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?
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?

A scatterplot is commonly used to display the relationship between two numerical variables.

ggplot(hfi, aes(x = pf_expression_control, y = pf_score)) +
  geom_point()

Based on the scatterplot, it appears that there is a strong positive linear relationship between pf_expression_control and pf_score. Therefore, it might be reasonable to use a linear model to predict pf_score from pf_expression_control.However, it’s important to note that we should also check for other assumptions of linear regression, such as linearity, normality, and constant variance of errors. We can do this by examining diagnostic plots or conducting hypothesis tests. Therefore, I would need to further investigate and test the assumptions before being comfortable using a linear model to predict the personal freedom score.

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.

Based on the scatterplot of pf_expression_control and pf_score, there appears to be a strong positive linear relationship between the two variables. As the score of pf_expression_control increases, so does the score of pf_score. There are no unusual observations, and the relationship is quite strong, with a correlation coefficient of 0.796.

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?
set.seed(7)

hfi_1 <- select(hfi, pf_expression_control, pf_score)

# Check for missing values
sum(is.na(hfi_1))
## [1] 160
# Drop NAs
hfi_2 <- na.omit(hfi_1)

# Plot
DATA606::plot_ss(x = hfi_2$pf_expression_control, y = hfi_2$pf_score)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##      4.6171       0.4914  
## 
## Sum of Squares:  952.153
set.seed(17)

DATA606::plot_ss(x = hfi_2$pf_expression_control, y = hfi_2$pf_score, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##      4.6171       0.4914  
## 
## Sum of Squares:  952.153

The smallest sum of squares that I got was 952.153 Since we used set.seed() before each call to plot_ss(), we generated different sets of random points each time. Therefore, we cannot compare the sum of squares across runs.

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?
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

Based on the output, the regression model equation for predicting hf_score using pf_expression_control as the predictor variable is

hf_score = 5.1537 + (0.3499 * 6.7)
hf_score
## [1] 7.49803

The slope of this regression line is 0.3499, which tells us that for every one-unit increase in pf_expression_control, we expect to see an increase of 0.3499 in hf_score, holding all other variables constant.

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?

The equation for the least squares regression line is

pf_score = 6.7578 -(0.3822 * 6.7)
pf_score
## [1] 4.19706

So the predicted pf_score for a country with a pf_expression_control rating of 6.7 is 4.19706

To determine if this prediction is an overestimate or an underestimate, we need to subtract the predicted pf_score from the actual pf_score for a country with a pf_expression_control rating of 6.7

we know that the actual pf_score for a country with a pf_expression_control rating is 6.7

Therefore, the residual for this prediction can be calculated as:

residual = 6.7 - 4.19706
residual
## [1] 2.50294

So the residual for this prediction is 2.50294.

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?

There appears to be a slight funnel shape in the residuals plot, with the points becoming more spread out as the fitted values increase. This pattern indicates that the variability of the residuals is not constant across the range of values, violating the assumption of constant variability. This can lead to issues with the accuracy and reliability of the model’s predictions, particularly for extreme values of the predictor variable.


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")

First of all we have to fix the code so we can look at the histogram in proper way.

ggplot(data = m1, aes(x = .resid)) +
  geom_histogram(binwidth = 1) +
  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?

Based on the histogram and the normal probability plot, the nearly normal residuals condition appears to be approximately met. The histogram appears to be somewhat bell-shaped, although there is some slight skewness to the right. The normal probability plot appears to show approximately a straight line, indicating that the residuals are reasonably normally distributed. However, there are a few outliers in the tails of the distribution, which may indicate some non-normality in the residuals. Overall, the nearly normal residuals condition appears to be reasonably met


Constant variability:

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

The residuals vs. fitted plot shows a fan-shaped pattern, which indicates that the constant variability condition may not be met. This pattern suggests that the variance of the residuals may be increasing or decreasing as the predicted values increase. In this case, the residuals appear to have a larger spread at the higher end of the predicted values compared to the lower end. This violation of the constant variability condition is known as heteroscedasticity.

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?
# Using new variables pf_association and pf_rank

hfi %>% 
  ggplot(aes( x = pf_association, y = pf_rank)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  xlab("Pf Association") +
  ylab("Pf Rank")

it looks like there’s a linear relationtion with a downward negative slope, as pf_association increases the pf_rank decreases.

  • 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?
# Model 1
lm_11 <- lm(hf_score ~ pf_expression_control, data = hfi)
summary(lm_11)
## 
## 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
# Model 2
lm_12 <- lm(hfi$pf_rank ~ hfi$pf_ss)

summary(lm_12)
## 
## Call:
## lm(formula = hfi$pf_rank ~ hfi$pf_ss)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -90.288 -23.554  -5.786  22.067  77.687 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 269.9610     5.0094   53.89   <2e-16 ***
## hfi$pf_ss   -23.5592     0.6039  -39.01   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 30.67 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.5252, Adjusted R-squared:  0.5248 
## F-statistic:  1522 on 1 and 1376 DF,  p-value: < 2.2e-16

It appears that the independent variable is not a strong predictor of the dependent variable, as evidenced by the lower R-squared value of Model 1 (57.75%) compared to that of Model 2 (52.52%)

  • What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship.
model_2 <- lm(ef_legal_integrity ~ ef_money_growth, data = hfi)

summary(model_2)
## 
## Call:
## lm(formula = ef_legal_integrity ~ ef_money_growth, data = hfi)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.2785 -1.6567  0.0264  1.7994  5.6234 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      1.97216    0.48134   4.097 4.47e-05 ***
## ef_money_growth  0.48825    0.05545   8.804  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.092 on 1176 degrees of freedom
##   (280 observations deleted due to missingness)
## Multiple R-squared:  0.06184,    Adjusted R-squared:  0.06104 
## F-statistic: 77.52 on 1 and 1176 DF,  p-value: < 2.2e-16
ggplot(data = hfi, aes(x = ef_money_growth, y = ef_legal_integrity)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  xlab("Economic Freedom: Money Growth") +
  ylab("Economic Freedom: Legal Integrity")

There is a positive linear relationship between the two variables, with countries that have higher levels of economic freedom in terms of money growth also tending to have higher levels of economic freedom in terms of legal integrity. The regression line (in blue) shows the overall trend in the data, with the slope indicating the strength of the relationship between the variables