10 Judges: estimating the ideological locations of Supreme Court justices
This program implements an ideal-point model (similar to the legislators
example), estimating both the locations of the justices on a latent ideological
dimension, and two parameters specific to each case (corresponding to the item
difficulty and item discrimination parameters of a two-parameter IRT
model).8 The data consist of the decisions of Justices Rehnquist,
Stevens, O’Connor, Scalia, Kennedy, Souter, Thomas, Ginsberg and Bryer, in that
order, \(i = 1, \dots , 9\). The decisions are coded 1 for votes with the
majority, and 0 for votes against the majority, and NA for abstentions.
In these models, the only observed data are votes, and the analyst wants to model those votes as a function of legislator- (\(\theta_i\)), and vote-specific (\(\alpha_i\), \(\lambda_i\)) parameters. The vote of legislator \(i\) on roll-call \(j\) (\(y_{i,j}\)) is a function of a the legislator’s ideal point (\(\theta_i\)), the vote’s difficulty parameter and the vote’s discrimination (\(\beta_j\)): \[ \begin{aligned}[t] y_{i,j} &\sim \mathsf{Bernoulli}(\pi_i) \\ \pi_i &= \frac{1}{1 + \exp(-\mu_{i,j})} \\ \mu_{i,j} &= \beta_j \theta_i - \alpha_j \end{aligned} \]
\[ \begin{aligned}[t] \beta_j &\sim \mathsf{Normal}(0, 2.5) \\ \alpha_j &\sim \mathsf{Normal}(0, 5) \\ \theta_i &\sim \mathsf{Normal}(0, 1) \\ \end{aligned} \]
To simplify the analysis, the outcomes will be aggregated to “Yes”, “No”, and missing values (which
sc9497_vote_data <- tibble(vote = colnames(sc9497$votes)) %>%
mutate(.vote_id = row_number())
sc9497_legis_data <- as.data.frame(sc9497$legis.names) %>%
rownames_to_column("judge") %>%
mutate(.judge_id = row_number())
sc9497_votes <- sc9497$votes %>%
as.data.frame() %>%
rownames_to_column("judge") %>%
gather(vote, yea, -judge) %>%
filter(!is.na(yea)) %>%
inner_join(dplyr::select(sc9497_vote_data, vote, .vote_id), by = "vote") %>%
inner_join(dplyr::select(sc9497_legis_data, judge, .judge_id), by = "judge")