lsirm1pl_normal_mar_ss {lsirm12pl}

1PL LSIRM with normal likelihood and model selection approach for missing at random data.

Description

lsirm1pl_normal_mar_ss is used to fit 1PL LSIRM with model selection approach based on spike-and-slab priors for continuous variable with 1pl in incomplete data assumed to be missing at random. lsirm1pl_normal_mar_ss factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_mar_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_a_eps = 0.001,
  pr_b_eps = 0.001,
  pr_xi_a = 0.001,
  pr_xi_b = 0.001,
  missing.val = 99,
  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_a_eps

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

pr_b_eps

Numeric; the scale parameter of inverse gamma prior for variance of data likelihood 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.

missing.val

Numeric; a number to replace missing values. Default is 99.

verbose

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

Details

lsirm1pl_normal_mar_ss models the continuous value of 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:

Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}

where the error e_{j,i} \sim N(0,\sigma^2) Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References. lsirm1pl_normal_mar_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_normal_mar_ss returns an object of list containing the following components:

data

Data frame or matrix containing the variables in the model.

missing.val

A number to replace missing values.

bic

Numeric value with the corresponding 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.

gamma

Posterior samples of the gamma parameter.

theta_sd

Posterior samples of the standard deviation of theta.

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.

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.

imp

Imputation for missing Values using posterior samples.

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.

imp_estimate

Probability of imputating a missing value with 1.

sigma_estimate

Posterior estimates of the standard deviation.

sigma

Posterior samples of the standard deviation.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons. Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: frequentist and Bayesian strategies. The Annals of Statistics, 33(2), 730-773.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_mar_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE,
missing_data = 'mar', missing = 99))