R/lsirm2pl_normal_mcar.R

In lsirm12pl: Latent Space Item Response Model

#' 2PL LSIRM with normal likelihood and missing completely at random data.
#'
#' @description \link{lsirm2pl_normal_mcar} is used to fit 2PL LSIRM for continuous variable in incomplete data assumed to be missing completely at random.
#' \link{lsirm2pl_normal_mcar} factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. Unlike 1PL model, 2PL model assumes the item effect can vary according to respondent, allowing additional parameter multiplied with respondent effect. The resulting latent space provides an interaction map that represents interactions between respondents and items.
#'
#' @inheritParams lsirm2pl
#' @param jump_gamma Numeric; the jumping rule for the gamma proposal density. Default is 0.025
#' @param pr_mean_gamma Numeric; mean of log normal prior for gamma. Default is 0.5.
#' @param pr_sd_gamma Numeric; standard deviation of log normal prior for gamma. Default is 1.0.
#' @param pr_a_eps Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
#' @param pr_b_eps Numeric; the scale parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
#' @param verbose Logical; If TRUE, MCMC samples are printed for each \code{nprint}. Default is FALSE.
#' @param missing.val Numeric; a number to replace missing values. Default is 99.
#'
#' @return \code{lsirm2pl_normal_mcar} returns an object of  list containing the following components:
#'  \item{data}{A data frame or matrix containing the variables used in the model.}
#'  \item{missing.val}{A number to replace missing values.}
#'  \item{bic}{Numeric value with the corresponding BIC.}
#' \item{mcmc_inf}{Details about the number of MCMC iterations, burn-in periods, and thinning intervals.}
#' \item{map_inf}{The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.}
#' \item{beta_estimate}{Posterior estimates of the beta parameter.}
#' \item{theta_estimate}{Posterior estimates of the theta parameter.}
#' \item{sigma_theta_estimate}{Posterior estimates of the standard deviation of theta.}
#' \item{gamma_estimate}{Posterior estimates of gamma parameter.}
#' \item{z_estimate}{Posterior estimates of the z parameter.}
#' \item{w_estimate}{Posterior estimates of the w parameter.}
#' \item{beta}{Posterior samples of the beta parameter.}
#' \item{theta}{Posterior samples of the theta parameter.}
#' \item{theta_sd}{Posterior samples of the standard deviation of theta.}
#' \item{gamma}{Posterior samples of the gamma parameter.}
#' \item{z}{Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.}
#' \item{w}{Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.}
#' \item{accept_beta}{Acceptance ratio for the beta parameter.}
#' \item{accept_theta}{Acceptance ratio for the theta parameter.}
#' \item{accept_z}{Acceptance ratio for the z parameter.}
#' \item{accept_w}{Acceptance ratio for the w parameter.}
#' \item{accept_gamma}{Acceptance ratio for the gamma parameter.}
#' \item{sigma_estimate}{Posterior estimates of the standard deviation.}
#' \item{sigma}{Posterior samples of the standard deviation.}
#' \item{alpha_estimate}{Posterior estimates of the alpha parameter.}
#'  \item{alpha}{Posterior estimates of the alpha parameter.}
#'  \item{accept_alpha}{Acceptance ratio for the alpha parameter.}
#'
#' @details \code{lsirm2pl_normal_mcar} models the continuous value of response by respondent \eqn{j} to item \eqn{i} with item effect \eqn{\beta_i}, respondent effect \eqn{\theta_j} and the distance between latent position \eqn{w_i} of item \eqn{i} and latent position \eqn{z_j} of respondent \eqn{j} in the shared metric space, with \eqn{\gamma} represents the weight of the distance term. For 2pl model, the the item effect is assumed to have additional discrimination parameter \eqn{\alpha_i} multiplied by \eqn{\theta_j}: \deqn{Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}} where the error \eqn{e_{j,i} \sim N(0,\sigma^2)} Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing completely at random assumption and data augmentation, see References.
#'
#' @references  Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.
#' @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 <- lsirm2pl_normal_mcar(data)
#'
#' # The code following can achieve the same result.
#' lsirm_result <- lsirm(data ~ lsirm2pl(spikenslab = FALSE, fixed_gamma = FALSE,
#'                       missing_data = "mcar"))
#'
#' @export
lsirm2pl_normal_mcar = function(data, ndim = 2, niter = 15000, nburn = 2500, nthin = 5, nprint = 500,
                                jump_beta = 0.4, jump_theta = 1.0, jump_alpha = 1.0, jump_gamma = 1.0, jump_z = 0.5, jump_w = 0.5,
                                pr_mean_beta = 0, pr_sd_beta = 1.0, pr_mean_theta = 0,
                                pr_mean_gamma = 0.5, pr_sd_gamma =1.0,
                                pr_mean_alpha = 0.5, pr_sd_alpha = 1,
                                pr_a_theta = 0.001, pr_b_theta = 0.001,pr_a_eps = 0.001, pr_b_eps = 0.001,
                                missing.val = 99, verbose=FALSE){
  if(niter < nburn){
    stop("niter must be greater than burn-in process.")
  }
  if(is.data.frame(data)){
    cname = colnames(data)
  }else{
    cname = paste("item", 1:ncol(data), sep=" ")
  }

  # cat("\n\nFitting with MCMC algorithm\n")


  output <- lsirm2pl_normal_mcar_cpp(as.matrix(data),
                                     ndim, niter, nburn, nthin, nprint,
                                     jump_beta, jump_theta, jump_alpha, jump_gamma, jump_z, jump_w,
                                     pr_mean_beta, pr_sd_beta, pr_a_theta, pr_b_theta, pr_mean_theta,
                                     pr_a_eps, pr_b_eps,
                                     pr_mean_gamma, pr_sd_gamma,
                                     pr_mean_alpha, pr_sd_alpha, missing.val, verbose=verbose)

  mcmc.inf = list(nburn=nburn, niter=niter, nthin=nthin)
  nsample <- nrow(data)
  nitem <- ncol(data)

  nmcmc = as.integer((niter - nburn) / nthin)
  max.address = min(which.max(output$map))
  map.inf = data.frame(value = output$map[which.max(output$map)], iter = which.max(output$map))
  w.star = output$w[max.address,,]
  z.star = output$z[max.address,,]
  w.proc = array(0,dim=c(nmcmc,nitem,ndim))
  z.proc = array(0,dim=c(nmcmc,nsample,ndim))

  # cat("\n\nProcrustes Matching Analysis\n")
cat("\n")

  for(iter in 1:nmcmc){
    z.iter = output$z[iter,,]
    w.iter = output$w[iter,,]

    if(ndim == 1){
      z.iter = as.matrix(z.iter)
      w.iter = as.matrix(w.iter)
      z.star = as.matrix(z.star)
      w.star = as.matrix(w.star)
    }

    if(iter != max.address) z.proc[iter,,] = procrustes(z.iter,z.star)$X.new
    else z.proc[iter,,] = z.iter

    if(iter != max.address) w.proc[iter,,] = procrustes(w.iter,w.star)$X.new
    else w.proc[iter,,] = w.iter
  }

  w.est = colMeans(w.proc, dims = 1)
  z.est = colMeans(z.proc, dims = 1)

  beta.estimate = apply(output$beta, 2, mean)
  theta.estimate = apply(output$theta, 2, mean)
  alpha.estimate = apply(output$alpha, 2, mean)
  sigma_theta.estimate = mean(output$sigma_theta)
  sigma.estimate = mean(output$sigma)
  gamma.estimate = mean(output$gamma)

  beta.summary = data.frame(cbind(apply(output$beta, 2, mean), t(apply(output$beta, 2, function(x) quantile(x, probs = c(0.025, 0.975))))))
  colnames(beta.summary) <- c("Estimate", "2.5%", "97.5%")
  rownames(beta.summary) <- cname

  # Calculate BIC
  # cat("\n\nCalculate BIC\n")
  log_like = log_likelihood_normal2pl_cpp(as.matrix(data), ndim, as.matrix(beta.estimate), as.matrix(alpha.estimate), as.matrix(theta.estimate), gamma.estimate, z.est, w.est, sigma.estimate, missing.val)
  p = 2 * nitem + nsample + 1 + 1 + ndim * nitem + ndim * nsample + 1
  bic = -2 * log_like[[1]] + p * log(nsample * nsample)

  result <- list(data = data,
              missing.val = missing.val,
              bic = bic,
                 mcmc_inf = mcmc.inf,
                 map_inf = map.inf,
                 beta_estimate  = beta.estimate,
                 beta_summary = beta.summary,
                 theta_estimate = theta.estimate,
                 sigma_theta_estimate    = sigma_theta.estimate,
                 sigma_estimate = sigma.estimate,
                 gamma_estimate = gamma.estimate,
                 alpha_estimate = alpha.estimate,
                 z_estimate     = z.est,
                 w_estimate     = w.est,
                 beta           = output$beta,
                 theta          = output$theta,
                 theta_sd       = output$sigma_theta,
                 sigma          = output$sigma,
                 gamma          = output$gamma,
                 alpha          = output$alpha,
                 z              = z.proc,
                 w              = w.proc,
                 accept_beta    = output$accept_beta,
                 accept_theta   = output$accept_theta,
                 accept_w       = output$accept_w,
                 accept_z       = output$accept_z,
                 accept_gamma   = output$accept_gamma,
                 accept_alpha   = output$accept_alpha)
  class(result) = "lsirm"

  return(result)
}