Update utilities and thresholds — utility

Low-level Gibbs sampler kernels for error covariance matrices, latent utilities, and ordered-response thresholds.

Usage

d_to_gamma(d)

log_likelihood_ordered(d, y, sys, Tvec)

update_Sigma(n_Sigma_0, V_Sigma_0, N, S)

update_U(U, y, sys, Sigma_inv, available = NULL)

update_U_ranked(U, sys, Sigma_inv)

update_d(d, y, sys, mu_d_0, Sigma_d_0, Tvec, step_scale)

Arguments

d: [numeric(J - 2)]
Log-increments between finite ordered thresholds.
y: [integer(1) | matrix(N, max(Tvec))]
Chosen alternative for update_U(), where J denotes the base alternative, or ordered responses by decider and occasion for the threshold functions.
sys: [numeric(J - 1) | matrix(N, max(Tvec))]
Systematic utilities matching U or y.
Tvec: [integer(N)]
Number of observed occasions for each decider.
n_Sigma_0: [integer(1)]
Prior degrees of freedom for the error covariance.
V_Sigma_0: [matrix(J - 1, J - 1)]
Prior scale matrix for the error covariance.
N: [integer(1)]
Number of independent sampling units.
S: [matrix(J - 1, J - 1)]
Error scatter matrix.
U: [numeric(J - 1)]
Current latent utility differences.
Sigma_inv: [matrix(J - 1, J - 1)]
Inverse error covariance matrix.
available: [logical(J) | NULL]
Availability of the J - 1 non-base alternatives followed by the base alternative. Utilities of unavailable alternatives are drawn without truncation. By default (NULL), all alternatives are available.
mu_d_0: [numeric(J - 2)]
Prior mean for threshold log-increments.
Sigma_d_0: [matrix(J - 2, J - 2)]
Prior covariance for threshold log-increments.
step_scale: [numeric(J - 2)]
Random-walk proposal standard deviations, one per log-increment.

Value

The functions return one sampler update or transformation:

update_Sigma(): a J - 1 by J - 1 covariance matrix.
update_U() and update_U_ranked(): J - 1 by 1 numeric latent utility matrices.
d_to_gamma(): a numeric column matrix containing ordered thresholds and their infinite bounds.
log_likelihood_ordered(): one numeric log-likelihood value.
update_d(): a numeric vector of updated log-increments.

References

Robert CP (1995). “Simulation of Truncated Normal Variables.” Statistics and Computing, 5(2), 121–125. doi:10.1007/BF00143942 .

Examples

### two deciders who choose twice on an ordered scale of four levels
set.seed(1)
d <- c(0, log(2))
y <- matrix(c(1, 2, 3, 2), nrow = 2)
sys <- matrix(0, nrow = 2, ncol = 2)
Tvec <- c(2, 2)

### the thresholds, their likelihood, and their random-walk update
d_to_gamma(d)
#>      [,1]
#> [1,] -Inf
#> [2,]    0
#> [3,]    1
#> [4,]    3
#> [5,]  Inf
log_likelihood_ordered(d, y, sys, Tvec)
#> [1] -4.692438
update_d(
  d, y, sys, mu_d_0 = c(0, 0), Sigma_d_0 = diag(2), Tvec = Tvec,
  step_scale = c(0.1, 0.1)
)
#>             [,1]
#> [1,] -0.06264538
#> [2,]  0.82612711

### the latent utilities of an unordered and of a ranked choice
update_U(
  c(0, 0), y = 1, sys = c(0, 0), Sigma_inv = diag(2),
  available = c(TRUE, TRUE, TRUE)
)
#>            [,1]
#> [1,]  0.4634427
#> [2,] -0.4874291
update_U_ranked(c(0, 0), sys = c(0, 0), Sigma_inv = diag(2))
#>            [,1]
#> [1,] -0.2615707
#> [2,] -0.9674915

### the error covariance
update_Sigma(n_Sigma_0 = 4, V_Sigma_0 = diag(2), N = 10, S = diag(2))
#>            [,1]       [,2]
#> [1,] 0.09405327 0.01900477
#> [2,] 0.01900477 0.14300195