Ideal point estimation is built on the idea that the ideology of each person (in our case, each Supreme Court justice) can be represented as a point, usually along a single line. Most commonly, as in our case, this line represents the liberal-conservative spectrum. Thus, the point representing a justice indicates how liberal or conservative the justice is, with larger (or further-to-the-right) ideal points indicating more conservatism. For example, if Justice Stevens’ ideal point is −3 and Justice Scalia’s ideal point is 2, we can say that Justice Stevens is more liberal than Justice Scalia.

The main objective of ideal point estimation is to obtain an estimated location for each justice on a liberal-conservative scale. We can get information about where the justices are located and how much more conservative or liberal each is relative to the others by looking at the voting patterns across all cases. In the end, our goal is to be able to order all of the justices on a line in from most liberal to most conservative.

To do this estimation, we use the justices’ votes on the disposition of cases. If Justices Scalia and Thomas dissent on an opinion in which all other justices join the majority opinion, we get an indication that they are likely to be different from the others. They are probably extremists on one side or the other that is, either more conservative or more liberal than the other justices. By looking at the coalitions of justices across different opinions, we are able to get an idea about where each justice is relative to the others on the liberal-conservative scale.

More formally, we assume that each justice has an ideal point along the real number line. We think of this line as going from liberal (represented by smaller or more negative values) to conservative (represented by larger or more positive values). When deciding how to vote on a given case, justices look at whether the majority or a dissenting position is closest to their ideal point. On cases for which the majority opinion is close to their own position, they are more likely to vote with the majority, either by joining the majority opinion or by writing or joining a concurring opinion. On cases where a justice disagrees with the majority position, he or she is likely to dissent.

We update our estimates of each justice’s ideal point after every vote he or she casts. Because looking at past votes gives us a very good idea about where Justices Stevens, Scalia, Kennedy, Souter, Thomas, Ginsburg and Breyer’s ideal points are, we mainly are looking to see where Roberts and Alito fall relative to these seven more senior justices. When, for example, Roberts joins Thomas and Scalia in a dissent against the other six justices’ position, we are more likely to believe that he is a conservative, being to the right of the moderates and liberals on the court. When he joins in a majority opinion against the dissent of Thomas, for example, this sends a signal that he is likely to be more liberal than Thomas.

See the technical section for more information on the technical details of our model and estimation technique.

Our ideal point estimation is done from a Bayesian statistical framework. In Bayesian statistics, we quantify our beliefs about the ideal points and other parameters of the model with probability distributions which tell us the probability that the parameters are within a given interval. Thus, if we were 90% sure that Justice Thomas is more conservative than Justice Scalia, the probability distribution would reflect this.

In our statistical model, we begin with a distribution that represents our initial beliefs about the justices’ ideal points and about other parameters. This distribution, which we form before looking at the data, is called our prior distribution. After this is done, we form our posterior beliefs—the probabilities given the data—using Bayes’ theorem and the data we have observed (in this case, the votes the justices have cast). As with our prior beliefs, our posterior beliefs are quantified as a probability distribution.

Finally, we can convert these posterior probability distributions to more easily interpreted information by forming credible intervals. Credible intervals are intervals of possible values which contain the parameter of interest with a specified probability. In our case, we find it useful to form an interval for a given justice which, according to our posterior beliefs, has a 95% probability of containing the ideal point for that justice. Thus, if our credible interval for Chief Justice Roberts were from 0 to 1, our posterior beliefs would tell us that there is a 95% chance that Chief Justice Roberts’ ideal point lies between 0 and 1—which, in our case, means that there would be a 95% chance that he is more conservative than Justice O’Connor and more liberal than Justice Rehnquist, since these two Justices are given to have ideal points 0 and 1, respectively.

Interested readers can refer to an article by Josh
Clinton, Simon Jackman, and Doug Rivers (“The Statistical Analysis of
Roll Call Data,” *American Political Science Review*, 2004, Vol. 98,
No. 2) for
more information on Bayesian ideal point estimation.