worth noticing, but they don’t seriously undermine the model. For the most part, the variance looks stable, so this assumption holds pretty well.
Normality Analysis:
Here the model shows a bit more trouble. The center of the Q–Q plot follows the expected pattern, but the ends drift away from the line, especially on the lower side. This suggests heavier tails, possibly coming from countries with extremely low life expectancies relative to their GDP. Because of this tail behavior, the normality assumption is weaker than the homoscedasticity one, and it raises more potential concerns.
Diagnose Model Fit with Metrics
Beyond summaries, compute RMSE for model.
# 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] 4.056417RMSE is how different the models predictions is from actual life expectancy. The value being roughly 4 means that it is off by 4 years. Residual is the difference between what the model predicted vs the real life life expectancy. If a country’s residual is really big, it means the model predicted way too high or way too low for that country. Large residuals makes the model less trustworthy. This could also be a call to include another factor that could be more important or missing. I would investigate further into outliers, countries that are extreme cases (very poor or very wealthy) might behave differently and cause big prediction errors.
Extra
cor (countries [, c("Energy", "Electricity")], use = "complete.obs")## Energy Electricity
## Energy 1.0000000 0.7970054
## Electricity 0.7970054 1.0000000Energy and Electricity have a correlation of 0.797, which is very high. Because they move together, the regression model can’t tell which one actually drives CO₂ emissions. This makes the coefficients unstable, hard to interpret, and sometimes misleading. The model can still predict CO₂ fairly well, but you can’t trust the individual effects of Energy vs. Electricity.