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)
library(ggplot2)
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?

Answer: The dimensions of the data set: 1,458 rows by 123 columns

glimpse(hfi)
## Rows: 1,458
## Columns: 123
## $ year                               <dbl> 2016, 2016, 2016, 2016, 2016, 2016,…
## $ ISO_code                           <chr> "ALB", "DZA", "AGO", "ARG", "ARM", …
## $ countries                          <chr> "Albania", "Algeria", "Angola", "Ar…
## $ region                             <chr> "Eastern Europe", "Middle East & No…
## $ pf_rol_procedural                  <dbl> 6.661503, NA, NA, 7.098483, NA, 8.4…
## $ pf_rol_civil                       <dbl> 4.547244, NA, NA, 5.791960, NA, 7.5…
## $ pf_rol_criminal                    <dbl> 4.666508, NA, NA, 4.343930, NA, 7.3…
## $ pf_rol                             <dbl> 5.291752, 3.819566, 3.451814, 5.744…
## $ pf_ss_homicide                     <dbl> 8.920429, 9.456254, 8.060260, 7.622…
## $ pf_ss_disappearances_disap         <dbl> 10, 10, 5, 10, 10, 10, 10, 10, 10, …
## $ pf_ss_disappearances_violent       <dbl> 10.000000, 9.294030, 10.000000, 10.…
## $ pf_ss_disappearances_organized     <dbl> 10.0, 5.0, 7.5, 7.5, 7.5, 10.0, 10.…
## $ pf_ss_disappearances_fatalities    <dbl> 10.000000, 9.926119, 10.000000, 10.…
## $ pf_ss_disappearances_injuries      <dbl> 10.000000, 9.990149, 10.000000, 9.9…
## $ pf_ss_disappearances               <dbl> 10.000000, 8.842060, 8.500000, 9.49…
## $ pf_ss_women_fgm                    <dbl> 10.0, 10.0, 10.0, 10.0, 10.0, 10.0,…
## $ pf_ss_women_missing                <dbl> 7.5, 7.5, 10.0, 10.0, 5.0, 10.0, 10…
## $ pf_ss_women_inheritance_widows     <dbl> 5, 0, 5, 10, 10, 10, 10, 5, NA, 0, …
## $ pf_ss_women_inheritance_daughters  <dbl> 5, 0, 5, 10, 10, 10, 10, 10, NA, 0,…
## $ pf_ss_women_inheritance            <dbl> 5.0, 0.0, 5.0, 10.0, 10.0, 10.0, 10…
## $ pf_ss_women                        <dbl> 7.500000, 5.833333, 8.333333, 10.00…
## $ pf_ss                              <dbl> 8.806810, 8.043882, 8.297865, 9.040…
## $ pf_movement_domestic               <dbl> 5, 5, 0, 10, 5, 10, 10, 5, 10, 10, …
## $ pf_movement_foreign                <dbl> 10, 5, 5, 10, 5, 10, 10, 5, 10, 5, …
## $ pf_movement_women                  <dbl> 5, 5, 10, 10, 10, 10, 10, 5, NA, 5,…
## $ pf_movement                        <dbl> 6.666667, 5.000000, 5.000000, 10.00…
## $ pf_religion_estop_establish        <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_religion_estop_operate          <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_religion_estop                  <dbl> 10.0, 5.0, 10.0, 7.5, 5.0, 10.0, 10…
## $ pf_religion_harassment             <dbl> 9.566667, 6.873333, 8.904444, 9.037…
## $ pf_religion_restrictions           <dbl> 8.011111, 2.961111, 7.455556, 6.850…
## $ pf_religion                        <dbl> 9.192593, 4.944815, 8.786667, 7.795…
## $ pf_association_association         <dbl> 10.0, 5.0, 2.5, 7.5, 7.5, 10.0, 10.…
## $ pf_association_assembly            <dbl> 10.0, 5.0, 2.5, 10.0, 7.5, 10.0, 10…
## $ pf_association_political_establish <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_political_operate   <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_political           <dbl> 10.0, 5.0, 2.5, 5.0, 5.0, 10.0, 10.…
## $ pf_association_prof_establish      <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_prof_operate        <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_prof                <dbl> 10.0, 5.0, 5.0, 7.5, 5.0, 10.0, 10.…
## $ pf_association_sport_establish     <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_sport_operate       <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA,…
## $ pf_association_sport               <dbl> 10.0, 5.0, 7.5, 7.5, 7.5, 10.0, 10.…
## $ pf_association                     <dbl> 10.0, 5.0, 4.0, 7.5, 6.5, 10.0, 10.…
## $ pf_expression_killed               <dbl> 10.000000, 10.000000, 10.000000, 10…
## $ pf_expression_jailed               <dbl> 10.000000, 10.000000, 10.000000, 10…
## $ pf_expression_influence            <dbl> 5.0000000, 2.6666667, 2.6666667, 5.…
## $ pf_expression_control              <dbl> 5.25, 4.00, 2.50, 5.50, 4.25, 7.75,…
## $ pf_expression_cable                <dbl> 10.0, 10.0, 7.5, 10.0, 7.5, 10.0, 1…
## $ pf_expression_newspapers           <dbl> 10.0, 7.5, 5.0, 10.0, 7.5, 10.0, 10…
## $ pf_expression_internet             <dbl> 10.0, 7.5, 7.5, 10.0, 7.5, 10.0, 10…
## $ pf_expression                      <dbl> 8.607143, 7.380952, 6.452381, 8.738…
## $ pf_identity_legal                  <dbl> 0, NA, 10, 10, 7, 7, 10, 0, NA, NA,…
## $ pf_identity_parental_marriage      <dbl> 10, 0, 10, 10, 10, 10, 10, 10, 10, …
## $ pf_identity_parental_divorce       <dbl> 10, 5, 10, 10, 10, 10, 10, 10, 10, …
## $ pf_identity_parental               <dbl> 10.0, 2.5, 10.0, 10.0, 10.0, 10.0, …
## $ pf_identity_sex_male               <dbl> 10, 0, 0, 10, 10, 10, 10, 10, 10, 1…
## $ pf_identity_sex_female             <dbl> 10, 0, 0, 10, 10, 10, 10, 10, 10, 1…
## $ pf_identity_sex                    <dbl> 10, 0, 0, 10, 10, 10, 10, 10, 10, 1…
## $ pf_identity_divorce                <dbl> 5, 0, 10, 10, 5, 10, 10, 5, NA, 0, …
## $ pf_identity                        <dbl> 6.2500000, 0.8333333, 7.5000000, 10…
## $ pf_score                           <dbl> 7.596281, 5.281772, 6.111324, 8.099…
## $ pf_rank                            <dbl> 57, 147, 117, 42, 84, 11, 8, 131, 6…
## $ ef_government_consumption          <dbl> 8.232353, 2.150000, 7.600000, 5.335…
## $ ef_government_transfers            <dbl> 7.509902, 7.817129, 8.886739, 6.048…
## $ ef_government_enterprises          <dbl> 8, 0, 0, 6, 8, 10, 10, 0, 7, 10, 7,…
## $ ef_government_tax_income           <dbl> 9, 7, 10, 7, 5, 5, 4, 9, 10, 10, 8,…
## $ ef_government_tax_payroll          <dbl> 7, 2, 9, 1, 5, 5, 3, 4, 10, 10, 8, …
## $ ef_government_tax                  <dbl> 8.0, 4.5, 9.5, 4.0, 5.0, 5.0, 3.5, …
## $ ef_government                      <dbl> 7.935564, 3.616782, 6.496685, 5.346…
## $ ef_legal_judicial                  <dbl> 2.6682218, 4.1867042, 1.8431292, 3.…
## $ ef_legal_courts                    <dbl> 3.145462, 4.327113, 1.974566, 2.930…
## $ ef_legal_protection                <dbl> 4.512228, 4.689952, 2.512364, 4.255…
## $ ef_legal_military                  <dbl> 8.333333, 4.166667, 3.333333, 7.500…
## $ ef_legal_integrity                 <dbl> 4.166667, 5.000000, 4.166667, 3.333…
## $ ef_legal_enforcement               <dbl> 4.3874441, 4.5075380, 2.3022004, 3.…
## $ ef_legal_restrictions              <dbl> 6.485287, 6.626692, 5.455882, 6.857…
## $ ef_legal_police                    <dbl> 6.933500, 6.136845, 3.016104, 3.385…
## $ ef_legal_crime                     <dbl> 6.215401, 6.737383, 4.291197, 4.133…
## $ ef_legal_gender                    <dbl> 0.9487179, 0.8205128, 0.8461538, 0.…
## $ ef_legal                           <dbl> 5.071814, 4.690743, 2.963635, 3.904…
## $ ef_money_growth                    <dbl> 8.986454, 6.955962, 9.385679, 5.233…
## $ ef_money_sd                        <dbl> 9.484575, 8.339152, 4.986742, 5.224…
## $ ef_money_inflation                 <dbl> 9.743600, 8.720460, 3.054000, 2.000…
## $ ef_money_currency                  <dbl> 10, 5, 5, 10, 10, 10, 10, 5, 0, 10,…
## $ ef_money                           <dbl> 9.553657, 7.253894, 5.606605, 5.614…
## $ ef_trade_tariffs_revenue           <dbl> 9.626667, 8.480000, 8.993333, 6.060…
## $ ef_trade_tariffs_mean              <dbl> 9.24, 6.22, 7.72, 7.26, 8.76, 9.50,…
## $ ef_trade_tariffs_sd                <dbl> 8.0240, 5.9176, 4.2544, 5.9448, 8.0…
## $ ef_trade_tariffs                   <dbl> 8.963556, 6.872533, 6.989244, 6.421…
## $ ef_trade_regulatory_nontariff      <dbl> 5.574481, 4.962589, 3.132738, 4.466…
## $ ef_trade_regulatory_compliance     <dbl> 9.4053278, 0.0000000, 0.9171598, 5.…
## $ ef_trade_regulatory                <dbl> 7.489905, 2.481294, 2.024949, 4.811…
## $ ef_trade_black                     <dbl> 10.00000, 5.56391, 10.00000, 0.0000…
## $ ef_trade_movement_foreign          <dbl> 6.306106, 3.664829, 2.946919, 5.358…
## $ ef_trade_movement_capital          <dbl> 4.6153846, 0.0000000, 3.0769231, 0.…
## $ ef_trade_movement_visit            <dbl> 8.2969231, 1.1062564, 0.1106256, 7.…
## $ ef_trade_movement                  <dbl> 6.406138, 1.590362, 2.044823, 4.697…
## $ ef_trade                           <dbl> 8.214900, 4.127025, 5.264754, 3.982…
## $ ef_regulation_credit_ownership     <dbl> 5, 0, 8, 5, 10, 10, 8, 5, 10, 10, 5…
## $ ef_regulation_credit_private       <dbl> 7.295687, 5.301526, 9.194715, 4.259…
## $ ef_regulation_credit_interest      <dbl> 9, 10, 4, 7, 10, 10, 10, 9, 10, 10,…
## $ ef_regulation_credit               <dbl> 7.098562, 5.100509, 7.064905, 5.419…
## $ ef_regulation_labor_minwage        <dbl> 5.566667, 5.566667, 8.900000, 2.766…
## $ ef_regulation_labor_firing         <dbl> 5.396399, 3.896912, 2.656198, 2.191…
## $ ef_regulation_labor_bargain        <dbl> 6.234861, 5.958321, 5.172987, 3.432…
## $ ef_regulation_labor_hours          <dbl> 8, 6, 4, 10, 10, 10, 6, 6, 8, 8, 10…
## $ ef_regulation_labor_dismissal      <dbl> 6.299741, 7.755176, 6.632764, 2.517…
## $ ef_regulation_labor_conscription   <dbl> 10, 1, 0, 10, 0, 10, 3, 1, 10, 10, …
## $ ef_regulation_labor                <dbl> 6.916278, 5.029513, 4.560325, 5.151…
## $ ef_regulation_business_adm         <dbl> 6.072172, 3.722341, 2.758428, 2.404…
## $ ef_regulation_business_bureaucracy <dbl> 6.000000, 1.777778, 1.333333, 6.666…
## $ ef_regulation_business_start       <dbl> 9.713864, 9.243070, 8.664627, 9.122…
## $ ef_regulation_business_bribes      <dbl> 4.050196, 3.765515, 1.945540, 3.260…
## $ ef_regulation_business_licensing   <dbl> 7.324582, 8.523503, 8.096776, 5.253…
## $ ef_regulation_business_compliance  <dbl> 7.074366, 7.029528, 6.782923, 6.508…
## $ ef_regulation_business             <dbl> 6.705863, 5.676956, 4.930271, 5.535…
## $ ef_regulation                      <dbl> 6.906901, 5.268992, 5.518500, 5.369…
## $ ef_score                           <dbl> 7.54, 4.99, 5.17, 4.84, 7.57, 7.98,…
## $ ef_rank                            <dbl> 34, 159, 155, 160, 29, 10, 27, 106,…
## $ hf_score                           <dbl> 7.568140, 5.135886, 5.640662, 6.469…
## $ hf_rank                            <dbl> 48, 155, 142, 107, 57, 4, 16, 130, …
## $ hf_quartile                        <dbl> 2, 4, 4, 3, 2, 1, 1, 4, 2, 2, 4, 2,…
  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?

Answer:

I would use a scatter polot to map the relationship.

ggplot(hfi, aes(x = pf_expression_control, y = pf_score, color = pf_ss_women_missing)) +
  geom_point() +
  scale_color_gradient(low = "blue", high = "red") +  # Adjust colors as needed
  labs(
    title = "Plotting Personal Freedom Score, Expression Control, and Women's Score",
    x = "Expression Control (pf_expression_control)",
    y = "Personal Freedom Score (pf_score)",
    color = "Women's Score (pf_ss_women_missing)"
  ) +
  theme_minimal()

To assess comfort with using a linear model to predict personal freedom scores, I ran a residual analysis:

hfi_res <- hfi %>%
  drop_na(pf_score, pf_expression_control)
model <- lm(pf_score ~ pf_expression_control, data = hfi_res)

hfi_res$residuals <- residuals(model)
hfi_res$fitted_values <- fitted(model)

ggplot(hfi_res, aes(x = fitted_values, y = residuals)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(
    title = "Residuals vs. Fitted Values",
    x = "Fitted Values",
    y = "Residuals"
  ) +
  theme_minimal()

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

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?

Answer: The lowest sum of residuals I got was 1156.485 for: Coefficients: (Intercept) x
3.635 0.643

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?

Answer:

Regression equation: \[ \hat{hf\_score} = 5.1537 + 0.3499 \times pf\_expression\_control \]

The slope of 0.3499 indicates that for every one-unit increase in the expression control score, the predicted total human freedom score increases by about 0.35 points. This means that as political pressure on media content decreases, overall human freedom tends to increase. The positive slope highlights that greater freedom of expression is associated with higher levels of total human freedom.

model_hf <- lm(hf_score ~ pf_expression_control, data = hfi_res)
summary(model_hf)
## 
## Call:
## lm(formula = hf_score ~ pf_expression_control, data = hfi_res)
## 
## 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
## Multiple R-squared:  0.5775, Adjusted R-squared:  0.5772 
## F-statistic:  1881 on 1 and 1376 DF,  p-value: < 2.2e-16

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?

To predict a country’s personal freedom score with a pf_expression_control of 6.7, we use the regression equation:

\[ \hat{pf\_score} = 4.975 + 0.452 \times 6.7 \approx 8.003 \]

If the actual pf_score is, for instance, 8.5, the residual would be:

\[ \text{Residual} = 8.5 - 8.003 \approx 0.497 \]

This indicates the model’s prediction is an by about points.

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?

Answer: There is no strong apparent pattern in the residuals plot, as the points appear to be randomly scattered around the horizontal line at \(y = 0\). This indicates that the linearity assumption of the model is reasonably met, suggesting that the linear model is an appropriate fit for the relationship between pf_expression_control and pf_score. However, there may be slight clustering at certain fitted values, which could hint at potential non-linearity or heteroscedasticity in specific areas, but overall, the plot supports the linearity of the relationship.

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?

Answer: The normal probability plot shows that most of the residuals follow a straight line, which suggests that the residuals are approximately normally distributed. Overall, the nearly normal residuals condition appears to be reasonably met for the bulk of the data, even though there are minor deviations at the extremes. This suggests that while the model’s assumptions are mostly satisfied, there may be some deviations in the tails that could warrant further investigation.

Constant variability:

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

Answer: Yes, the constant variability condition appears to be met. The residuals have a fairly consistent spread across the fitted values, without any clear funneling or expanding patterns.

There is no strong indication of increasing or decreasing spread, which suggests that the assumption of constant variance is generally upheld. While there may be slight deviations in specific regions, overall, the residuals exhibit a relatively uniform variance. This supports the use of a linear model.

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?

Answer: I chose total freedom score and freedom of movement.

On spec, there appears to be a positive linear relationship between pf_movement and hf_score. The points generally align with the regression line, indicating that as freedom of movement increases, the total human freedom score tends to increase as well.Overall, the plot suggests that a linear model is a reasonable fit, but there might be potential for further investigation into the variability at specific points.

model_movement <- lm(hf_score ~ pf_movement, data = hfi_res)
ggplot(hfi_res, aes(x = pf_movement, y = hf_score)) +
  geom_point(color = "black") +
  geom_smooth(method = "lm", color = "blue") +
  labs(
    title = "Scatterplot of Total Human Freedom Score vs. Freedom of Movement",
    x = "Freedom of Movement (pf_movement)",
    y = "Total Human Freedom Score (hf_score)"
  ) +
  theme_minimal()

  • 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?

Answer:

The \(R^2\) value for the model using pf_expression_control to predict pf_score is 0.63, which indicates it explains 63% of the variability in personal freedom scores. In contrast, the model using pf_movement to predict hf_score has a lower \(R^2\) of 0.52, showing it accounts for 51.58% of the variability in total human freedom scores. This comparison suggests that pf_expression_control is a stronger predictor of its corresponding outcome, likely because it captures aspects more directly linked to personal freedom than pf_movement does to overall human freedom.

summary(model_movement)
## 
## Call:
## lm(formula = hf_score ~ pf_movement, data = hfi_res)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.93512 -0.48806  0.01532  0.55176  2.37263 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 4.808176   0.060229   79.83   <2e-16 ***
## pf_movement 0.279320   0.007295   38.29   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.714 on 1376 degrees of freedom
## Multiple R-squared:  0.5158, Adjusted R-squared:  0.5155 
## F-statistic:  1466 on 1 and 1376 DF,  p-value: < 2.2e-16
  • What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship.

Answer: One surprising relationship was between pf_ss_women_missing (women’s rights score) and hf_score (total human freedom score). It was unexpected that the correlation between women’s rights and total human freedom was not as strong as anticipated, given that women’s freedom is a crucial part of societal liberty.

model_women <- lm(hf_score ~ pf_ss_women_missing, data = hfi_res)

summary(model_women)
## 
## Call:
## lm(formula = hf_score ~ pf_ss_women_missing, data = hfi_res)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.96195 -0.62771 -0.03183  0.84821  2.46280 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          5.16605    0.14716   35.10   <2e-16 ***
## pf_ss_women_missing  0.19966    0.01586   12.59   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9793 on 1336 degrees of freedom
##   (40 observations deleted due to missingness)
## Multiple R-squared:  0.1061, Adjusted R-squared:  0.1054 
## F-statistic: 158.5 on 1 and 1336 DF,  p-value: < 2.2e-16
par(mfrow = c(2, 2))  # Set up a 2x2 plotting area
plot(model_women)