R/original_par_2GM.R

In IRTest: Parameter Estimation of Item Response Theory with Estimation of Latent Distribution

#' Recovering original parameters of two-component Gaussian mixture distribution from re-parameterized values
#'
#' @param prob The \eqn{\pi = \frac{n_1}{N}} parameter of two-component Gaussian mixture distribution, where \eqn{n_1} is the estimated number of examinees belonging to the first Gaussian component and \eqn{N} is the total number of examinees (Li, 2021).
#' @param d The \eqn{\delta = \frac{\mu_2 - \mu_1}{\bar{\sigma}}} parameter of two-component Gaussian mixture distribution,
#' where \eqn{\mu_1} and \eqn{\mu_2} are the estimated means of the first and second Gaussian components, respectively.
#' And \eqn{\bar{\sigma}} is the overall standard deviation of the latent distribution (Li, 2021).
#' Without loss of generality, \eqn{\mu_2 \ge \mu_1} is assumed, thus \eqn{\delta \ge 0}.
#' @param sd_ratio A numeric value of \eqn{\zeta = \frac{\sigma_2}{\sigma_1}} parameter of two-component Gaussian mixture distribution, where \eqn{\sigma_1} and \eqn{\sigma_2} are the estimated standard deviations of the first and second Gaussian components, respectively (Li, 2021).
#' @param overallmean A numeric value of \eqn{\bar{\mu}} that determines the overall mean of two-component Gaussian mixture distribution.
#' @param overallsd A numeric value of \eqn{\bar{\sigma}} that determines the overall standard deviation of two-component Gaussian mixture distribution.
#'
#' @details
#' \describe{
#' \item{Original two-component Gaussian mixture distribution}{
#' \deqn{f(x)=\pi\times \phi(x | \mu_1, \sigma_1)+(1-\pi)\times \phi(x | \mu_2, \sigma_2)}
#' , where \eqn{\phi} is a Gaussian component.
#' }
#' \item{Re-parameterized two-component Gaussian mixture distribution}{
#' \deqn{f(x)=2GM(x|\pi, \delta, \zeta, \bar{\mu}, \bar{\sigma})}
#' , where \eqn{\bar{\mu}} is overall mean and \eqn{\bar{\sigma}} is overall standard deviation of the distribution.
#' }
#'
#' \item{
#' The original parameters retrieved from re-parameterized values}{
#' 1) Mean of the first Gaussian component (\code{m1}).
#' \deqn{\mu_1=-(1-\pi)\delta\bar{\sigma}+\bar{\mu}}
#'
#' 2) Mean of the second Gaussian component (\code{m2}).
#' \deqn{\mu_2=\pi\delta\bar{\sigma}+\bar{\mu}}
#'
#' 3) Standard deviation of the first Gaussian component (\code{s1}).
#' \deqn{\sigma_1^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\pi+(1-\pi)\zeta^2}\right)}
#'
#' 4) Standard deviation of the second Gaussian component (\code{s2}).
#' \deqn{\sigma_2^2=\bar{\sigma}^2\left(\frac{1-\pi(1-\pi)\delta^2}{\frac{1}{\zeta^2}\pi+(1-\pi)}\right)=\zeta^2\sigma_1^2}
#' }
#' }
#'
#' @return This function returns a vector of length 4: \code{c(m1,m2,s1,s2)}.
#' \item{m1}{The location parameter (mean) of the first Gaussian component.}
#' \item{m2}{The location parameter (mean) of the second Gaussian component.}
#' \item{s1}{The scale parameter (standard deviation) of the first Gaussian component.}
#' \item{s2}{The scale parameter (standard deviation) of the second Gaussian component.}
#'
#' @author Seewoo Li \email{cu@@yonsei.ac.kr}
#'
#' @references
#' Li, S. (2021). Using a two-component normal mixture distribution as a latent distribution in estimating parameters of item response models. \emph{Journal of Educational Evaluation, 34}(4), 759-789.
#'
#' @export
#'
original_par_2GM <- function(prob = 0.5, d = 0, sd_ratio = 1, overallmean=0, overallsd=1){
  m1 <- -(1-prob)*d*overallsd+overallmean
  m2 <- prob*d*overallsd+overallmean
  s1 <- sqrt((overallsd^2-prob*(1-prob)*(d*overallsd)^2)/(prob+(1-prob)*sd_ratio^2))
  s2 <- s1*sd_ratio
  return(c(m1,m2,s1,s2))
}