lsirm1pl_ss {lsirm12pl}

1PL LSIRM with model selection approach.

Description

lsirm1pl_ss is used to fit 1PL LSIRM with model selection approach based on spike-and-slab priors. LSIRM factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_ss(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_spike_mean = -3,
  pr_spike_sd = 1,
  pr_slab_mean = 0.5,
  pr_slab_sd = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_xi_a = 1,
  pr_xi_b = 1,
  verbose = FALSE
)

Arguments

data

Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.

ndim

Integer; the dimension of the latent space. Default is 2.

niter

Integer; the total number of MCMC iterations to run. Default is 15000.

nburn

Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.

nthin

Integer; the number of MCMC iterations to thin. Default is 5.

nprint

Integer; the interval at which MCMC samples are displayed during execution. Default is 500.

jump_beta

Numeric; the jumping rule for the beta proposal density. Default is 0.4.

jump_theta

Numeric; the jumping rule for the theta proposal density. Default is 1.0.

jump_gamma

Numeric; the jumping rule for the theta proposal density. Default is 1.0.

jump_z

Numeric; the jumping rule for the z proposal density. Default is 0.5.

jump_w

Numeric; the jumping rule for the w proposal density. Default is 0.5.

pr_mean_beta

Numeric; the mean of the normal prior for beta. Default is 0.

pr_sd_beta

Numeric; the standard deviation of the normal prior for beta. Default is 1.0.

pr_mean_theta

Numeric; the mean of the normal prior for theta. Default is 0.

pr_spike_mean

Numeric; the mean of spike prior for log gamma. Default is -3.

pr_spike_sd

Numeric; the standard deviation of spike prior for log gamma. Default is 1.

pr_slab_mean

Numeric; the mean of spike prior for log gamma. Default is 0.5.

pr_slab_sd

Numeric; the standard deviation of spike prior for log gamma. Default is is 1.

pr_a_theta

Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.

pr_b_theta

Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.

pr_xi_a

Numeric; the first shape parameter of beta prior for latent variable xi. Default is 1.

pr_xi_b

Numeric; the second shape parameter of beta prior for latent variable xi. Default is 1.

verbose

Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_ss models the probability of correct response by respondent j to item i with item effect \beta_i, respondent effect \theta_j and the distance between latent position w_i of item i and latent position z_j of respondent j in the shared metric space, with \gamma represents the weight of the distance term:

logit(P(Y_{j,i} = 1|\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||

lsirm1pl_ss model include model selection approach based on spike-and-slab priors for log gamma. For detail of spike-and-slab priors, see References.

Value

lsirm1pl_ss returns an object of list containing the following components:

data

Data frame or matrix containing the variables used in the model.

bic

A numeric value representing the Bayesian Information Criterion (BIC).

mcmc_inf

Details about the number of MCMC iterations, burn-in periods, and thinning intervals.

map_inf

The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.

beta_estimate

Posterior estimates of the beta parameter.

theta_estimate

Posterior estimates of the theta parameter.

sigma_theta_estimate

Posterior estimates of the standard deviation of theta.

gamma_estimate

Posterior estimates of gamma parameter.

z_estimate

Posterior estimates of the z parameter.

w_estimate

Posterior estimates of the w parameter.

beta

Posterior samples of the beta parameter.

theta

Posterior samples of the theta parameter.

theta_sd

Posterior samples of the standard deviation of theta.

gamma

Posterior samples of the gamma parameter.

z

Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.

w

Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.

accept_beta

Acceptance ratio for the beta parameter.

accept_theta

Acceptance ratio for the theta parameter.

accept_z

Acceptance ratio for the z parameter.

accept_w

Acceptance ratio for the w parameter.

accept_gamma

Acceptance ratio for the gamma parameter.

pi_estimate

Posterior estimation of phi. inclusion probability of gamma. if estimation of phi is less than 0.5, choose Rasch model with gamma = 0, otherwise latent space model with gamma > 0.

pi

Posterior samples of phi which is indicator of spike and slab prior. If phi is 1, log gamma follows the slab prior, otherwise follows the spike prior.

References

Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: Frequentist and Bayesian strategies (Vol. 33). The Annals of Statistics

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

lsirm_result <- lsirm1pl_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE))