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.

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
names(hfi)
##   [1] "year"                               "ISO_code"                          
##   [3] "countries"                          "region"                            
##   [5] "pf_rol_procedural"                  "pf_rol_civil"                      
##   [7] "pf_rol_criminal"                    "pf_rol"                            
##   [9] "pf_ss_homicide"                     "pf_ss_disappearances_disap"        
##  [11] "pf_ss_disappearances_violent"       "pf_ss_disappearances_organized"    
##  [13] "pf_ss_disappearances_fatalities"    "pf_ss_disappearances_injuries"     
##  [15] "pf_ss_disappearances"               "pf_ss_women_fgm"                   
##  [17] "pf_ss_women_missing"                "pf_ss_women_inheritance_widows"    
##  [19] "pf_ss_women_inheritance_daughters"  "pf_ss_women_inheritance"           
##  [21] "pf_ss_women"                        "pf_ss"                             
##  [23] "pf_movement_domestic"               "pf_movement_foreign"               
##  [25] "pf_movement_women"                  "pf_movement"                       
##  [27] "pf_religion_estop_establish"        "pf_religion_estop_operate"         
##  [29] "pf_religion_estop"                  "pf_religion_harassment"            
##  [31] "pf_religion_restrictions"           "pf_religion"                       
##  [33] "pf_association_association"         "pf_association_assembly"           
##  [35] "pf_association_political_establish" "pf_association_political_operate"  
##  [37] "pf_association_political"           "pf_association_prof_establish"     
##  [39] "pf_association_prof_operate"        "pf_association_prof"               
##  [41] "pf_association_sport_establish"     "pf_association_sport_operate"      
##  [43] "pf_association_sport"               "pf_association"                    
##  [45] "pf_expression_killed"               "pf_expression_jailed"              
##  [47] "pf_expression_influence"            "pf_expression_control"             
##  [49] "pf_expression_cable"                "pf_expression_newspapers"          
##  [51] "pf_expression_internet"             "pf_expression"                     
##  [53] "pf_identity_legal"                  "pf_identity_parental_marriage"     
##  [55] "pf_identity_parental_divorce"       "pf_identity_parental"              
##  [57] "pf_identity_sex_male"               "pf_identity_sex_female"            
##  [59] "pf_identity_sex"                    "pf_identity_divorce"               
##  [61] "pf_identity"                        "pf_score"                          
##  [63] "pf_rank"                            "ef_government_consumption"         
##  [65] "ef_government_transfers"            "ef_government_enterprises"         
##  [67] "ef_government_tax_income"           "ef_government_tax_payroll"         
##  [69] "ef_government_tax"                  "ef_government"                     
##  [71] "ef_legal_judicial"                  "ef_legal_courts"                   
##  [73] "ef_legal_protection"                "ef_legal_military"                 
##  [75] "ef_legal_integrity"                 "ef_legal_enforcement"              
##  [77] "ef_legal_restrictions"              "ef_legal_police"                   
##  [79] "ef_legal_crime"                     "ef_legal_gender"                   
##  [81] "ef_legal"                           "ef_money_growth"                   
##  [83] "ef_money_sd"                        "ef_money_inflation"                
##  [85] "ef_money_currency"                  "ef_money"                          
##  [87] "ef_trade_tariffs_revenue"           "ef_trade_tariffs_mean"             
##  [89] "ef_trade_tariffs_sd"                "ef_trade_tariffs"                  
##  [91] "ef_trade_regulatory_nontariff"      "ef_trade_regulatory_compliance"    
##  [93] "ef_trade_regulatory"                "ef_trade_black"                    
##  [95] "ef_trade_movement_foreign"          "ef_trade_movement_capital"         
##  [97] "ef_trade_movement_visit"            "ef_trade_movement"                 
##  [99] "ef_trade"                           "ef_regulation_credit_ownership"    
## [101] "ef_regulation_credit_private"       "ef_regulation_credit_interest"     
## [103] "ef_regulation_credit"               "ef_regulation_labor_minwage"       
## [105] "ef_regulation_labor_firing"         "ef_regulation_labor_bargain"       
## [107] "ef_regulation_labor_hours"          "ef_regulation_labor_dismissal"     
## [109] "ef_regulation_labor_conscription"   "ef_regulation_labor"               
## [111] "ef_regulation_business_adm"         "ef_regulation_business_bureaucracy"
## [113] "ef_regulation_business_start"       "ef_regulation_business_bribes"     
## [115] "ef_regulation_business_licensing"   "ef_regulation_business_compliance" 
## [117] "ef_regulation_business"             "ef_regulation"                     
## [119] "ef_score"                           "ef_rank"                           
## [121] "hf_score"                           "hf_rank"                           
## [123] "hf_quartile"
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,…
summary(hfi$pf_score)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   2.167   6.197   7.189   7.201   8.449   9.568      80
summary(hfi$pf_expression_control)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   0.000   3.750   5.250   5.259   7.250   9.250      80

The dataset contains 1,458 rows and 123 columns

  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?
hfi %>%
  ggplot(aes(x = pf_expression_control, y = pf_score)) +
  geom_point(alpha = 0.5) +
  labs(
    x = "Political pressures / media expression control",
    y = "Personal freedom score",
    title = "pf_score vs pf_expression_control"
  )

A scatterplot is the right type of plot for showing the relationship between two numerical variables. When I plotted pf_score against pf_expression_control, most of the dots formed a cloud that moves upward and to the right. The lower values are in the bottom-left, and the higher values show up in the top-right. This tells me there is a positive relationship between the two variables. The trend looks fairly straight overall, so I would feel comfortable using a linear model to predict personal freedom based on expression control.

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.

The relationship between pf_expression_control (x) and pf_score (y) is roughly linear, positive (as media control scores go up, personal freedom tends to go up), and fairly strong (sample correlation ≈ 0.796). The scatter looks like a tilted “cloud” rising from left-low to right-high, with most points concentrated around x ≈ 2.5–8.25 and y ≈ 5–8. I don’t see dramatic outliers, though there are a few countries on the edges. Overall, a straight line seems like a reasonable first model.

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?

I tried several lines. My smallest sum of squared residuals was 972.73 (slope ≈ 0.4935, intercept ≈ 4.4840). My earlier attempts ranged from about 1,859 to 3,520, so 972.73 was much better. (I can’t compare to classmates here, but mine clearly improved as I aligned the line with the center of the cloud.)

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 pf_free is explained by pf_expression_control.

  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

Fitted model: (hf_score) ̂=5.1537+0.3499×pf_expression_control Slope meaning: For a 1-point increase in the media-control index, the expected total human freedom score increases by about 0.35 points, on average.

Interpretation: The slope tells us how much the total human freedom score is expected to change when the media expression control score increases by one point. In this model, the slope is about 0.35, so a 1-unit increase in pf_expression_control is associated with an average increase of about 0.35 points in the human freedom score.

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?
predict(m1, newdata = data.frame(pf_expression_control = 6.7))
##        1 
## 7.909663

observed country. From your code: y ̂at x=6.7: 7.9097 Nearest observed row (Belize in 2016): Predicted y ̂≈ 7.9342 Actual y= 7.4309 Residual e=y-y ̂=-0.5034

Using the least-squares line, the predicted personal freedom score at pf_expression_control=6.7is 7.9097. For the nearest observed country (Belize, 2016), y ̂≈ 7.9342, actual y= 7.4309, so the residual is −0.5034. Because the residual is negative, the line overestimates the actual score by about 0.50 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?

The residuals vs. fitted plot does not show a strong pattern. The points are mostly scattered around the horizontal line at 0 with no obvious curve or bend. That suggests the relationship between pf_expression_control and pf_score is reasonably linear, and a straight-line model is appropriate.

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

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

The residuals appear roughly bell-shaped in the histogram, and the QQ plot shows most points falling close to the straight line. There are small deviations in the tails, but nothing extreme. Overall, the nearly normal residuals condition seems to be met.

Constant variability:

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

The spread of residuals looks fairly consistent across the fitted values. I don’t see a funnel shape or any major change in variability from left to right. This suggests that the constant variability condition is reasonably satisfied.

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

I chose (pf_religion) and (pf_score) because religious freedom is an important part of overall personal freedom. The scatterplot shows a clear upward trend, and the points form a pretty tight, rising cloud. At a glance, this looks like a strong and fairly linear relationship.

  • 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?

The earlier model using pf_expression_control explained about 63.42% of the variation in personal freedom (R² = 0.6342).

My new model using pf_religion explained about 59.23% of the variation (R² = 0.5923).

This means pf_religion still has a strong positive relationship with personal freedom, but it does not predict pf_score as well as pf_expression_control does. The slightly lower R² tells me that religious freedom explains a little less of the variation in personal freedom compared to media-expression control.

In short, both variables are helpful predictors, but pf_expression_control does a better job based on the R² values.

  • What’s one freedom relationship you were most surprised about and why? Display the model diagnostics for the regression model analyzing this relationship.
ggplot(hfi, aes(x = pf_identity, y = pf_score)) +
  geom_point() +
  stat_smooth(method = "lm", se = FALSE)

m3 <- lm(pf_score ~ pf_identity, data = hfi)
summary(m3)
## 
## Call:
## lm(formula = pf_score ~ pf_identity, data = hfi)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.64199 -0.59317  0.04541  0.71058  1.84541 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 4.771116   0.060732   78.56   <2e-16 ***
## pf_identity 0.331348   0.007605   43.57   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8917 on 1376 degrees of freedom
##   (80 observations deleted due to missingness)
## Multiple R-squared:  0.5797, Adjusted R-squared:  0.5794 
## F-statistic:  1898 on 1 and 1376 DF,  p-value: < 2.2e-16

Diagnostics

# Residuals vs Fitted
ggplot(m3, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed")

# Histogram
ggplot(m3, aes(x = .resid)) +
  geom_histogram(bins = 25)

# QQ Plot
ggplot(m3, aes(sample = .resid)) +
  stat_qq()

One relationship that surprised me was the connection between internet freedom (pf_internet) and personal freedom (pf_score). I expected internet freedom to match personal freedom very closely, since open access to information usually reflects a more open society.

However, the scatterplot showed more spread than I expected, meaning some countries have relatively high internet freedom but lower personal freedom, or the reverse. The linear model is still positive, but the pattern is not as tight as the other models I looked at.

The diagnostics for this model support that observation. The residuals vs. fitted plot shows a roughly centered cloud but with more variability than earlier models. The histogram of residuals is somewhat bell-shaped, and the QQ plot mostly follows the straight line with some deviation at the ends. This tells me the model is usable, but the relationship between internet freedom and overall personal freedom is weaker and less predictable than I expected.

#Conclusion

In this lab, I analyzed how different types of freedom relate to one another using the Human Freedom Index dataset. The strongest relationship I examined was between expression control and personal freedom, which showed a clear upward trend and an R² of about 63%. This means expression freedom explains a large portion of personal freedom across countries.

The linear model performed well overall: the residuals showed no major patterns, they were roughly normally distributed, and their variability stayed fairly constant. Together, this supports the use of a straight-line model for these variables.

When I tested other freedoms, such as religious freedom and identity rights, I found that these also had positive relationships with personal freedom but slightly lower R² values. This suggests that while these freedoms matter, they do not predict personal freedom as strongly as expression control.

One of the more surprising findings was how unevenly identity rights connected to personal freedom. I expected that strong identity protections would match high personal freedom, but the relationship was weaker and more variable than expected.

Overall, the analysis showed that different types of freedom are connected, but not always equally. Expression freedom stood out as one of the strongest predictors of personal freedom, and the diagnostics showed that the linear model was an appropriate way to describe this relationship. * * *