Dynamic IRT: ideal points that drift by a random walk (Martin and Quinn 2002)

The one-dimensional dynamic item response model of Martin and Quinn (2002), fitted by their Gibbs sampler: a vote is cast when the latent utility \(z_{jk} = -\alpha_k + \beta_k \theta_{j,t(k)} + \varepsilon_{jk}\) is positive, \(\varepsilon \sim N(0, 1)\); the ideal points follow a random walk, \(\theta_{j,0} \sim N(e_0, E_0)\), \(\theta_{j,t} \sim N(\theta_{j,t-1}, \tau^2_j)\); and the roll-call parameters have normal priors \(\alpha_k \sim N(a_0, 1/A_0)\), \(\beta_k \sim N(b_0, 1/B_0)\). Each sweep draws the latent utilities (truncated normals), every roll call's \((\alpha, \beta)\) (conjugate normal), every legislator's whole path by forward filtering and backward sampling, and, when c0 and d0 are positive, the evolution variances from \(IG((c_0 + T)/2, (d_0 + SS)/2)\). The sign is fixed by reflecting a draw whenever the anchor legislator's mean ideal point is negative. This is the model of MCMCpack::MCMCdynamicIRT1d, with the same default priors; both recover the same paths on simulated data.

Usage

morie_spatial_voting_dynamic_irt(
  votes,
  time_periods,
  n_samples = 500L,
  burn_in = 100L,
  seed = 42L,
  thin = 1L,
  tau2 = 1,
  c0 = -1,
  d0 = -1,
  e0 = 0,
  E0 = 1,
  a0 = 0,
  A0 = 0.1,
  b0 = 0,
  B0 = 0.1,
  anchor = NULL
)

Arguments

votes

Legislators-by-roll-calls matrix of 0/1 votes (-1/1 is read as 0/1); NA for absent.

time_periods

Period of each roll call (one entry per column).

n_samples

Posterior draws kept.

burn_in

Sweeps discarded first.

seed

RNG seed.

thin

Keep every thin-th sweep.

tau2

Evolution variance (the starting value when it is estimated).

c0, d0

Inverse-gamma prior on \(\tau^2\); \(\tau^2\) is held at tau2 unless both are positive.

e0, E0

Prior mean and variance of the initial ideal point.

a0, A0, b0, B0

Prior means and precisions of \(\alpha\) and \(\beta\).

anchor

Legislator (row index) whose ideal point is kept positive; the largest first principal-component score by default.

Value

A list with theta (legislators by periods, posterior means), theta_sd, alpha, beta, tau2, per_period (one entry per period with its ideal_points), periods, n_periods, n_legislators, n_samples, anchor and engine.

References

Martin, A. D. and Quinn, K. M. (2002). Dynamic ideal point estimation via Markov chain Monte Carlo for the U.S. Supreme Court, 1953-1999. Political Analysis 10(2), 134-153.

Examples

# \donttest{
set.seed(7)
N <- 12; Tn <- 3; kper <- 15
period <- rep(1:Tn, each = kper)
truth <- t(sapply(rnorm(N), function(s) s + cumsum(c(0, rnorm(Tn - 1, 0, 0.3)))))
a <- rnorm(Tn * kper, 0, 0.5); b <- rnorm(Tn * kper, 1.5, 0.5)
p <- pnorm(-rep(a, each = N) + rep(b, each = N) * truth[, period])
V <- matrix(rbinom(length(p), 1, p), N)
fit <- morie_spatial_voting_dynamic_irt(V, period, n_samples = 400, burn_in = 100)
abs(cor(as.numeric(fit$theta), as.numeric(truth)))   # the paths are recovered
#> [1] 0.9440225
# }