20251MBI013_Olga_Isengwe_Assignment
Olga Isengwe
2026-06-04
HW1: Find a data set of which you can fit multiple linear regression and interpret the results.
Definition
Multiple linear regression (MLR) is a method for estimating how several independent factors together influence a single outcome. It fits a straight-line equation to data points to reveal how each variable contributes when the others are held steady.
When you are conducting Multiple Linear Regression Model, this formula is used:
Y = β0 + β1X1 + β2X2 + … + βnXn + ε
# We are going to first import the data set to use
library()
world_population <- read.csv("world_population.csv", check.names = FALSE)
# Renaming the selected columns in the data set so that the regression formula is easier to read
names(world_population)[names(world_population) == "Country/Territory"] <- "Country"
names(world_population)[names(world_population) == "2022 Population"] <- "Pop2022"
names(world_population)[names(world_population) == "Area (km²)"] <- "Area_km2"
names(world_population)[names(world_population) == "Area (km²)"] <- "Area_km2"
names(world_population)[names(world_population) == "Density (per km²)"] <- "Density_per_km2"
names(world_population)[names(world_population) == "Density (per km²)"] <- "Density_per_km2"
names(world_population)[names(world_population) == "Growth Rate"] <- "Growth_Rate"
names(world_population)## [1] "Rank" "CCA3"
## [3] "Country" "Capital"
## [5] "Continent" "Pop2022"
## [7] "2020 Population" "2015 Population"
## [9] "2010 Population" "2000 Population"
## [11] "1990 Population" "1980 Population"
## [13] "1970 Population" "Area_km2"
## [15] "Density_per_km2" "Growth_Rate"
## [17] "World Population Percentage"# Let's now keep the variables which we will use in the model and remove missing values
head(world_population)## Rank CCA3 Country Capital Continent Pop2022 2020 Population
## 1 36 AFG Afghanistan Kabul Asia 41128771 38972230
## 2 138 ALB Albania Tirana Europe 2842321 2866849
## 3 34 DZA Algeria Algiers Africa 44903225 43451666
## 4 213 ASM American Samoa Pago Pago Oceania 44273 46189
## 5 203 AND Andorra Andorra la Vella Europe 79824 77700
## 6 42 AGO Angola Luanda Africa 35588987 33428485
## 2015 Population 2010 Population 2000 Population 1990 Population
## 1 33753499 28189672 19542982 10694796
## 2 2882481 2913399 3182021 3295066
## 3 39543154 35856344 30774621 25518074
## 4 51368 54849 58230 47818
## 5 71746 71519 66097 53569
## 6 28127721 23364185 16394062 11828638
## 1980 Population 1970 Population Area_km2 Density_per_km2 Growth_Rate
## 1 12486631 10752971 652230 63.0587 1.0257
## 2 2941651 2324731 28748 98.8702 0.9957
## 3 18739378 13795915 2381741 18.8531 1.0164
## 4 32886 27075 199 222.4774 0.9831
## 5 35611 19860 468 170.5641 1.0100
## 6 8330047 6029700 1246700 28.5466 1.0315
## World Population Percentage
## 1 0.52
## 2 0.04
## 3 0.56
## 4 0.00
## 5 0.00
## 6 0.45model_data <- na.omit(world_population[, c(
"Country",
"Continent",
"Pop2022",
"Area_km2",
"Density_per_km2",
"Growth_Rate"
)])
model_data$Continent <- factor(model_data$Continent)
model_data$Continent <- relevel(model_data$Continent, ref = "Africa")
# Log-transform population and area because both are highly skewed.
model_data$log_Pop2022 <- log(model_data$Pop2022)
model_data$log_Area_km2 <- log(model_data$Area_km2)
mlr_model <- lm(
log_Pop2022 ~ log_Area_km2 + Density_per_km2 + Growth_Rate + Continent,
data = model_data
)
summary(mlr_model)##
## Call:
## lm(formula = log_Pop2022 ~ log_Area_km2 + Density_per_km2 + Growth_Rate +
## Continent, data = model_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.6847 -0.5672 0.1473 0.7306 2.0665
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.742e+00 6.863e+00 0.982 0.327004
## log_Area_km2 7.206e-01 2.886e-02 24.973 < 2e-16 ***
## Density_per_km2 1.896e-04 4.088e-05 4.637 5.98e-06 ***
## Growth_Rate 6.110e-01 6.750e+00 0.091 0.927960
## ContinentAsia 7.533e-01 2.406e-01 3.130 0.001976 **
## ContinentEurope 9.719e-02 2.611e-01 0.372 0.710071
## ContinentNorth America -2.419e-01 2.807e-01 -0.862 0.389610
## ContinentOceania -1.074e+00 3.240e-01 -3.315 0.001067 **
## ContinentSouth America -1.224e+00 3.596e-01 -3.405 0.000784 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.16 on 225 degrees of freedom
## Multiple R-squared: 0.8321, Adjusted R-squared: 0.8261
## F-statistic: 139.4 on 8 and 225 DF, p-value: < 2.2e-16anova(mlr_model)## Analysis of Variance Table
##
## Response: log_Pop2022
## Df Sum Sq Mean Sq F value Pr(>F)
## log_Area_km2 1 1370.96 1370.96 1018.9886 < 2.2e-16 ***
## Density_per_km2 1 57.53 57.53 42.7610 4.103e-10 ***
## Growth_Rate 1 0.01 0.01 0.0058 0.9392
## Continent 5 71.59 14.32 10.6421 3.430e-09 ***
## Residuals 225 302.72 1.35
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Convert log-scale coefficients into approximate percentage effects.
coef_table <- summary(mlr_model)$coefficients
percent_effect <- (exp(coef(mlr_model)) - 1) * 100
interpretation_table <- data.frame(
Term = names(coef(mlr_model)),
Estimate = coef(mlr_model),
Percent_Effect = percent_effect,
P_Value = coef_table[, "Pr(>|t|)"],
row.names = NULL
)
print(interpretation_table)## Term Estimate Percent_Effect P_Value
## 1 (Intercept) 6.7418483514 8.461251e+04 3.270042e-01
## 2 log_Area_km2 0.7206219728 1.055711e+02 8.538853e-67
## 3 Density_per_km2 0.0001895945 1.896125e-02 5.982233e-06
## 4 Growth_Rate 0.6109670574 8.422121e+01 9.279601e-01
## 5 ContinentAsia 0.7533191370 1.124038e+02 1.976384e-03
## 6 ContinentEurope 0.0971935585 1.020737e+01 7.100714e-01
## 7 ContinentNorth America -0.2419244333 -2.148845e+01 3.896101e-01
## 8 ContinentOceania -1.0742006188 -6.584293e+01 1.066903e-03
## 9 ContinentSouth America -1.2243572316 -7.060534e+01 7.840724e-04# Basic diagnostic plots for checking linear regression assumptions.
par(mfrow = c(2, 2))
plot(mlr_model)
Interpretation
log_Area_km2 is strongly significant. A 1% increase in area is associated with about a 0.72% increase in 2022 population, holding other variables constant.
Density_per_km2 is also significant. An increase of 100 people per km² is associated with about a 1.9% higher population, holding other variables constant.
Growth_Rate is not statistically significant in this model, meaning it does not add much explanatory power after accounting for area, density, and continent. Compared with Africa:
Asia has significantly higher population levels, about 112% higher on average, holding other predictors constant.
Oceania and South America have significantly lower population levels.
Europe and North America are not significantly different from Africa in this model.
HW2: Read about variable selection method
Variable Selection
Variable selection is the process of identifying and choosing the most important independent variables (predictors) to include in a regression model. The goal is to retain variables that significantly contribute to explaining or predicting the dependent variable while excluding irrelevant or redundant variables.
Variable selection plays a crucial role in building effective statistical and machine learning models because it helps:
- Improve model performance by focusing on the variables that have the strongest relationship with the outcome.
- Reduce over-fitting, where a model performs well on training data but poorly on new data due to excessive complexity.
- Enhance model interpretability by making the model simpler and easier to understand.
- Reduce computational cost by decreasing the number of variables that need to be processed.
- Minimize multicollinearity by removing highly correlated predictors that can negatively affect coefficient estimates.
Common variable selection methods include:
- Forward Selection – Starts with no predictors and adds variables one at a time based on their contribution to the model.
- Backward Elimination – Starts with all candidate variables and removes the least significant variables step by step.
- Stepwise Selection – Combines forward selection and backward elimination by adding and removing variables iteratively.
- Regularization Methods – Techniques such as LASSO and Ridge Regression automatically shrink or eliminate less important variables.
By selecting only the most relevant variables, researchers and analysts can develop regression models that are more accurate, robust, and easier to interpret while avoiding the problems associated with overfitting.