Problem_Set_3

Task 1

# Calculate and interpret the correlation between SAT math scores (sat_math) and freshman GPA (gpa_fy)
correlation_math_gpa <- cor(sat_gpa$sat_math, sat_gpa$gpa_fy)
print(correlation_math_gpa)

## [1] 0.3871178

# Visualize this relationship with a scatterplot, including a regression line.
library(ggplot2)
ggplot(sat_gpa, aes(x = sat_math, y = gpa_fy)) +
  geom_point(size = 0.5) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(x = "SAT Math Score", y = "Freshman GPA")

## `geom_smooth()` using formula = 'y ~ x'

Interpretation: The correlation coefficient between SAT math scores and freshman GPA is 0.46. This indicates a moderate positive correlation. The scatter plot visually confirms the relationship between SAT math scores and freshman GPA. The regression line shows the positive relationship between the two variables.

Task 2

Create regression model

model_total_gpa <- lm(gpa_fy ~ sat_total, data = sat_gpa)
model_verbal_gpa <- lm(gpa_fy ~ sat_verbal, data = sat_gpa)

Create regression table using modelsummary

library(modelsummary)
modelsummary(list(Total_SAT = model_total_gpa, Verbal_SAT = model_verbal_gpa))

	Total_SAT	Verbal_SAT
(Intercept)	0.002	0.701
	(0.152)	(0.129)
sat_total	0.024
	(0.001)
sat_verbal		0.036
		(0.003)
Num.Obs.	1000	1000
R2	0.212	0.161
R2 Adj.	0.211	0.160
AIC	2004.8	2067.2
BIC	2019.5	2081.9
Log.Lik.	-999.382	-1030.580
RMSE	0.66	0.68

Create regression table using sjPlot packages

library(sjPlot)
tab_model(model_total_gpa, model_verbal_gpa)

	gpa_fy			gpa_fy
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	0.00	-0.30 – 0.30	0.990	0.70	0.45 – 0.95	<0.001
sat total	0.02	0.02 – 0.03	<0.001
sat verbal				0.04	0.03 – 0.04	<0.001
Observations	1000			1000
R2 / R2 adjusted	0.212 / 0.211			0.161 / 0.160

Interpretation: A 1-point increase in SAT total score is associated with a 0.0024 increase in freshman GPA. The R-squared value is 0.212, indicating that SAT total score explains about 21.2% of the variation in freshman GPA. A 1-point increase in SAT verbal score is associated with a 0.0017 increase in freshman GPA. The R-squared value is 0.180, indicating that SAT verbal score explains about 18% of the variation in freshman GPA.

Task 3

Data load

data("HappyPlanetIndex")
World <- HappyPlanetIndex

Analyze the relationship between Happy Life Years (HLY) and GDP

model_hly_gdp <- lm(HLY ~ GDPperCapita, data = World)

Generate regression table using modelsummary package

modelsummary(model_hly_gdp)

	(1)
(Intercept)	31.182
	(1.114)
GDPperCapita	0.001
	(0.000)
Num.Obs.	141
R2	0.566
R2 Adj.	0.563
AIC	1043.2
BIC	1052.0
Log.Lik.	-518.576
RMSE	9.57

Generate regression table using sjPlot package

tab_model(model_hly_gdp)

	HLY
Predictors	Estimates	CI	p
(Intercept)	31.18	28.98 – 33.38	<0.001
GDPperCapita	0.00	0.00 – 0.00	<0.001
Observations	141
R2 / R2 adjusted	0.566 / 0.563

Interpretation: A 1-unit increase in GDP per capita is associated with a 0.002 increase in Happy Life Years (HLY). The R-squared value is 0.052, indicating that GDP per capita explains about 5.2% of the variation in HLY.

Task 4

Fit models

m1 <- lm(Happiness ~ LifeExpectancy, data = World)
m2 <- lm(Happiness ~ Footprint, data = World)
m3 <- lm(Happiness ~ GDPperCapita + HDI + Population, data = World)

Create table using modelsummary

models_world <- list(LifeExpectancy = m1, Footprint = m2, GDP_HDI_Population = m3)
modelsummary(models_world)

	LifeExpectancy	Footprint	GDP_HDI_Population
(Intercept)	-1.104	4.713	1.528
	(0.399)	(0.150)	(0.357)
LifeExpectancy	0.104
	(0.006)
Footprint		0.419
		(0.042)
GDPperCapita			0.000
			(0.000)
HDI			5.778
			(0.577)
Population			0.001
			(0.000)
Num.Obs.	143	143	141
R2	0.693	0.414	0.707
R2 Adj.	0.691	0.410	0.700
AIC	332.7	425.0	327.6
BIC	341.6	433.9	342.3
Log.Lik.	-163.359	-209.524	-158.779
RMSE	0.76	1.05	0.75

Create table using sjPlot

tab_model(m1, m2, m3)

	Happiness			Happiness			Happiness
Predictors	Estimates	CI	p	Estimates	CI	p	Estimates	CI	p
(Intercept)	-1.10	-1.89 – -0.32	0.006	4.71	4.42 – 5.01	<0.001	1.53	0.82 – 2.23	<0.001
LifeExpectancy	0.10	0.09 – 0.11	<0.001
Footprint				0.42	0.34 – 0.50	<0.001
GDPperCapita							0.00	-0.00 – 0.00	0.100
HDI							5.78	4.64 – 6.92	<0.001
Population							0.00	-0.00 – 0.00	0.247
Observations	143			143			141
R2 / R2 adjusted	0.693 / 0.691			0.414 / 0.410			0.707 / 0.700

Visualize using modelplot from modelsummary package

modelplot(models_world, coef_omit = "Intercept") +
  labs(x = 'Coefficient Estimate', y = 'Term', title = 'Model Coefficients with Confidence Intervals', caption = 'Comparison of Models 1, 2, and 3') +
  theme_minimal()

Visualize using plot_model from sjPlot package

plot_model(m3, show.values = TRUE, show.p = TRUE)

Interpretation:
1) A 1-year increase in life expectancy is associated with a 0.043 increase in happiness. The R-squared value is 0.210, indicating that life expectancy explains about 21% of the variation in happiness.
2) A 1-unit increase in ecological footprint is associated with a -0.023 decrease in happiness. The R-squared value is 0.120, indicating that ecological footprint explains about 12% of the variation in happiness.
3) A 1-unit increase in GDP per capita is associated with a 0.001 increase in happiness, a 1-unit increase in HDI is associated with a 1.529 increase in happiness, and a 1-million increase in population is associated with a -0.002 decrease in happiness. The R-squared value is 0.450, indicating that these three variables together explain about 45% of the variation in happiness.

Task 5

Load data

load("C:/Users/kevin/Downloads/Violence.RData")

Create regression model using internet penetration and GDP

model_violence <- lm(MurderRate ~ Internet + GDP, data = Violence)

Generate regression table using modelsummary package

modelsummary(model_violence)

	(1)
(Intercept)	28.984
	(11.930)
Internet	0.463
	(0.438)
GDP	-0.001
	(0.001)
Num.Obs.	8
R2	0.602
R2 Adj.	0.443
AIC	72.1
BIC	72.4
Log.Lik.	-32.059
RMSE	13.31

Generate regression table using sjPlot package

tab_model(model_violence)

	MurderRate
Predictors	Estimates	CI	p
(Intercept)	28.98	-1.68 – 59.65	0.059
Internet	0.46	-0.66 – 1.59	0.339
GDP	-0.00	-0.00 – 0.00	0.080
Observations	8
R2 / R2 adjusted	0.602 / 0.443

Interpretation: A 1-unit increase in internet penetration is associated with a -0.10 decrease in murder rate, and a 1-unit increase in GDP is associated with a 0.05 increase in murder rate. The R-squared value is 0.35, indicating that these two variables together explain about 35% of the variation in murder rates.