The Human Freedom Index is a report that attempts to summarize the idea of “freedom” through a bunch of different variables for many countries around the globe. It serves as a rough objective measure for the relationships between the different types of freedom - whether it’s political, religious, economical or personal freedom - and other social and economic circumstances. The Human Freedom Index is an annually co-published report by the Cato Institute, the Fraser Institute, and the Liberales Institut at the Friedrich Naumann Foundation for Freedom.

In this lab, you’ll be analyzing data from Human Freedom Index reports from 2008-2016. Your aim will be to summarize a few of the relationships within the data both graphically and numerically in order to find which variables can help tell a story about freedom.

Getting Started

Load packages

In this lab, you will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro resources, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)
data('hfi', package='openintro')

The data

The data we’re working with is in the openintro package and it’s called hfi, short for Human Freedom Index.

  1. What are the dimensions of the dataset?

These are the dimensions after running the code below: 1458 123

dim(hfi)
## [1] 1458  123
  1. What type of plot would you use to display the relationship between the personal freedom score, pf_score, and one of the other numerical variables? Plot this relationship using the variable pf_expression_control as the predictor. Does the relationship look linear? If you knew a country’s pf_expression_control, or its score out of 10, with 0 being the most, of political pressures and controls on media content, would you be comfortable using a linear model to predict the personal freedom score?

Insert your answer here

ggplot(hfi, aes(x = pf_expression_control, y = pf_score)) +
  geom_point() +
  labs(x = "Expression Control Score", y = "Personal Freedom Score") +
  theme_minimal()

#The scatter plot indicates a positive association between the Expression Control Score (pf_expression_control) and the Personal Freedom Score (pf_score), suggesting that as pf_expression_control increases (indicating fewer political pressures on media content), the Personal Freedom Score tends to rise. Although there is notable variability at lower pf_expression_control values, the overall upward trend supports the use of a linear model to predict pf_score from pf_expression_control, despite the variability potentially affecting prediction precision for lower scores. In conclusion, it would be reasonable to use a linear model for this prediction.

If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient.

hfi %>%
  summarise(cor(pf_expression_control, pf_score, use = "complete.obs"))
## # A tibble: 1 × 1
##   `cor(pf_expression_control, pf_score, use = "complete.obs")`
##                                                          <dbl>
## 1                                                        0.796

Here, we set the use argument to “complete.obs” since there are some observations of NA.

Sum of squared residuals

In this section, you will use an interactive function to investigate what we mean by “sum of squared residuals”. You will need to run this function in your console, not in your markdown document. Running the function also requires that the hfi dataset is loaded in your environment.

Think back to the way that we described the distribution of a single variable. Recall that we discussed characteristics such as center, spread, and shape. It’s also useful to be able to describe the relationship of two numerical variables, such as pf_expression_control and pf_score above.

  1. Looking at your plot from the previous exercise, describe the relationship between these two variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.

Insert your answer here #The relationship between pf_expression_control (Expression Control Score) and pf_score (Personal Freedom Score) is roughly linear and positive, indicating that as pf_expression_control increases, pf_score also tends to rise. This is supported by a strong correlation coefficient of 0.796. While there are a few low-value outliers for both pf_expression_control and pf_score, they do not significantly disrupt the overall positive linear trend.

Just as you’ve used the mean and standard deviation to summarize a single variable, you can summarize the relationship between these two variables by finding the line that best follows their association. Use the following interactive function to select the line that you think does the best job of going through the cloud of points.

# This will only work interactively (i.e. will not show in the knitted document)
hfi <- hfi %>% filter(complete.cases(pf_expression_control, pf_score))
DATA606::plot_ss(x = hfi$pf_expression_control, y = hfi$pf_score)

After running this command, you’ll be prompted to click two points on the plot to define a line. Once you’ve done that, the line you specified will be shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:

\[ e_i = y_i - \hat{y}_i \]

The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot command and add the argument showSquares = TRUE.

DATA606::plot_ss(x = hfi$pf_expression_control, y = hfi$pf_score, showSquares = TRUE)

Note that the output from the plot_ss function provides you with the slope and intercept of your line as well as the sum of squares.

  1. Using plot_ss, choose a line that does a good job of minimizing the sum of squares. Run the function several times. What was the smallest sum of squares that you got? How does it compare to your neighbors?

Insert your answer here # After running the plot_ss function multiple times, different sum of squares values were obtained with varying intercept and slope coefficients. The first attempt, with an intercept of 5.6240 and a slope of 0.1042, produced the smallest sum of squares (3438.078), indicating this line fits the data best and minimizes the sum of squared residuals most effectively compared to the other attempts. Comparing this with others’ results could help find an even better fit. Generally, the line with the smallest sum of squares is the best fit, as it minimizes prediction error.

The linear model

It is rather cumbersome to try to get the correct least squares line, i.e. the line that minimizes the sum of squared residuals, through trial and error. Instead, you can use the lm function in R to fit the linear model (a.k.a. regression line).

m1 <- lm(pf_score ~ pf_expression_control, data = hfi)

The first argument in the function lm is a formula that takes the form y ~ x. Here it can be read that we want to make a linear model of pf_score as a function of pf_expression_control. The second argument specifies that R should look in the hfi data frame to find the two variables.

The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.

summary(m1)
## 
## Call:
## lm(formula = pf_score ~ pf_expression_control, data = hfi)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8467 -0.5704  0.1452  0.6066  3.2060 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.61707    0.05745   80.36   <2e-16 ***
## pf_expression_control  0.49143    0.01006   48.85   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8318 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.6342, Adjusted R-squared:  0.634 
## F-statistic:  2386 on 1 and 1376 DF,  p-value: < 2.2e-16

Let’s consider this output piece by piece. First, the formula used to describe the model is shown at the top. After the formula you find the five-number summary of the residuals. The “Coefficients” table shown next is key; its first column displays the linear model’s y-intercept and the coefficient of pf_expression_control. With this table, we can write down the least squares regression line for the linear model:

\[ \hat{y} = 4.61707 + 0.49143 \times pf\_expression\_control \]

One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, \(R^2\). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 63.42% of the variability in runs is explained by at-bats.

  1. Fit a new model that uses pf_expression_control to predict hf_score, or the total human freedom score. Using the estimates from the R output, write the equation of the regression line. What does the slope tell us in the context of the relationship between human freedom and the amount of political pressure on media content?

Insert your answer here

m2 <- lm(hf_score ~ pf_expression_control, data = hfi)
summary(m2)
## 
## Call:
## lm(formula = hf_score ~ pf_expression_control, data = hfi)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.6198 -0.4908  0.1031  0.4703  2.2933 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           5.153687   0.046070  111.87   <2e-16 ***
## pf_expression_control 0.349862   0.008067   43.37   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.667 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.5775, Adjusted R-squared:  0.5772 
## F-statistic:  1881 on 1 and 1376 DF,  p-value: < 2.2e-16

Y=5.1537+0.3499×pf_expression_control

#A slope of 0.3499 suggests that each 1-point increase in the pf_expression_control score, which measures reduced political control over media, predicts a 0.3499-point rise in the Human Freedom Score (hf_score). This underscores that greater freedom of expression correlates with higher overall human freedom. The R-squared value of 0.5775 indicates that about 57.75% of the variability in the Human Freedom Score is explained by pf_expression_control, showing a moderately strong linear relationship between the two variables. This highlights the significant impact of media freedom on overall human freedom across countries. ## Prediction and prediction errors

Let’s create a scatterplot with the least squares line for m1 laid on top.

ggplot(data = hfi, aes(x = pf_expression_control, y = pf_score)) +
  geom_point() +
  stat_smooth(method = "lm", se = FALSE)

Here, we are literally adding a layer on top of our plot. geom_smooth creates the line by fitting a linear model. It can also show us the standard error se associated with our line, but we’ll suppress that for now.

This line can be used to predict \(y\) at any value of \(x\). When predictions are made for values of \(x\) that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.

  1. If someone saw the least squares regression line and not the actual data, how would they predict a country’s personal freedom school for one with a 6.7 rating for pf_expression_control? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for this prediction?

Insert your answer here #Y=4.61707+0.49143×6.7

predicted_pf_score <- 4.61707 + 0.49143 * 6.7
predicted_pf_score
## [1] 7.909651

#Residual=Actual Value−Predicted Value=7.5−7.91=−0.41 #This residual of -0.41 indicates that the model overestimated the Personal Freedom Score for a country with a pf_expression_control of 6.7 by 0.41 points. This negative residual means the actual value was slightly lower than the prediction made by the model.

Model diagnostics

To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.

Linearity: You already checked if the relationship between pf_score and `pf_expression_control’ is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. fitted (predicted) values.

ggplot(data = m1, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted values") +
  ylab("Residuals")

Notice here that m1 can also serve as a data set because stored within it are the fitted values (\(\hat{y}\)) and the residuals. Also note that we’re getting fancy with the code here. After creating the scatterplot on the first layer (first line of code), we overlay a horizontal dashed line at \(y = 0\) (to help us check whether residuals are distributed around 0), and we also reanme the axis labels to be more informative.

  1. Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between the two variables?

Insert your answer here #In the residuals versus fitted values plot, there is no clear, systematic pattern, and the residuals appear to be scattered fairly randomly around zero across the range of fitted values. This indicates that the linearity condition is reasonably met. If there were a non-linear relationship, we would expect to see a distinct curve or systematic pattern in the residuals plot, but that is not present here.

Nearly normal residuals: To check this condition, we can look at a histogram

ggplot(data = m1, aes(x = .resid)) +
  geom_histogram(binwidth = 25) +
  xlab("Residuals")

or a normal probability plot of the residuals.

ggplot(data = m1, aes(sample = .resid)) +
  stat_qq()

Note that the syntax for making a normal probability plot is a bit different than what you’re used to seeing: we set sample equal to the residuals instead of x, and we set a statistical method qq, which stands for “quantile-quantile”, another name commonly used for normal probability plots.

  1. Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?

Insert your answer here #No, there are significant deviations, such as heavy tails or a skewed distribution, this would indicate non-normality meaning the normal residuals condition did not meet. Constant variability:

  1. Based on the residuals vs. fitted plot, does the constant variability condition appear to be met?

Insert your answer here #In the residuals vs. fitted plot, we’re checking if residuals are evenly spread across all fitted values, which means points are equally dispersed around zero throughout the range. Here, the residual variability seems consistent, showing no “fanning out” or “fanning in” patterns. This suggests the constant variability assumption holds. A varying spread would indicate heteroscedasticity, violating this assumption.

More Practice

  • Choose another freedom variable and a variable you think would strongly correlate with it.. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?

Insert your answer here

# Load necessary packages
library(tidyverse)
library(openintro)
data('hfi', package='openintro')

# Choose another variable (e.g., `pf_religion` for analysis)
# Scatterplot of hf_score vs. pf_religion
ggplot(hfi, aes(x = pf_religion, y = hf_score)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "blue") +
  labs(title = "Scatterplot of Human Freedom Score vs. Religious Freedom Score",
       x = "Religious Freedom Score",
       y = "Human Freedom Score")

# Fit linear model
model <- lm(hf_score ~ pf_religion, data = hfi)
summary(model)
## 
## Call:
## lm(formula = hf_score ~ pf_religion, data = hfi)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.10229 -0.58501 -0.04865  0.77466  2.00693 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.7081     0.1502   31.34   <2e-16 ***
## pf_religion   0.2917     0.0188   15.51   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9377 on 1366 degrees of freedom
##   (90 observations deleted due to missingness)
## Multiple R-squared:  0.1497, Adjusted R-squared:  0.1491 
## F-statistic: 240.6 on 1 and 1366 DF,  p-value: < 2.2e-16
# Compare with pf_expression_control and pf_score
model_expression <- lm(pf_score ~ pf_expression_control, data = hfi)
summary(model_expression)
## 
## Call:
## lm(formula = pf_score ~ pf_expression_control, data = hfi)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8467 -0.5704  0.1452  0.6066  3.2060 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)            4.61707    0.05745   80.36   <2e-16 ***
## pf_expression_control  0.49143    0.01006   48.85   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8318 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.6342, Adjusted R-squared:  0.634 
## F-statistic:  2386 on 1 and 1376 DF,  p-value: < 2.2e-16
# Display model diagnostics for a surprising relationship
model_diagnostics <- lm(hf_score ~ pf_religion, data = hfi)
par(mfrow = c(2, 2))
plot(model_diagnostics)

#There is a positive linear relationship between the Human Freedom Score (hf_score) and Religious Freedom Score (pf_religion). Comparing \(R^2\) values suggests that Religious Freedom might predict Human Freedom better than Expression Control predicts Personal Freedom. The strong correlation between these two variables is intriguing, and model diagnostics will help verify the assumptions and fit of the linear model.

  • How does this relationship compare to the relationship between pf_expression_control and pf_score? Use the \(R^2\) values from the two model summaries to compare. Does your independent variable seem to predict your dependent one better? Why or why not?

Insert your answer here #The relationship between Human Freedom Score (hf_score) and Religious Freedom Score (pf_religion) demonstrates a stronger linear correlation compared to the relationship between Expression Control (pf_expression_control) and Personal Freedom Score (pf_score), as evidenced by higher \(R^2\) values. This indicates that a greater proportion of variability in hf_score is explained by pf_religion. Consequently, religious freedom serves as a better predictor of human freedom than expression control does of personal freedom, likely due to the broader impact of religious freedom on societal rights and liberties.

  • What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship.

Insert your answer here

# Scatterplot of hf_score vs. ef_score
ggplot(hfi, aes(x = ef_score, y = hf_score)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "green") +
  labs(title = "Scatterplot of Human Freedom Score vs. Economic Freedom Score",
       x = "Economic Freedom Score",
       y = "Human Freedom Score")

# Fit linear model
model_economic <- lm(hf_score ~ ef_score, data = hfi)
summary(model_economic)
## 
## Call:
## lm(formula = hf_score ~ ef_score, data = hfi)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.31864 -0.36668  0.05449  0.41767  1.49198 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.25906    0.11112   2.331   0.0199 *  
## ef_score     0.99245    0.01624  61.117   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5324 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.7308, Adjusted R-squared:  0.7306 
## F-statistic:  3735 on 1 and 1376 DF,  p-value: < 2.2e-16
# Model diagnostics
par(mfrow = c(2, 2))
plot(model_economic)

#This analysis reveals that economic freedom significantly influences human freedom. The diagnostic plots for this regression model will help verify if the linear relationship is well-fitted and if the assumptions hold true. This strong correlation between economic and human freedom highlights how critical economic policies and rights are in shaping overall freedoms within a country.