% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/fit.FCMIRT.R
\name{fit.FCMIRT}
\alias{fit.FCMIRT}
\title{Fit the Forced-Choice Multidimensional IRT (FCMIRT) Model}
\usage{
fit.FCMIRT(
  data,
  model = "2PL",
  Q.matrix = NULL,
  block.items = NULL,
  D = NULL,
  fc.type = NULL,
  method = c("iStEM", "stan"),
  control.model = NULL,
  control.method = NULL
)
}
\arguments{
\item{data}{An \eqn{N \times B} forced-choice data object. Each entry is
either a ranking string (e.g., \code{"2>1>3"}) or a 1-based pattern
index into the block's ranking patterns. If pattern indices are used,
\code{block.items} must be supplied.}

\item{model}{MIRT model type: \code{"m1pl"}, \code{"m2pl"} (default),
\code{"m3pl"}, or \code{"m4pl"}.}

\item{Q.matrix}{An optional \eqn{I \times D} binary (0/1) matrix
indicating active item loadings.}

\item{block.items}{A list of length \eqn{B} where each element is an
integer vector of global item indices belonging to that block. If
\code{NULL} and \code{data} contains ranking strings, automatically
parsed from the data.}

\item{D}{Integer; number of latent dimensions. If \code{NULL}, inferred
from \code{Q.matrix}; if both are \code{NULL}, an error is raised.}

\item{fc.type}{Character vector (length 1 or \eqn{B}): \code{"RANK"}
(full ranking, default), \code{"MOLE"} (most-least), or \code{"PICK"}
(best only). Recycled to all blocks if scalar.}

\item{method}{Estimation method: \code{"iStEM"} (default) or
\code{"stan"}.}

\item{control.model}{A named list of model-level hyperparameters.
Supported entries:
\describe{
\item{\code{a.mu}, \code{a.sigma}}{Prior location and scale for
\eqn{\log a_{id}} (log-normal). Defaults: 0, 1.0.
\strong{Important:} The Stan default for \code{a.sigma}
is deliberately set to 1.0 (wider than the MIRT default of
0.25) because forced-choice blocks identify relative
utilities; tight common slope priors can spuriously
shrink within-block slopes toward equality, degrading
trait recovery. For iStEM, the prior defaults match the
standard MIRT values (\code{a.mu = 0.25}, \code{a.sigma = 0.25})
unless overridden.}
\item{\code{b.mu}, \code{b.sigma}}{Prior mean and SD for \eqn{b_i}
(normal). Defaults: 0, 1. Under the block zero-sum constraint
\eqn{\sum_{i \in \text{block } b} b_i = 0}, these priors
regularize the free difficulty parameters.}
\item{\code{c.mu}, \code{c.sigma}}{Uniform support bounds
\eqn{[c_{\min}, c_{\max}]} for lower-asymptote parameters
\eqn{c_i}. Defaults: 0, 0.35. Only used for 3PL/4PL models.}
\item{\code{d.mu}, \code{d.sigma}}{Uniform support bounds
\eqn{[d_{\min}, d_{\max}]} for upper-asymptote parameters
\eqn{d_i}. Defaults: 0.65, 1. Only used for 4PL models.}
\item{\code{theta.mu}}{Prior mean vector for
\eqn{\boldsymbol{\theta}_j}. Default: \code{rep(0, D)}.}
\item{\code{L}}{Theta grid size per dimension for marginal
log-likelihood computation and iStEM block Gibbs sampling.
Default adapts to \eqn{D}
(e.g., 61 for D = 1, 31 for D = 2, 15 for D = 3).}
\item{\code{theta.lower}, \code{theta.upper}}{Bounds for the
theta grid used by marginal log-likelihood computation and
iStEM block Gibbs sampling. Defaults: -6, 6.}
\item{\code{use.prior}}{Logical (iStEM only). If \code{FALSE}, the
item-prior penalty term is omitted from the item update,
recovering an approximate maximum-likelihood item step.
Default is \code{TRUE}.}
}}

\item{control.method}{A named list of method-specific tuning parameters.
Common entries (used by both Stan and iStEM):
\describe{
\item{\code{cores}}{Number of CPU cores for parallel chains
(Stan) or ignored (iStEM). Default: the number of
\code{chains}.}
\item{\code{vis}}{Logical; if \code{TRUE} (default), prints progress
information to the console.}
\item{\code{seed}}{Random seed for reproducibility.
Default: a random integer.}
}
Stan-specific entries:
\describe{
\item{\code{chains}}{Number of MCMC chains (default: 2).}
\item{\code{iter}}{Total iterations per chain (default: 5000).}
\item{\code{warmup}}{Warmup/burn-in iterations per chain
(default: \code{iter / 2}).}
\item{\code{thin}}{Thinning interval (default: 1).}
\item{\code{init}}{Initial values: \code{"random"} (default)
for uniform(-2, 2) initialization, or a list of initial
values per chain.}
\item{\code{algorithm}}{MCMC algorithm: \code{"HMC"} (default),
\code{"HMC"}, or \code{"Fixed_param"}.}
\item{\code{adapt_delta}}{Target average acceptance probability
(NUTS; default: 0.95). Forced-choice models often have
challenging posterior geometries; consider increasing to
0.9 or higher if divergent transitions appear.}
\item{\code{max_treedepth}}{Maximum tree depth (NUTS; default: 10).
Increase if "max treedepth exceeded" warnings appear.}
\item{\code{stepsize}}{Initial step size for the leapfrog
integrator (auto-tuned by Stan if not set).}
\item{\code{int_time}}{Total integration time for HMC trajectories
(only when \code{algorithm = "HMC"}).}
\item{\code{metric}}{Mass matrix type: \code{"unit_e"},
\code{"diag_e"} (default), or \code{"dense_e"}.}
\item{\code{adapt_engaged}}{Logical; if \code{TRUE} (default),
warmup adaptation is enabled.}
\item{\code{adapt_init_buffer}, \code{adapt_term_buffer},
\code{adapt_window}}{Warmup adaptation scheduling parameters
(defaults: 25, 50, 25).}
}
iStEM-specific entries:
\describe{
\item{\code{M}}{Number of burn-in batches retained for Geweke
convergence diagnosis (default: 10; must be \eqn{\ge 2}).}
\item{\code{B}}{Batch size: MCMC iterations per batch (default: 20).}
\item{\code{burnin.maxitr}}{Maximum burn-in batches (default: 100).}
\item{\code{maxitr}}{Maximum total batches (default: 2000).}
\item{\code{eps1}}{Geweke z-score convergence threshold for
burn-in (default: 1.5).}
\item{\code{eps2}}{Monte Carlo error tolerance for the final
chain (default: 0.4).}
\item{\code{frac1}, \code{frac2}}{Fractions for the Geweke
diagnostic (defaults: 0.1, 0.5).}
\item{\code{corr.optim.maxit}}{Maximum L-BFGS-B iterations for the
constrained unit-diagonal correlation update (default: 50).}
\item{\code{optim.maxit}}{Maximum L-BFGS-B iterations per item
(default: 50).}
\item{\code{fix.corr}}{Logical; fix correlations to identity
(default: \code{FALSE}).}
\item{\code{estimate.se}}{Logical; compute standard errors from
final MC chain (default: \code{TRUE}).}
\item{\code{a.lower}, \code{a.upper}}{Bounds on \eqn{a_{id}}
during optimization (defaults: 1e-4, 6).}
\item{\code{b.lower}, \code{b.upper}}{Bounds on \eqn{b_i}
during optimization (defaults: -6, 6).}
\item{\code{c.lower}, \code{c.upper}}{Bounds on \eqn{c_i} for
3PL/4PL models. Defaults from prior support.}
\item{\code{d.lower}, \code{d.upper}}{Bounds on \eqn{d_i} for
4PL models. Defaults from prior support.}
}}
}
\value{
An object of class \code{"FCMIRT"} containing:
\describe{
\item{\code{npar}}{Number of free parameters.}
\item{\code{method}}{\code{"stan"} or \code{"iStEM"}.}
\item{\code{theta}}{List with \code{est}, \code{se}, \code{Rhat}
(\eqn{N \times D}); person trait estimates.}
\item{\code{par}}{List with \code{est}, \code{se}, \code{Rhat},
\code{free} (\eqn{I \times (D+3)}); item parameters
\code{a1..aD, b, c, d}.}
\item{\code{Corr}}{List with \code{est}, \code{se}, \code{Rhat}
(\eqn{D \times D}).}
\item{\code{block.items}, \code{patterns}, \code{patterns.total},
\code{response}}{Block structure and response data.}
\item{\code{logLik}}{Marginal log-likelihood (class \code{"logLik"}).}
}
}
\description{
Fits a forced-choice variant of the multidimensional item response theory
model to comparative ranking data. In forced-choice (FC) formats,
respondents compare items within blocks rather than rating each item in
isolation. The FCMIRT model couples an item-level MIRT endorsement model
with a block-level ranking mechanism to recover latent trait estimates
from comparative responses.
}
\section{Model Specification}{


\strong{Item-level endorsement.}
Each statement \eqn{i = 1, \dots, I} is governed by a MIRT model
(1PL through 4PL). For a person with trait \eqn{\boldsymbol{\theta}_j},
the probability of endorsing statement \eqn{i} in isolation is:
\deqn{
  P_i(\boldsymbol{\theta}_j) =
  c_i + (d_i - c_i) \times
  \frac{1}{1 + \exp\bigl[-\bigl(\sum_{d=1}^{D} a_{id}\,\theta_{jd} - b_i\bigr)\bigr]}.
}

\strong{Block-level ranking.}
Let block \eqn{b = 1, \dots, B} consist of items
\eqn{\{i_1, i_2, \dots, i_{K_b}\}} and define the implemented
statement utility as the logit of the item-level endorsement probability,
\deqn{
  u_i(\boldsymbol{\theta}_j) =
  \log\left[
    \frac{P_i(\boldsymbol{\theta}_j)}
         {1-P_i(\boldsymbol{\theta}_j)}
  \right].
}
The probability that an examinee
produces the ranking \eqn{i_{(1)} \succ i_{(2)} \succ \dots \succ i_{(K_b)}}
(read as "\eqn{i_{(1)}} is preferred over \eqn{i_{(2)}} over ...") is:
\deqn{
  P\bigl(i_{(1)} \succ \dots \succ i_{(K_b)} \mid \boldsymbol{\theta}_j\bigr) =
  \prod_{m=1}^{K_b-1}
  \frac{
    \exp\{u_{i_{(m)}}(\boldsymbol{\theta}_j)\}
  }{
    \sum_{r=m}^{K_b} \exp\{u_{i_{(r)}}(\boldsymbol{\theta}_j)\}
  }.
}
The Stan code evaluates the same choice kernel as
\eqn{\log P_i + \sum_{h \ne i}\log(1-P_h)} within each remaining set,
which is algebraically equivalent to a softmax over
\eqn{\mathrm{logit}(P_i)}.

\strong{Partial rankings (MOLE / PICK).} For MOLE (most-least) data, only
the best and worst items in each block are identified; for PICK data,
only the best item. The probability of a partial ranking is obtained by
marginalizing (summing) the full-ranking probabilities over all
completions consistent with the partial constraint; the C++ probability
helper returns probabilities normalized over the observed RANK/MOLE/PICK
patterns for each block.

\strong{Identification constraint (2PLM-RANK).} Block intercepts
\eqn{b_i} are constrained to sum to zero within each block:
\deqn{
  \sum_{i \in \text{block } b} b_i = 0, \qquad b = 1, \dots, B,
}
so that only \eqn{K_b - 1} difficulties are freely estimated per block.
}

\section{Estimation Methods}{


\describe{
\item{\strong{Stan} (\code{method = "stan"}):}{
Full Bayesian inference via HMC. The forced-choice likelihood is
evaluated over the ranking pattern probabilities. By default,
\code{a.sigma} is set to \code{1.0} (wider prior) to avoid spurious
shrinkage of within-block slopes toward equality.}
\item{\strong{iStEM} (\code{method = "iStEM"}):}{
Improved Stochastic EM with finite-grid block Gibbs person-sampling
and item optimization (L-BFGS-B). Block-level identification
constraints are enforced during each item-parameter update.}
}
}

\examples{
# Simulate forced-choice data
sim <- sim.data.FCMIRT(N.person = 20, N.block = 3, I.block = 2,
                       D = 2, model = "m2pl", fc.type = "RANK")

# Fit via iStEM
fit <- fit.FCMIRT(sim$data, model = "m2pl",
                  Q.matrix = sim$Q.matrix,
                  block.items = sim$block.items,
                  D = 2, fc.type = "RANK",
                  method = "iStEM",
                  control.method = list(
                    vis = FALSE, seed = 123,
                    M = 2, B = 2, burnin.maxitr = 2,
                    maxitr = 3, eps1 = 10, eps2 = 10,
                    estimate.se = FALSE))

# Examine trait recovery
cor(fit$theta$est, sim$theta)

# Fit indices (nominal binary expansion)
gof <- get.fit.index(fit)
summary(gof)

}
\references{
Brown, A., & Maydeu-Olivares, A. (2011). Item response modeling of
forced-choice questionnaires. \emph{Educational and Psychological
Measurement}, 71(3), 460--502. \doi{10.1177/0013164410375112}

Luce, R. D. (1959). \emph{Individual choice behavior: A theoretical
analysis}. Wiley.

Plackett, R. L. (1975). The analysis of permutations. \emph{Journal of
the Royal Statistical Society: Series C}, 24(2), 193--202.
}
\seealso{
\code{\link{sim.data.FCMIRT}}, \code{\link{get.fit.index.FCMIRT}},
\code{\link{logLik.FCMIRT}}, \code{\link{fit.MIRT}}
}
