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R/irtQ-package.R
In irtQ: Unidimensional Item Response Theory Modeling
#' irtQ: Unidimensional Item Response Theory Modeling
#'
#' @description
#' Fit unidimensional item response theory (IRT) models to a mixture of dichotomous and polytomous data,
#' calibrate online item parameters (i.e., pretest and operational items), estimate examinees' abilities,
#' and provide useful functions related to unidimensional IRT such as IRT model-data fit evaluation and
#' differential item functioning analysis.
#'
#' For the item parameter estimation, the marginal maximum likelihood estimation via the expectation-maximization (MMLE-EM) algorithm
#' (Bock & Aitkin, 1981) is used. Also, the fixed item parameter calibration (FIPC) method (Kim, 2006) and
#' the fixed ability parameter calibration (FAPC) method, (Ban, Hanson, Wang, Yi, & Harris, 2001; stocking, 1988),
#' often called Stocking's Method A, are provided. For the ability estimation, several popular scoring methods (e.g., ML, EAP, and MAP)
#' are implemented.
#'
#' In addition, there are many useful functions related to IRT analyses such as evaluating IRT model-data fit,
#' analyzing differential item functioning (DIF), importing item and/or ability parameters from popular IRT software,
#' running flexMIRT (Cai, 2017) through R, generating simulated data, computing the conditional distribution of observed scores
#' using the Lord-Wingersky recursion formula, computing item and test information functions, computing item and test characteristic
#' curve functions, and plotting item and test characteristic curves and item and test information functions.
#'
#' \tabular{ll}{ Package: \tab irtQ\cr Version: \tab 0.2.1\cr Date: \tab
#' 2024-08-23\cr Depends: \tab R (>= 4.1)\cr License: \tab GPL (>= 2)\cr }
#'
#' @details
#' Following five sections describe a) how to implement the online item calibration using FIPC, a) how to implement the online item
#' calibration using Method A, b) two illustrations of the online calibration, and c) IRT Models used in \pkg{irtQ} package.
#'
#'
#' @section Online item calibration with the fixed item parameter calibration method (e.g., Kim, 2006):
#'
#' The fixed item parameter calibration (FIPC) is one of useful online item calibration methods for computerized adaptive testing (CAT)
#' to put the parameter estimates of pretest items on the same scale of operational item parameter estimates without post hoc
#' linking/scaling (Ban, Hanson, Wang, Yi, & Harris, 2001; Chen & Wang, 2016). In FIPC, the operational item parameters are fixed to
#' estimate the characteristic of the underlying latent variable prior distribution when calibrating the pretest items. More specifically,
#' the underlying latent variable prior distribution of the operational items is estimated during the calibration of the pretest
#' items to put the item parameters of the pretest items on the scale of the operational item parameters (Kim, 2006). In the \pkg{irtQ}
#' package, FIPC is implemented with two main steps:
#'
#' \enumerate{
#' \item Prepare a response data set and the item metadata of the fixed (or operational) items.
#' \item Implement FIPC to estimate the item parameters of pretest items using the \code{\link{est_irt}} function.
#' }
#'
#' \describe{
#' \item{1. Preparing a data set}{
#' To run the \code{\link{est_irt}} function, it requires two data sets:
#'
#' \enumerate{
#' \item Item metadata set (i.e., model, score category, and item parameters. see the desciption of the argument \code{x} in the function \code{\link{est_irt}}).
#' \item Examinees' response data set for the items. It should be a matrix format where a row and column indicate the examinees and the items, respectively.
#' The order of the columns in the response data set must be exactly the same as the order of rows of the item metadata.
#' }
#' }
#'
#' \item{2. Estimating the pretest item parameters}{
#' When FIPC is implemented in \code{\link{est_irt}} function, the pretest item parameters are estimated by fixing the operational item parameters. To estimate the item
#' parameters, you need to provide the item metadata in the argument \code{x} and the response data in the argument \code{data}.
#'
#' It is worthwhile to explain about how to prepare the item metadata set in the argument \code{x}. A specific form of a data frame should be used for
#' the argument \code{x}. The first column should have item IDs, the second column should contain the number of score categories of the items, and the third
#' column should include IRT models. The available IRT models are "1PLM", "2PLM", "3PLM", and "DRM" for dichotomous items, and "GRM" and "GPCM" for polytomous
#' items. Note that "DRM" covers all dichotomous IRT models (i.e, "1PLM", "2PLM", and "3PLM") and "GRM" and "GPCM" represent the graded response model and
#' (generalized) partial credit model, respectively. From the fourth column, item parameters should be included. For dichotomous items, the fourth, fifth,
#' and sixth columns represent the item discrimination (or slope), item difficulty, and item guessing parameters, respectively. When "1PLM" or "2PLM" is
#' specified for any items in the third column, NAs should be inserted for the item guessing parameters. For polytomous items, the item discrimination (or slope)
#' parameters should be contained in the fourth column and the item threshold (or step) parameters should be included from the fifth to the last columns.
#' When the number of categories differs between items, the empty cells of item parameters should be filled with NAs. See `est_irt` for more details about
#' the item metadata.
#'
#' Also, you should specify in the argument \code{fipc = TRUE} and a specific FIPC method in the argument \code{fipc.method}. Finally, you should provide
#' a vector of the location of the items to be fixed in the argument \code{fix.loc}. For more details about implementing FIPC, see the
#' description of the function \code{\link{est_irt}}.
#'
#' When implementing FIPC, you can estimate both the emprical histogram and the scale of latent variable prior distribution by setting \code{EmpHist = TRUE}.
#' If \code{EmpHist = FALSE}, the normal prior distribution is used during the item parameter estimation and the scale of the normal prior distribution is
#' updated during the EM cycle.
#'
#' The \code{\link{est_item}} function requires a vector of the number of score categories for the items in the argument \code{cats}. For example, a dichotomous item has
#' two score categories. If a single numeric value is specified, that value will be recycled across all items. If NULL and all items are binary items
#' (i.e., dichotomous items), it assumes that all items have two score categories.
#'
#' If necessary, you need to specify whether prior distributions of item slope and guessing parameters (only for the IRT 3PL model) are used in the arguments of
#' \code{use.aprior} and \code{use.gprior}, respectively. If you decide to use the prior distributions, you should specify what distributions will be used for the prior
#' distributions in the arguments of \code{aprior} and \code{gprior}, respectively. Currently three probability distributions of Beta, Log-normal, and Normal
#' distributions are available.
#'
#' In addition, if the response data include missing values, you must indicate the missing value in argument \code{missing}.
#'
#' Once the \code{\link{est_irt}} function has been implemented, you'll get a list of several internal objects such as the item parameter estimates,
#' standard error of the parameter estimates.
#' }
#' }
#'
#'
#' @section Online item calibration with the fixed ability parameter calibration method (e.g., Stocking, 1988):
#' In CAT, the fixed ability parameter calibration (FAPC), often called Stocking's Method A, is the relatively simplest
#' and most straightforward online calibration method, which is the maximum likelihood estimation of the item parameters
#' given the proficiency estimates. In CAT, FAPC can be used to put the parameter estimates of pretest items on
#' the same scale of operational item parameter estimates and recalibrate the operational items to evaluate the parameter
#' drifts of the operational items (Chen & Wang, 2016; Stocking, 1988). Also, FAPC is known to result in accurate, unbiased
#' item parameters calibration when items are randomly rather than adaptively administered to examinees, which occurs most
#' commonly with pretest items (Ban, Hanson, Wang, Yi, & Harris, 2001; Chen & Wang, 2016). Using \pkg{irtQ} package,
#' the FAPC is implemented to calibrate the items with two main steps:
#'
#' \enumerate{
#' \item Prepare a data set for the calibration of item parameters (i.e., item response data and ability estimates).
#' \item Implement the FAPC to estimate the item parameters using the \code{\link{est_item}} function.
#' }
#'
#' \describe{
#' \item{1. Preparing a data set}{
#' To run the \code{\link{est_item}} function, it requires two data sets:
#'
#' \enumerate{
#' \item Examinees' ability (or proficiency) estimates. It should be in the format of a numeric vector.
#' \item response data set for the items. It should be in the format of matrix where a row and column indicate
#' the examinees and the items, respectively. The order of the examinees in the response data set must be exactly the same as that of the examinees' ability estimates.
#' }
#' }
#'
#' \item{2. Estimating the pretest item parameters}{
#' The \code{\link{est_item}} function estimates the pretest item parameters given the proficiency estimates. To estimate the item parameters,
#' you need to provide the response data in the argument \code{data} and the ability estimates in the argument \code{score}.
#'
#' Also, you should provide a string vector of the IRT models in the argument \code{model} to indicate what IRT model is used to calibrate each item.
#' Available IRT models are "1PLM", "2PLM", "3PLM", and "DRM" for dichotomous items, and "GRM" and "GPCM" for polytomous items. "GRM" and "GPCM" represent
#' the graded response model and (generalized) partial credit model, respectively. Note that "DRM" is considered as "3PLM" in this function. If a single
#' character of the IRT model is specified, that model will be recycled across all items.
#'
#' The \code{\link{est_item}} function requires a vector of the number of score categories for the items in the argument \code{cats}. For example, a dichotomous item has
#' two score categories. If a single numeric value is specified, that value will be recycled across all items. If NULL and all items are binary items
#' (i.e., dichotomous items), it assumes that all items have two score categories.
#'
#' If necessary, you need to specify whether prior distributions of item slope and guessing parameters (only for the IRT 3PL model) are used in the arguments of
#' \code{use.aprior} and \code{use.gprior}, respectively. If you decide to use the prior distributions, you should specify what distributions will be used for the prior
#' distributions in the arguments of \code{aprior} and \code{gprior}, respectively. Currently three probability distributions of Beta, Log-normal, and Normal
#' distributions are available.
#'
#' In addition, if the response data include missing values, you must indicate the missing value in argument \code{missing}.
#'
#' Once the \code{\link{est_item}} function has been implemented, you'll get a list of several internal objects such as the item parameter estimates,
#' standard error of the parameter estimates.
#' }
#' }
#'
#'
#' @section Three examples of R script:
#'
#' The example code below shows how to implement the online calibration and how to evaluate the IRT model-data fit:\preformatted{
#' ##---------------------------------------------------------------
#' # Attach the packages
#' library(irtQ)
#'
#' ##----------------------------------------------------------------------------
#' # 1. The example code below shows how to prepare the data sets and how to
#' # implement the fixed item parameter calibration (FIPC):
#' ##----------------------------------------------------------------------------
#'
#' ## Step 1: prepare a data set
#' ## In this example, we generated examinees' true proficiency parameters and simulated
#' ## the item response data using the function "simdat".
#'
#' ## import the "-prm.txt" output file from flexMIRT
#' flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
#'
#' # select the item metadata
#' x <- bring.flexmirt(file=flex_sam, "par")$Group1$full_df
#'
#' # generate 1,000 examinees' latent abilities from N(0.4, 1.3)
#' set.seed(20)
#' score <- rnorm(1000, mean=0.4, sd=1.3)
#'
#' # simulate the response data
#' sim.dat <- simdat(x=x, theta=score, D=1)
#'
#' ## Step 2: Estimate the item parameters
#' # fit the 3PL model to all dichotmous items, fit the GRM model to all polytomous data,
#' # fix the five 3PL items (1st - 5th items) and three GRM items (53th to 55th items)
#' # also, estimate the empirical histogram of latent variable
#' fix.loc <- c(1:5, 53:55)
#' (mod.fix1 <- est_irt(x=x, data=sim.dat, D=1, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 16)), EmpHist=TRUE, Etol=1e-3,
#' fipc=TRUE, fipc.method="MEM", fix.loc=fix.loc))
#' summary(mod.fix1)
#'
#' # plot the estimated empirical histogram of latent variable prior distribution
#' (emphist <- getirt(mod.fix1, what="weights"))
#' plot(emphist$weight ~ emphist$theta, xlab="Theta", ylab="Density")
#'
#'
#' ##----------------------------------------------------------------------------
#' # 2. The example code below shows how to prepare the data sets and how to estimate
#' # the item parameters using the fixed abilit parameter calibration (FAPC):
#' ##----------------------------------------------------------------------------
#'
#' ## Step 1: prepare a data set
#' ## In this example, we generated examinees' true proficiency parameters and simulated
#' ## the item response data using the function "simdat". Because, the true
#' ## proficiency parameters are not known in reality, however, the true proficiencies
#' ## would be replaced with the proficiency estimates for the calibration.
#'
#' # import the "-prm.txt" output file from flexMIRT
#' flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
#'
#' # select the item metadata
#' x <- bring.flexmirt(file=flex_sam, "par")$Group1$full_df
#'
#' # modify the item metadata so that some items follow 1PLM, 2PLM and GPCM
#' x[c(1:3, 5), 3] <- "1PLM"
#' x[c(1:3, 5), 4] <- 1
#' x[c(1:3, 5), 6] <- 0
#' x[c(4, 8:12), 3] <- "2PLM"
#' x[c(4, 8:12), 6] <- 0
#' x[54:55, 3] <- "GPCM"
#'
#' # generate examinees' abilities from N(0, 1)
#' set.seed(23)
#' score <- rnorm(500, mean=0, sd=1)
#'
#' # simulate the response data
#' data <- simdat(x=x, theta=score, D=1)
#'
#' ## Step 2: Estimate the item parameters
#' # 1) item parameter estimation: constrain the slope parameters of the 1PLM to be equal
#' (mod1 <- est_item(x, data, score, D=1, fix.a.1pl=FALSE, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
#' summary(mod1)
#'
#' # 2) item parameter estimation: fix the slope parameters of the 1PLM to 1
#' (mod2 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, a.val.1pl=1, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
#' summary(mod2)
#'
#' # 3) item parameter estimation: fix the guessing parameters of the 3PLM to 0.2
#' (mod3 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, fix.g=TRUE, a.val.1pl=1, g.val=.2,
#' use.startval=FALSE))
#' summary(mod3)
#'
#'
#' }
#'
#'
#' @section IRT Models:
#'
#' In the \pkg{irtQ} package, both dichotomous and polytomous IRT models are available.
#' For dichotomous items, IRT one-, two-, and three-parameter logistic models (1PLM, 2PLM, and 3PLM) are used.
#' For polytomous items, the graded response model (GRM) and the (generalized) partial credit model (GPCM) are used.
#' Note that the item discrimination (or slope) parameters should be fixed to 1 when the partial credit model is fit to data.
#'
#' In the following, let \eqn{Y} be the response of an examinee with latent ability \eqn{\theta} on an item and suppose that there
#' are \eqn{K} unique score categories for each polytomous item.
#'
#' \describe{
#' \item{IRT 1-3PL models}{
#' For the IRT 1-3PL models, the probability that an examinee with \eqn{\theta} provides a correct answer for an item is given by,
#' \deqn{P(Y = 1|\theta) = g + \frac{(1 - g)}{1 + exp(-Da(\theta - b))},}
#' where \eqn{a} is the item discrimination (or slope) parameter, \eqn{b} represents the item difficulty parameter,
#' \eqn{g} refers to the item guessing parameter. \eqn{D} is a scaling factor in IRT models to make the logistic function
#' as close as possible to the normal ogive function when \eqn{D = 1.702}. When the 1PLM is used, \eqn{a} is either fixed to a constant
#' value (e.g., \eqn{a=1}) or constrained to have the same value across all 1PLM item data. When the IRT 1PLM or 2PLM is fit to data,
#' \eqn{g = 0} is set to 0.
#' }
#' \item{GRM}{
#' For the GRM, the probability that an examinee with latent ability \eqn{\theta} responds to score category \eqn{k} (\eqn{k=0,1,...,K-1})
#' of an item is a given by,
#' \deqn{P(Y = k | \theta) = P^{*}(Y \ge k | \theta) - P^{*}(Y \ge k + 1 | \theta),}
#' \deqn{P^{*}(Y \ge k | \theta) = \frac{1}{1 + exp(-Da(\theta - b_{k}))}, and}
#' \deqn{P^{*}(Y \ge k + 1 | \theta) = \frac{1}{1 + exp(-Da(\theta - b_{k+1}))}, }
#'
#' where \eqn{P^{*}(Y \ge k | \theta} refers to the category boundary (threshold) function for score category \eqn{k} of an item
#' and its formula is analogous to that of 2PLM. \eqn{b_{k}} is the difficulty (or threshold) parameter for category boundary
#' \eqn{k} of an item. Note that \eqn{P(Y = 0 | \theta) = 1 - P^{*}(Y \ge 1 | \theta)}
#' and \eqn{P(Y = K-1 | \theta) = P^{*}(Y \ge K-1 | \theta)}.
#' }
#' \item{GPCM}{
#' For the GPCM, the probability that an examinee with latent ability \eqn{\theta} responds to score category \eqn{k} (\eqn{k=0,1,...,K-1})
#' of an item is a given by,
#' \deqn{P(Y = k | \theta) = \frac{exp(\sum_{v=0}^{k}{Da(\theta - b_{v})})}{\sum_{h=0}^{K-1}{exp(\sum_{v=0}^{h}{Da(\theta - b_{v})})}},}
#' where \eqn{b_{v}} is the difficulty parameter for category boundary \eqn{v} of an item. In other contexts, the difficulty parameter \eqn{b_{v}}
#' can also be parameterized as \eqn{b_{v} = \beta - \tau_{v}}, where \eqn{\beta} refers to the location (or overall difficulty) parameter
#' and \eqn{\tau_{jv}} represents a threshold parameter for score category \eqn{v} of an item. In the \pkg{irtQ} package, \eqn{K-1} difficulty
#' parameters are necessary when an item has \eqn{K} unique score categories because \eqn{b_{0}=0}. When a partial credit model is fit to data, \eqn{a}
#' is fixed to 1.
#' }
#'
#' }
#'
#' @author Hwanggyu Lim \email{hglim83@@gmail.com}
#'
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#'
#' Wells, C. S., & Bolt, D. M. (2008). Investigation of a nonparametric procedure for assessing goodness-of-fit in
#' item response theory. \emph{Applied Measurement in Education, 21}(1), 22-40.
#'
#' Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference.
#' \emph{Journal of the American Statistical Association, 22}(158), 209-212.
#'
#' Woods, C. M. (2007). Empirical histograms in item response theory with ordinal data. \emph{Educational and Psychological Measurement, 67}(1), 73-87.
#'
#' Yen, W. M. (1981). Using simulation results to choose a latent trait model. \emph{Applied Psychological Measurement, 5}, 245-262.
#'
#' Zimowski, M. F., Muraki, E., Mislevy, R. J., & Bock, R. D. (2003). BILOG-MG 3: Multiple-group
#' IRT analysis and test maintenance for binary items [Computer Program]. Chicago, IL: Scientific
#' Software International. URL http://www.ssicentral.com
#'
#' @name irtQ-package
#' @keywords package
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#' irtQ: Unidimensional Item Response Theory Modeling
#'
#' @description
#' Fit unidimensional item response theory (IRT) models to a mixture of dichotomous and polytomous data,
#' calibrate online item parameters (i.e., pretest and operational items), estimate examinees' abilities,
#' and provide useful functions related to unidimensional IRT such as IRT model-data fit evaluation and
#' differential item functioning analysis.
#'
#' For the item parameter estimation, the marginal maximum likelihood estimation via the expectation-maximization (MMLE-EM) algorithm
#' (Bock & Aitkin, 1981) is used. Also, the fixed item parameter calibration (FIPC) method (Kim, 2006) and
#' the fixed ability parameter calibration (FAPC) method, (Ban, Hanson, Wang, Yi, & Harris, 2001; stocking, 1988),
#' often called Stocking's Method A, are provided. For the ability estimation, several popular scoring methods (e.g., ML, EAP, and MAP)
#' are implemented.
#'
#' In addition, there are many useful functions related to IRT analyses such as evaluating IRT model-data fit,
#' analyzing differential item functioning (DIF), importing item and/or ability parameters from popular IRT software,
#' running flexMIRT (Cai, 2017) through R, generating simulated data, computing the conditional distribution of observed scores
#' using the Lord-Wingersky recursion formula, computing item and test information functions, computing item and test characteristic
#' curve functions, and plotting item and test characteristic curves and item and test information functions.
#'
#' \tabular{ll}{ Package: \tab irtQ\cr Version: \tab 0.2.1\cr Date: \tab
#' 2024-08-23\cr Depends: \tab R (>= 4.1)\cr License: \tab GPL (>= 2)\cr }
#'
#' @details
#' Following five sections describe a) how to implement the online item calibration using FIPC, a) how to implement the online item
#' calibration using Method A, b) two illustrations of the online calibration, and c) IRT Models used in \pkg{irtQ} package.
#'
#'
#' @section Online item calibration with the fixed item parameter calibration method (e.g., Kim, 2006):
#'
#' The fixed item parameter calibration (FIPC) is one of useful online item calibration methods for computerized adaptive testing (CAT)
#' to put the parameter estimates of pretest items on the same scale of operational item parameter estimates without post hoc
#' linking/scaling (Ban, Hanson, Wang, Yi, & Harris, 2001; Chen & Wang, 2016). In FIPC, the operational item parameters are fixed to
#' estimate the characteristic of the underlying latent variable prior distribution when calibrating the pretest items. More specifically,
#' the underlying latent variable prior distribution of the operational items is estimated during the calibration of the pretest
#' items to put the item parameters of the pretest items on the scale of the operational item parameters (Kim, 2006). In the \pkg{irtQ}
#' package, FIPC is implemented with two main steps:
#'
#' \enumerate{
#' \item Prepare a response data set and the item metadata of the fixed (or operational) items.
#' \item Implement FIPC to estimate the item parameters of pretest items using the \code{\link{est_irt}} function.
#' }
#'
#' \describe{
#' \item{1. Preparing a data set}{
#' To run the \code{\link{est_irt}} function, it requires two data sets:
#'
#' \enumerate{
#' \item Item metadata set (i.e., model, score category, and item parameters. see the desciption of the argument \code{x} in the function \code{\link{est_irt}}).
#' \item Examinees' response data set for the items. It should be a matrix format where a row and column indicate the examinees and the items, respectively.
#' The order of the columns in the response data set must be exactly the same as the order of rows of the item metadata.
#' }
#' }
#'
#' \item{2. Estimating the pretest item parameters}{
#' When FIPC is implemented in \code{\link{est_irt}} function, the pretest item parameters are estimated by fixing the operational item parameters. To estimate the item
#' parameters, you need to provide the item metadata in the argument \code{x} and the response data in the argument \code{data}.
#'
#' It is worthwhile to explain about how to prepare the item metadata set in the argument \code{x}. A specific form of a data frame should be used for
#' the argument \code{x}. The first column should have item IDs, the second column should contain the number of score categories of the items, and the third
#' column should include IRT models. The available IRT models are "1PLM", "2PLM", "3PLM", and "DRM" for dichotomous items, and "GRM" and "GPCM" for polytomous
#' items. Note that "DRM" covers all dichotomous IRT models (i.e, "1PLM", "2PLM", and "3PLM") and "GRM" and "GPCM" represent the graded response model and
#' (generalized) partial credit model, respectively. From the fourth column, item parameters should be included. For dichotomous items, the fourth, fifth,
#' and sixth columns represent the item discrimination (or slope), item difficulty, and item guessing parameters, respectively. When "1PLM" or "2PLM" is
#' specified for any items in the third column, NAs should be inserted for the item guessing parameters. For polytomous items, the item discrimination (or slope)
#' parameters should be contained in the fourth column and the item threshold (or step) parameters should be included from the fifth to the last columns.
#' When the number of categories differs between items, the empty cells of item parameters should be filled with NAs. See `est_irt` for more details about
#' the item metadata.
#'
#' Also, you should specify in the argument \code{fipc = TRUE} and a specific FIPC method in the argument \code{fipc.method}. Finally, you should provide
#' a vector of the location of the items to be fixed in the argument \code{fix.loc}. For more details about implementing FIPC, see the
#' description of the function \code{\link{est_irt}}.
#'
#' When implementing FIPC, you can estimate both the emprical histogram and the scale of latent variable prior distribution by setting \code{EmpHist = TRUE}.
#' If \code{EmpHist = FALSE}, the normal prior distribution is used during the item parameter estimation and the scale of the normal prior distribution is
#' updated during the EM cycle.
#'
#' The \code{\link{est_item}} function requires a vector of the number of score categories for the items in the argument \code{cats}. For example, a dichotomous item has
#' two score categories. If a single numeric value is specified, that value will be recycled across all items. If NULL and all items are binary items
#' (i.e., dichotomous items), it assumes that all items have two score categories.
#'
#' If necessary, you need to specify whether prior distributions of item slope and guessing parameters (only for the IRT 3PL model) are used in the arguments of
#' \code{use.aprior} and \code{use.gprior}, respectively. If you decide to use the prior distributions, you should specify what distributions will be used for the prior
#' distributions in the arguments of \code{aprior} and \code{gprior}, respectively. Currently three probability distributions of Beta, Log-normal, and Normal
#' distributions are available.
#'
#' In addition, if the response data include missing values, you must indicate the missing value in argument \code{missing}.
#'
#' Once the \code{\link{est_irt}} function has been implemented, you'll get a list of several internal objects such as the item parameter estimates,
#' standard error of the parameter estimates.
#' }
#' }
#'
#'
#' @section Online item calibration with the fixed ability parameter calibration method (e.g., Stocking, 1988):
#' In CAT, the fixed ability parameter calibration (FAPC), often called Stocking's Method A, is the relatively simplest
#' and most straightforward online calibration method, which is the maximum likelihood estimation of the item parameters
#' given the proficiency estimates. In CAT, FAPC can be used to put the parameter estimates of pretest items on
#' the same scale of operational item parameter estimates and recalibrate the operational items to evaluate the parameter
#' drifts of the operational items (Chen & Wang, 2016; Stocking, 1988). Also, FAPC is known to result in accurate, unbiased
#' item parameters calibration when items are randomly rather than adaptively administered to examinees, which occurs most
#' commonly with pretest items (Ban, Hanson, Wang, Yi, & Harris, 2001; Chen & Wang, 2016). Using \pkg{irtQ} package,
#' the FAPC is implemented to calibrate the items with two main steps:
#'
#' \enumerate{
#' \item Prepare a data set for the calibration of item parameters (i.e., item response data and ability estimates).
#' \item Implement the FAPC to estimate the item parameters using the \code{\link{est_item}} function.
#' }
#'
#' \describe{
#' \item{1. Preparing a data set}{
#' To run the \code{\link{est_item}} function, it requires two data sets:
#'
#' \enumerate{
#' \item Examinees' ability (or proficiency) estimates. It should be in the format of a numeric vector.
#' \item response data set for the items. It should be in the format of matrix where a row and column indicate
#' the examinees and the items, respectively. The order of the examinees in the response data set must be exactly the same as that of the examinees' ability estimates.
#' }
#' }
#'
#' \item{2. Estimating the pretest item parameters}{
#' The \code{\link{est_item}} function estimates the pretest item parameters given the proficiency estimates. To estimate the item parameters,
#' you need to provide the response data in the argument \code{data} and the ability estimates in the argument \code{score}.
#'
#' Also, you should provide a string vector of the IRT models in the argument \code{model} to indicate what IRT model is used to calibrate each item.
#' Available IRT models are "1PLM", "2PLM", "3PLM", and "DRM" for dichotomous items, and "GRM" and "GPCM" for polytomous items. "GRM" and "GPCM" represent
#' the graded response model and (generalized) partial credit model, respectively. Note that "DRM" is considered as "3PLM" in this function. If a single
#' character of the IRT model is specified, that model will be recycled across all items.
#'
#' The \code{\link{est_item}} function requires a vector of the number of score categories for the items in the argument \code{cats}. For example, a dichotomous item has
#' two score categories. If a single numeric value is specified, that value will be recycled across all items. If NULL and all items are binary items
#' (i.e., dichotomous items), it assumes that all items have two score categories.
#'
#' If necessary, you need to specify whether prior distributions of item slope and guessing parameters (only for the IRT 3PL model) are used in the arguments of
#' \code{use.aprior} and \code{use.gprior}, respectively. If you decide to use the prior distributions, you should specify what distributions will be used for the prior
#' distributions in the arguments of \code{aprior} and \code{gprior}, respectively. Currently three probability distributions of Beta, Log-normal, and Normal
#' distributions are available.
#'
#' In addition, if the response data include missing values, you must indicate the missing value in argument \code{missing}.
#'
#' Once the \code{\link{est_item}} function has been implemented, you'll get a list of several internal objects such as the item parameter estimates,
#' standard error of the parameter estimates.
#' }
#' }
#'
#'
#' @section Three examples of R script:
#'
#' The example code below shows how to implement the online calibration and how to evaluate the IRT model-data fit:\preformatted{
#' ##---------------------------------------------------------------
#' # Attach the packages
#' library(irtQ)
#'
#' ##----------------------------------------------------------------------------
#' # 1. The example code below shows how to prepare the data sets and how to
#' # implement the fixed item parameter calibration (FIPC):
#' ##----------------------------------------------------------------------------
#'
#' ## Step 1: prepare a data set
#' ## In this example, we generated examinees' true proficiency parameters and simulated
#' ## the item response data using the function "simdat".
#'
#' ## import the "-prm.txt" output file from flexMIRT
#' flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
#'
#' # select the item metadata
#' x <- bring.flexmirt(file=flex_sam, "par")$Group1$full_df
#'
#' # generate 1,000 examinees' latent abilities from N(0.4, 1.3)
#' set.seed(20)
#' score <- rnorm(1000, mean=0.4, sd=1.3)
#'
#' # simulate the response data
#' sim.dat <- simdat(x=x, theta=score, D=1)
#'
#' ## Step 2: Estimate the item parameters
#' # fit the 3PL model to all dichotmous items, fit the GRM model to all polytomous data,
#' # fix the five 3PL items (1st - 5th items) and three GRM items (53th to 55th items)
#' # also, estimate the empirical histogram of latent variable
#' fix.loc <- c(1:5, 53:55)
#' (mod.fix1 <- est_irt(x=x, data=sim.dat, D=1, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 16)), EmpHist=TRUE, Etol=1e-3,
#' fipc=TRUE, fipc.method="MEM", fix.loc=fix.loc))
#' summary(mod.fix1)
#'
#' # plot the estimated empirical histogram of latent variable prior distribution
#' (emphist <- getirt(mod.fix1, what="weights"))
#' plot(emphist$weight ~ emphist$theta, xlab="Theta", ylab="Density")
#'
#'
#' ##----------------------------------------------------------------------------
#' # 2. The example code below shows how to prepare the data sets and how to estimate
#' # the item parameters using the fixed abilit parameter calibration (FAPC):
#' ##----------------------------------------------------------------------------
#'
#' ## Step 1: prepare a data set
#' ## In this example, we generated examinees' true proficiency parameters and simulated
#' ## the item response data using the function "simdat". Because, the true
#' ## proficiency parameters are not known in reality, however, the true proficiencies
#' ## would be replaced with the proficiency estimates for the calibration.
#'
#' # import the "-prm.txt" output file from flexMIRT
#' flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
#'
#' # select the item metadata
#' x <- bring.flexmirt(file=flex_sam, "par")$Group1$full_df
#'
#' # modify the item metadata so that some items follow 1PLM, 2PLM and GPCM
#' x[c(1:3, 5), 3] <- "1PLM"
#' x[c(1:3, 5), 4] <- 1
#' x[c(1:3, 5), 6] <- 0
#' x[c(4, 8:12), 3] <- "2PLM"
#' x[c(4, 8:12), 6] <- 0
#' x[54:55, 3] <- "GPCM"
#'
#' # generate examinees' abilities from N(0, 1)
#' set.seed(23)
#' score <- rnorm(500, mean=0, sd=1)
#'
#' # simulate the response data
#' data <- simdat(x=x, theta=score, D=1)
#'
#' ## Step 2: Estimate the item parameters
#' # 1) item parameter estimation: constrain the slope parameters of the 1PLM to be equal
#' (mod1 <- est_item(x, data, score, D=1, fix.a.1pl=FALSE, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
#' summary(mod1)
#'
#' # 2) item parameter estimation: fix the slope parameters of the 1PLM to 1
#' (mod2 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, a.val.1pl=1, use.gprior=TRUE,
#' gprior=list(dist="beta", params=c(5, 17)), use.startval=FALSE))
#' summary(mod2)
#'
#' # 3) item parameter estimation: fix the guessing parameters of the 3PLM to 0.2
#' (mod3 <- est_item(x, data, score, D=1, fix.a.1pl=TRUE, fix.g=TRUE, a.val.1pl=1, g.val=.2,
#' use.startval=FALSE))
#' summary(mod3)
#'
#'
#' }
#'
#'
#' @section IRT Models:
#'
#' In the \pkg{irtQ} package, both dichotomous and polytomous IRT models are available.
#' For dichotomous items, IRT one-, two-, and three-parameter logistic models (1PLM, 2PLM, and 3PLM) are used.
#' For polytomous items, the graded response model (GRM) and the (generalized) partial credit model (GPCM) are used.
#' Note that the item discrimination (or slope) parameters should be fixed to 1 when the partial credit model is fit to data.
#'
#' In the following, let \eqn{Y} be the response of an examinee with latent ability \eqn{\theta} on an item and suppose that there
#' are \eqn{K} unique score categories for each polytomous item.
#'
#' \describe{
#' \item{IRT 1-3PL models}{
#' For the IRT 1-3PL models, the probability that an examinee with \eqn{\theta} provides a correct answer for an item is given by,
#' \deqn{P(Y = 1|\theta) = g + \frac{(1 - g)}{1 + exp(-Da(\theta - b))},}
#' where \eqn{a} is the item discrimination (or slope) parameter, \eqn{b} represents the item difficulty parameter,
#' \eqn{g} refers to the item guessing parameter. \eqn{D} is a scaling factor in IRT models to make the logistic function
#' as close as possible to the normal ogive function when \eqn{D = 1.702}. When the 1PLM is used, \eqn{a} is either fixed to a constant
#' value (e.g., \eqn{a=1}) or constrained to have the same value across all 1PLM item data. When the IRT 1PLM or 2PLM is fit to data,
#' \eqn{g = 0} is set to 0.
#' }
#' \item{GRM}{
#' For the GRM, the probability that an examinee with latent ability \eqn{\theta} responds to score category \eqn{k} (\eqn{k=0,1,...,K-1})
#' of an item is a given by,
#' \deqn{P(Y = k | \theta) = P^{*}(Y \ge k | \theta) - P^{*}(Y \ge k + 1 | \theta),}
#' \deqn{P^{*}(Y \ge k | \theta) = \frac{1}{1 + exp(-Da(\theta - b_{k}))}, and}
#' \deqn{P^{*}(Y \ge k + 1 | \theta) = \frac{1}{1 + exp(-Da(\theta - b_{k+1}))}, }
#'
#' where \eqn{P^{*}(Y \ge k | \theta} refers to the category boundary (threshold) function for score category \eqn{k} of an item
#' and its formula is analogous to that of 2PLM. \eqn{b_{k}} is the difficulty (or threshold) parameter for category boundary
#' \eqn{k} of an item. Note that \eqn{P(Y = 0 | \theta) = 1 - P^{*}(Y \ge 1 | \theta)}
#' and \eqn{P(Y = K-1 | \theta) = P^{*}(Y \ge K-1 | \theta)}.
#' }
#' \item{GPCM}{
#' For the GPCM, the probability that an examinee with latent ability \eqn{\theta} responds to score category \eqn{k} (\eqn{k=0,1,...,K-1})
#' of an item is a given by,
#' \deqn{P(Y = k | \theta) = \frac{exp(\sum_{v=0}^{k}{Da(\theta - b_{v})})}{\sum_{h=0}^{K-1}{exp(\sum_{v=0}^{h}{Da(\theta - b_{v})})}},}
#' where \eqn{b_{v}} is the difficulty parameter for category boundary \eqn{v} of an item. In other contexts, the difficulty parameter \eqn{b_{v}}
#' can also be parameterized as \eqn{b_{v} = \beta - \tau_{v}}, where \eqn{\beta} refers to the location (or overall difficulty) parameter
#' and \eqn{\tau_{jv}} represents a threshold parameter for score category \eqn{v} of an item. In the \pkg{irtQ} package, \eqn{K-1} difficulty
#' parameters are necessary when an item has \eqn{K} unique score categories because \eqn{b_{0}=0}. When a partial credit model is fit to data, \eqn{a}
#' is fixed to 1.
#' }
#'
#' }
#'
#' @author Hwanggyu Lim \email{hglim83@@gmail.com}
#'
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#' for Scores on Tests Including Polytomous Items with Ordered Responses. \emph{Applied Psychological
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#' Weeks, J. P. (2010). plink: An R Package for Linking Mixed-Format Tests Using IRT-Based Methods.
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#'
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#' @name irtQ-package
#' @keywords package
"_PACKAGE"
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