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 for 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

The data

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

  1. What are the dimensions of the dataset? (How many rows, how many columns?) [Hint: Move your mouse over the dataset name in the Data toolbox, and you will see this information.]

Subsetting the data

This dataset has a lot of variables, but we are only interested in a few. There are also nine years of data in this dataset, but we are only interested in 2016. We can use Rguroo’s Subset function to obtain only the variables of interest for 2016, but it is simpler to use the built-in filters in the Data Editor.

  • In the Data toolbox, right-click the dataset hfi and select Edit.

  • On the very right, select the Filters tab filters and open the year dropdown.

  • Uncheck the (Select All) to de-select everything, then check the 2016 box to select only the year 2016.

  • Again, on the very right, select columns to choose columns in the dataset. Uncheck the checkbox next to the Search textbox to de-select everything, then check the boxes next to the variables we will use in this lab: pf_score, pf_expression_control, hf_score, and countries. It may be easiest to search for these variables individually in the search box rather than scroll through the entire list.

  • In the Save As textbox, enter hfi_2016 as the name of the dataset. Finally, click the Save As save as button to save the dataset.

  1. How many rows and columns are in the new hfi_2016 dataset?
*A view of the *hfi_2016* dataset after it has been saved.*

A view of the hfi_2016* dataset after it has been saved.*

Exploring the data

  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. Use the appropriate function in Rguroo’s Plots toolbox. Does the relationship look linear? If you knew a country’s value for pf_expression_control (how much political pressure and control the country has on media content, with 0 being the most control and 10 being no control), would you be comfortable using a linear model to predict the personal freedom score in that country?

If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient. In the Rguroo tab showing your graph, click Basics and check the Show Correlation and LS Equation box.

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. However, it is quite easy to obtain the model in Rguroo. In your plot, click Basics and check the LS Line button. Your plot should now show the least squares line and, if you also checked the box earlier to show the correlation, the equation of the line should be shown above the plot.

To get more information about the line, in the Analytics toolbox section, select Linear Regression then Simple and Multiple Regression. In the dialog, select hfi_2016 as the Dataset. You can specify the Response variable for our model by either typing pf_score in the text box or selecting pf_score from the dropdown menu. The Formula box specifies the explanatory variable(s) in the model. You can either type pf_expression_control in the box, or double-click pf_expression_control in the Variables list on the right to move it into the box automatically.

Once your dialog looks like the screenshot below, Preview eye the output.

*Creating the least-squares model in Rguroo*

Creating the least-squares model in Rguroo

The default Rguroo output contains a table reporting the number of cases (rows) used in the analysis, the Regression Coefficients Estimates table, a Model Summary giving additional information about the model, an Analysis of Variance table, and two diagnostic plots (Residuals Versus Fit and Normal Probability Plot: Residuals) useful for checking conditions for inference.

We will start by investigating the Regression Coefficients Estimates table:

*The Regression Coefficients Estimates table for the least-squares model*

The Regression Coefficients Estimates table for the least-squares model

The first column in this table identifies each Term in our model. A simple linear regression model has two terms: a y-intercept and a term containing the predictor variable. So according to the Term column, the first row contains information about the y-intercept term and the second row contains information about the pf_expression_control term. The second column displays the actual values of the y-intercept and the slope coefficient corresponding to pf_expression_control. With this table, we can write down the equation of the least squares regression line for the linear model:

\[ \widehat{pf\_score} = 4.2838 + 0.54185 \times pf\_expression\_control \]

One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, \(R^2\). This value can be found in the table helpfully labeled Model Summary: Coefficient of Determination (R-Squared). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 71.4% of the variability in pf_score is explained by pf_expression_control.

In Rguroo, the model and report need to be saved separately. Click the Save button, then save both the Model and Report as hfi_model1.

  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 Rguroo 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?

Prediction and prediction errors

Go back to the scatterplot with the least squares line laid on top.

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 score for a country with a 7 rating for pf_expression_control?

  2. The United States in 2016 had a 7 rating for pf_expression_control and a pf_score of 8.75. Use the equation of your least-squares regression line to predict the personal freedom score for the United States. Is this prediction an overestimate or an underestimate, and by how much? How is your answer related to the residual for this prediction?

Now we will check your answer to the previous exercise.

Go back to your Linear Regression output for the hfi_model1 model. Click Details, go to the Diagnostic Indices Table section, and check Include Diagnostics Table. First, we will select countries as our ID Variable, which will make it easy to identify which country each residual corresponds to. Then click Preview eye to see the updated output, which now contains a table labeled Predicted Values, Residuals, and Diagnostic Indices.

*Obtaining a table including the residuals*

Obtaining a table including the residuals

  1. In the Predicted Values, Residuals, and Diagnostic Indices table, find the residual (Residuals column) for the United States. Does this agree with the residual you computed earlier?

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 of residuals. We also need to check for (4) independent observations, but that is more difficult to check using model diagnostics, so we will not perform that check in this lab.

Checking 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.

This plot can be found in Rguroo’s Linear Regression output as the plot labeled Residuals Versus Fit.

*The Residuals Versus Fit plot generated by Rguroo*

The Residuals Versus Fit plot generated by Rguroo

The red dashed line is a horizontal line at \(y = 0\) (to help us check whether the residuals are distributed around 0), and the green line is a moving average of sorts (to help us look for underlying patterns).

  1. Is there any apparent pattern in the Residuals Versus Fit plot? What does this indicate about the linearity of the relationship between the predictor and the response?

Checking Nearly Normal Residuals

To check the nearly normal residuals condition, we can look at a histogram of the residuals. To do this, we need to save the residuals as an Rguroo dataset. Click Details, open the Diagnostics Indices Table section again and Save Table as hfi_residuals.

Once you have the dataset saved, you can make a Histogram of the hfi_residuals dataset, using the Variable Residuals.

In Rguroo, it is much less work to look at a normal probability plot of the residuals, as the plot is shown in the default Rguroo output and labeled Normal Probability Plot: Residuals.

*The Normal Probability Plot generated by Rguroo*

The Normal Probability Plot generated by Rguroo

In the Normal Distribution lab, you compared the normal probability plot to plots of simulated data from known normal distributions. This is a lot of work to just check an assumption visually, so the Rguroo output helps to streamline this process. If we simulated from a normal distribution with mean 0 and standard deviation equal to the standard deviation of the residuals, we would expect 95% of the points to lie within the dashed red lines.

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

Checking Constant Variability

To check the constant variability condition, look again at the Residuals Versus Fit plot you examined earlier.

  1. Are the points on the Residuals Versus Fit plot roughly evenly distributed around the red horizontal line \(y = 0\), or are there regions along the x-axis where there is much more scatter in the y-direction? What does this indicate about whether the constant variability condition appears to be met?

More Practice

  1. Choose another freedom variable and a variable you think would strongly correlate with it. Create a Subset of the hfi dataset containing only those two variables and the variable countries for the year 2016. Using this subset, produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?

  2. 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 predictor variable in this model seem to predict your response better than in the model we fit earlier? Why or why not?

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