Statistics in general – and logit models in particular – are based on comparisons. All statements are done by comparison to a reference group. Since all the GLMs in this class draw from a probability distribution in the exponential family, it also means that the effect size of one variable depends on the value of the other variables in the regression. These exercices are intended to help you see that.

We will be going back to work on the likelihood that judges at the Court of Justice of the European Union (CJEU) leave office at the expiry of their mandate. They serve 6 year-terms that can be renewed. The question is whether their government will want to renew their mandate. The dependent variable exit reports whether a judge leaves or remains for another term. The data is the same as in the R notebook.

#Download data

download.file(url = "https://siljehermansen.github.io/teaching/beyond-linear-models/Reappointments.rda",
              destfile = "Reappointments.rda")
library(stargazer);library(dplyr); library(ggeffects)

#Load in data
load("Reappointments.rda")
df <- Reappointments
mod1 <- glm(exit ~
             #Political difference between governments
             free_economy_diff
           #Performance of judge 
           + performance
           + age 
           + tenure
           + attendance
           + court,
           #Recoding strategy (logit-transformation) and binomial probability distribution
           family = binomial(link = "logit"),
           #Data
           df)

free_economy_diff captures the political distance between the government that appointed the judge at the beginning of the term and the government that might re-appoint the judge today.

performance captures the share of salient/high-impact assignment the judge received in the previous term compared to the share of high-impact assignments on the Court.

Exit decisions among judges at the CJEU (a binomial logit)
Dependent variable
exit
(1) (2)
free_economy_diff 1.490*** 1.570***
(0.443) (0.608)
performance -0.556* -3.144**
(0.307) (1.241)
age 0.132*** 0.170***
(0.027) (0.040)
tenure 0.045 0.034
(0.036) (0.047)
attendance -0.015* -0.022*
(0.009) (0.011)
courtGC 0.809**
(0.376)
Constant -9.746*** -9.629***
(1.759) (2.608)
Observations 235 142
Log Likelihood -110.922 -60.560
Akaike Inf. Crit. 235.844 133.121
Note: p<0.1; p<0.05; p<0.01

Exercise 1: Interpretation (the logistic transformation)

In the paper, we argue that judges’ performance matters in the upper-level Court of Justice (CJ), but not in the General Court.

  1. Subset the data to observation relating to the Court of Justice. Consider the variation, then the bivariate relationship between the outcome variable and governments’ preference distance and judges’ performance.

  2. What is the rationale behind the control variables?

  3. Re-estimate model 1 on the subset of the data relating to the Court of Justice. What is the effect of judges past performance on the likelihood that they will be replaced at the end of their term?

  4. What would you say is a good increment for a “high-performing” judge? Can you make a partial scenario and report the marginal effect of judges’ performance on governments’ decision to replace a judge?

  5. Calculate the first difference: Consider four scenarios and compare the predicted probabilities that a judge exits the Court for high and low levels of both performance and political distance.

    • what are the predicted probabilities for the two groups?
    • what is the effect of ideology among high-performing judges? What is the effect among low-performing judges?
    • who stands the most to win by performing better?

  6. Explore the compensation threshold: How much better must a judge perform to compensate for a median political distance in government preferences (i.e. free_economy_diff == 0.22)? Given the distribution of performance, how realistic is it for judges to survive based on merit alone?

  7. Illustrate the effect of performance graphically. Illustrate where the actual data points are on the x-axis using for example a rug plot.

  8. What is the marginal effect of a left-right overturn in government during a judge’s mandate on their probability of being replaced? To find a good increment, draw on the descriptive statistics.

    • use the filter() function to find the appointing and reappointing prime minister’s party family (you might want to eyeball the data using the View() function first) (family_id, family_id_ren).
    • use the group_by and reframe() functions to calculate a measure for a “typical” left-right shift
    • fill in the partial scenario and make a catchy sentence!

To consider the compensation threshold, you can follow the three steps:

Step 1: The logistic regression model

The log-odds of exit are given by:

\[ \log\left(\frac{P(\text{exit} = 1)}{1 - P(\text{exit} = 1)}\right) = \beta_0 + \beta_1 \cdot \text{free_economy_diff} + \beta_2 \cdot \text{performance} + \dots \]

We are interested in finding how much performance is needed to offset a given level of free_economy_diff.

Step 2: Neutralize the effect

To “compensate,” we set the combined effect of free_economy_diff and performance to zero, meaning they cancel each other out:

\[ \beta_1 \cdot \text{free_economy_diff} + \beta_2 \cdot \text{performance} = 0 \]

Step 3: Solve for performance Rearrange for performance:

\[ \text{performance} = -\frac{\beta_1}{\beta_2} \cdot \text{free_economy_diff} \]

This is the compensation equation: for each unit increase in free_economy_diff, performance must increase by \(-\beta_1 / \beta_2\) to keep the probability of exit unchanged.

Literature