Recovers respondent ideal points from an issue-scale response matrix
by Poole's (1998) decomposition
\(X_0 = \Psi W' + J_n c' + E_0\) over the
observed cells. Respondents with fewer than minscale responses are
not scaled (their rows are NA). The fit is a native port of the
BLACKB routine of basicspace::blackbox: each dimension is started
from sign-corrected respondent means (column polarities from the
pairwise correlation matrix) and refined by four alternating least
squares sweeps on the residuals of the earlier dimensions; up to five
joint sweeps follow, stopping once the two half-step error sums of
squares differ by less than 0.01; and the fitted \(\Psi W'\) is
re-expressed by its singular value decomposition,
\(\Psi = U D^{1/2}\), \(W = V D^{1/2}\). The results equal
basicspace::blackbox's (which rounds them to three decimals), with
or without missing cells; like it, the alternating fit is stopped
early, so on weakly structured data it need not be the fully converged
least-squares (truncated SVD) solution.
Value
A list with ideal_points (\(\Psi\), NA rows for unscaled
respondents), stimuli_weights (\(W\)), eigenvalues (squared
singular values), singular_values (of the fitted \(\Psi W'\),
basicspace's fits$singular), explained_variance (\(1 -
SSE/SS\), SS taken about the grand mean of the observed cells: the sum
of basicspace's fits$percent over the dimensions, divided by 100),
col_means (the issue intercepts \(c\), basicspace's c column),
n_dims, and engine ("native").
Examples
set.seed(1)
X <- matrix(rnorm(30 * 2), 30, 2) %*% matrix(rnorm(2 * 8), 2, 8) +
matrix(rnorm(240, 0, 0.3), 30, 8)
X[sample(240, 10)] <- NA
fit <- morie_spatial_voting_blackbox(X, n_dims = 2)
fit$engine
#> [1] "native"
head(fit$ideal_points)
#> [,1] [,2]
#> [1,] -1.0393690 -0.96223469
#> [2,] 0.2901126 0.24491438
#> [3,] -0.7818754 -0.02044986
#> [4,] 1.3506573 -0.15572170
#> [5,] 0.7512586 1.04767730
#> [6,] NA NA