For our statistical model, we follow Clinton, Jackman and Rivers (American Political Science Review, 2004, Vol. 98 No. 2) in assuming that the justices have quadratic utility functions. This means that their utility from taking a given position is equal to the negative squared distance from their ideal point to that position. We assume that a justice votes for the position—majority or dissent—that is closest to his ideal point. This process, however, is subject to some error. Accordingly, we assume that there is a random error that is added to the justice’s perception of each position's location. He is more likely to vote for the position that is closest to him, but also has some probability of voting for the farther position. As the difference in utility between the two positions grows, he becomes more and more likely to vote for the closer position. Formally, the logit of the probability of justice i voting with the majority on case j is equal to βjxi − αj, where βj equals twice the difference between the yea and nay positions on case j and αj equals the difference between the squared yea and nay positions on case j.
Our graphs all begin at the start of the 2005 term. Chief Justice Roberts took office just before the start of this term, so the graphs also begin at the start of his tenure on the Court. Our data begins ten years before this point in order to include existing knowledge about sitting justices and particularly to allow Chief Justice Rehnquist and Justice O’Connor to be compared to their replacements and others appointed after they left the Court. Because unanimous decisions tell us little about the location of the justices, we only include non-unanimous decisions. This applies both to the graphs and to the underlying data
To estimate ideal points from the justices’ voting decisions, we adopt a Bayesian approach and use a Gibbs sampler to sample from the posterior distribution representing our beliefs about the various parameter values. Credible intervals are formed based on the regions with the highest posterior density.
We begin with an informative prior for the ideal points of all justices who served in the court under Chief Justice Rehnquist since the appointment of Justice Breyer and semi-informative priors about Chief Justice Roberts and Justice Alito based on what we can infer from their past behavior and from what we know about President Bush's judicial philosophy. We also use vague independent normal priors for the mean and standard deviation of the case parameters αj and βj.
We run our ideal point model with vague priors and the identifying restriction that Justice O’Connor has an ideal point of 0 and Chief Justice Rehnquist has an ideal point of 1. To obtain the informative priors over the justices who served before the appointment of Chief Justice Roberts, we run our ideal point model on all cases decided during the last two years of the Rehnquist court.
We impose the identifying restriction that Chief Justice Rehnquist’s ideal point is 1 and Justice O’Connor’s ideal point is 0. This identifies the model and also allows for easy comparison between the ideal points of these former justices’ ideal points and those of the current justices.
The formal model is written below:
where i indexes the justice, j indexes the case, yi,j is the vote of justice i on case j, xi is the ideal point of justice i, and Λ is the logistic function.
The prior distributions are:
The prior distributions on the justices present during the tenure of Chief Justice Rehnquist were given a N(0,100) prior before updating the beliefs based on the last two years of the Rehnquist court, with the exception of Justice O’Connor and Chief Justice Rehnquist. The ideal points of Justice O’Connor and Chief Justice Rehnquist were 0 and 1, respectively, and are used as an identifying restriction.
The prior distributions on Justice Alito and Chief Justice Roberts were both N(1,2.5), giving mean and modal values equal to the position of the former Chief Justice Rehnquist. The prior distributions for Justices Sotomayor and Kagan were N(−1,2.5), giving a mean and modal value similar to Justices Breyer and Ginsburg.