lsirm1pl_mar {lsirm12pl}

1pl LSIRM model for missing at random data.

Description

lsirm1pl_mar is used to fit 1pl LSIRM model in incomplete data assumed to be missing at random. lsirm1pl_mar factorizes 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_mar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 0.025,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_mean_gamma = 0.5,
  pr_sd_gamma = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data

Matrix; binary item response matrix to be analyzed. Each row is assumed to be respondent and its column values are assumed to be response to the corresponding item.

ndim

Numeric; dimension of latent space. default value is 2.

niter

Numeric; number of iterations to run MCMC sampling. default value is 15000.

nburn

Numeric; number of initial, pre-thinning, MCMC iterations to discard. default value is 2500.

nthin

Numeric;number of thinning, MCMC iterations to discard. default value is 5.

nprint

Numeric; MCMC samples is displayed during execution of MCMC chain for each nprint. default value is 500.

jump_beta

Numeric; jumping rule of the proposal density for beta. default value is 0.4.

jump_theta

Numeric; jumping rule of the proposal density for theta. default value is 1.0.

jump_gamma

Numeric; jumping rule of the proposal density for gamma. default value is 0.025.

jump_z

Numeric; jumping rule of the proposal density for z. default value is 0.5.

jump_w

Numeric; jumping rule of the proposal density for w. default value is 0.5.

pr_mean_beta

Numeric; mean of normal prior for beta. default value is 0.

pr_sd_beta

Numeric; standard deviation of normal prior for beta. default value is 1.0.

pr_mean_theta

Numeric; mean of normal prior for theta. default value is 0.

pr_mean_gamma

Numeric; mean of log normal prior for gamma. default value is 0.5.

pr_sd_gamma

Numeric; standard deviation of log normal prior for gamma. default value is 1.0.

pr_a_theta

Numeric; shape parameter of inverse gamma prior for variance of theta. default value is 0.001.

pr_b_theta

Numeric; scale parameter of inverse gamma prior for variance of theta. default value is 0.001.

missing.val

Numeric; a number to replace missing values. default value is 99.

verbose

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

Details

lsirm1pl_mar 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||

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.

Value

lsirm1pl_mar 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

number of mcmc iteration, burn-in periods, and thinning intervals.

map_inf

value of log maximum a posterior and iteration number which have log maximum a posterior.

beta_estimate

posterior estimation of beta.

theta_estimate

posterior estimation of theta.

sigma_theta_estimate

posterior estimation of standard deviation of theta.

gamma_estimate

posterior estimation of gamma.

z_estimate

posterior estimation of z.

w_estimate

posterior estimation of w.

imp_estimate

probability of imputating a missing value with 1.

beta

posterior samples of beta.

theta

posterior samples of theta.

theta_sd

posterior samples of standard deviation of theta.

gamma

posterior samples of gamma.

z

posterior samples of z. The output is 3-dimensional matrix with last axis represent the dimension of latent space.

w

posterior samples of w. The output is 3-dimensional matrix with last axis represent the dimension of latent space.

imp

imputation for missing Values using posterior samples.

accept_beta

accept ratio of beta.

accept_theta

accept ratio of theta.

accept_z

accept ratio of z.

accept_w

accept ratio of w.

accept_gamma

accept ratio of gamma.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),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_mar(data)

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