Introduction to ForceChoice

Overview

The ForceChoice package provides a unified framework for fitting, simulating, and evaluating forced-choice and traditional item response theory (IRT) models. It supports eight model families with full Bayesian estimation via Stan (Hamiltonian Monte Carlo), a fast iterative stochastic EM (iStEM) algorithm, and a deterministic EM backend for FCGDINA.

This vignette is intended as a reproducible user guide rather than a complete methodological review. Model formulas and object components are described using the parameterization implemented in ForceChoice. The reference list gives DOI links where available so readers can verify the underlying psychometric sources.

Model Families

Family	Description	Data Type
MIRT	Multidimensional IRT (1PL–4PL)	Binary
MGPCM	Multidimensional Generalized Partial Credit	Polytomous
MGGUM	Multidimensional Generalized Graded Unfolding	Polytomous
FCMIRT	Forced-Choice MIRT with Luce–Plackett ranking	Ranking/MOLE/PICK
FCGGUM	Forced-Choice GGUM with ranking	Ranking/MOLE/PICK
TIRT	Thurstonian IRT with pairwise probit	Ranking/MOLE/PICK
FCDCM	Forced-Choice Diagnostic Classification	Paired comparison
FCGDINA	Forced-Choice GDINA diagnostic model	Ranking/MOLE/PICK

The forced-choice families accept full ranking ("RANK"), most-least ("MOLE"), or best-only ("PICK") data when supported by the corresponding fitting function. FCDCM is the exception: it is a paired-comparison model where every block contains exactly two statements.

Model and Backend Coverage

Family	Main response process	Backends	Primary outputs
MIRT	Dominance IRT for binary items	Stan, iStEM	item parameters, \(\theta\), factor correlations
MGPCM	Dominance IRT for ordered categories	Stan, iStEM	category intercepts, \(\theta\), factor correlations
MGGUM	Ideal-point/unfolding IRT	Stan, iStEM	slopes, locations, thresholds, \(\theta\)
FCMIRT	MIRT endorsement plus Luce–Plackett ranking	Stan, iStEM	item parameters, \(\theta\), block fit
FCGGUM	GGUM endorsement plus Luce–Plackett ranking	Stan, iStEM	unfolding item parameters, \(\theta\), block fit
TIRT	Pairwise Thurstonian probit comparisons	Stan, iStEM	loadings, uniquenesses, \(\theta\)
FCDCM	Higher-order DCM for two-statement FC blocks	Stan, iStEM	attribute profiles, higher-order parameters
FCGDINA	GDINA/DINA/DINO/ACDM plus FC ranking	Stan, iStEM, EM	attribute profiles, CDM item parameters

Estimation Backends

Stan (method = "stan"): Full Bayesian inference via HMC/NUTS. Provides posterior means, standard deviations, and R-hat convergence diagnostics. Suitable for final inference with small-to-moderate datasets.
iStEM (method = "iStEM"): Iterative Stochastic EM combining Metropolis-within-Gibbs person sampling with L-BFGS-B item optimization. Scales to large datasets. Convergence monitored via Geweke diagnostics.
EM (method = "EM"): Deterministic posterior-weight EM for FCGDINA.

Installation

# Development version from GitHub
remotes::install_github("Naidantu/ForceChoice")

Quick Start: Traditional IRT

Simulating and Fitting a MIRT Model

library(ForceChoice)

# Simulate binary response data
sim <- sim.data.MIRT(N = 20, I = 6, D = 2, model = "2PL")

# Fit via iStEM
fit <- fit.MIRT(sim$data, model = "2PL", D = 2,
                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 results
print(fit)
summary(fit)

# Item parameter estimates (first 6 items)
head(coef(fit))

# Factor correlation matrix
fit$Corr$est

# Trait recovery
diag(cor(fit$theta$est, sim$theta))

Goodness-of-Fit Evaluation

# Compute comprehensive fit indices
gof <- get.fit.index(fit)

# Summary of fit indices
summary(gof)

# Extract specific indices
gof$M2       # Limited-information M2 statistic
gof$RMSEA    # RMSEA with 90% CI
gof$CFI      # Comparative Fit Index
gof$TLI      # Tucker-Lewis Index
gof$SRMSR    # Standardized Root Mean Square Residual
gof$AIC      # Akaike Information Criterion
gof$BIC      # Bayesian Information Criterion

Forced-Choice Modeling

Simulating and Fitting FCMIRT

Forced-choice data uses ranking strings (e.g., "2>1>3" means item 2 is preferred over item 1 over item 3).

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

# The data contains ranking strings
head(sim$data)

# Fit: block.items and fc.type are auto-detected
fit <- fit.FCMIRT(sim$data, model = "2PL", D = 2,
                  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))

# Trait recovery
cor(fit$theta$est, sim$theta)

# Goodness-of-fit (uses nominal binary expansion)
gof <- get.fit.index(fit)
summary(gof)

Thurstonian IRT (TIRT)

sim <- sim.data.TIRT(N.person = 20, N.block = 3, I.block = 2,
                     D = 2, fc.type = "RANK")

fit <- fit.TIRT(sim$data, Q.matrix = sim$Q.matrix,
                block.items = sim$block.items,
                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))

# Structural parameters: loadings and uniquenesses
head(coef(fit))

# Gamma matrix (pairwise intercepts)
fit$gamma.matrix$est[1:5, 1:5]

Forced-Choice DCM (FCDCM)

sim <- sim.data.FCDCM(N.person = 20, N.block = 3, D = 2,
                      dcm.type = "DINA")

fit <- fit.FCDCM(sim$data, Q.matrix = sim$Q.matrix,
                 block.items = sim$block.items,
                 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))

# Posterior attribute mastery probabilities
head(fit$alpha$prob)

# Attribute mastery proportions
colMeans(fit$alpha$prob > 0.5)

# Higher-order IRT parameters
fit$delta$est

Forced-Choice GDINA (FCGDINA)

FCGDINA is the diagnostic-classification counterpart for multi-statement forced-choice blocks. Unlike FCDCM, which is restricted to paired comparisons under a higher-order DCM structure, FCGDINA supports "GDINA", "DINA", "DINO", and "ACDM" statement-level models and can be fitted to full ranking, most-least, or best-only forced-choice data.

sim <- sim.data.FCGDINA(N.person = 20, N.block = 2, I.block = 2,
                        D = 2, model = "GDINA", fc.type = "RANK")

fit <- fit.FCGDINA(sim$data, Q.matrix = sim$Q.matrix,
                   block.items = sim$block.items, model = "GDINA",
                   fc.type = sim$fc.type, method = "EM",
                   control.method = list(vis = FALSE, seed = 123,
                                         maxitr = 2,
                                         estimate.se = FALSE))

# Posterior attribute mastery probabilities
head(fit$alpha$est)

# CDM item-parameter estimates
coef(fit, type = "delta")

Working with Polytomous Data

MGPCM (Dominance)

sim <- sim.data.MGPCM(N = 20, I = 6, D = 2, length.poly = 4)

fit <- fit.MGPCM(sim$data, D = 2, 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))

# Category threshold parameters
coef(fit)

MGGUM (Ideal-Point / Unfolding)

sim <- sim.data.MGGUM(N = 20, I = 6, D = 2, length.poly = 4)

fit <- fit.MGGUM(sim$data, D = 2, 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))

coef(fit)

Solution Rotation

For exploratory MIRT analyses, post-hoc rotation helps achieve simple structure:

fit_rot <- rotate.MIRT(fit, rotate = "oblimin")

# Compare original and rotated loadings
head(coef(fit))
head(coef(fit_rot))

Diagnostic Plots

# Item characteristic curves (ICC)
plot(fit, type = "icc", items = 1:8)

# Person parameter distributions
plot(fit, type = "theta")

# iStEM convergence trace
plot(fit, type = "trace")

Estimation Control

Common Parameters (top-level)

fit <- fit.MIRT(data, model = "2PL", D = 2, method = "iStEM",
                control.method = list(
                  vis = FALSE,
                  seed = 123,      # Reproducibility
                  M = 2, B = 2, burnin.maxitr = 2,
                  maxitr = 3, eps1 = 10, eps2 = 10,
                  estimate.se = FALSE))

Stan-Specific Control

# stan code, long time
fit <- fit.MIRT(data, model = "2PL", D = 2, method = "stan",
                control.method = list(chains = 1, iter = 200,
                                      warmup = 100, cores = 1,
                                      seed = 123))

iStEM-Specific Control (advanced)

fit <- fit.MIRT(data, model = "2PL", D = 2, method = "iStEM",
                control.method = list(
                  seed = 123,
                  M = 2,            # Burn-in batches for convergence
                  B = 2,            # Iterations per batch
                  burnin.maxitr = 2,
                  maxitr = 3,
                  eps1 = 1.5,       # Geweke convergence threshold
                  eps2 = 0.4        # MC error tolerance
                ))

References

Brown, A., & Maydeu-Olivares, A. (2011). Item response modeling of forced-choice questionnaires. Educational and Psychological Measurement, 71(3), 460–502. https://doi.org/10.1177/0013164410375112
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199. https://doi.org/10.1007/s11336-011-9207-7
Huang, H.-Y. (2022). Diagnostic classification model for forced-choice items and noncognitive tests. Educational and Psychological Measurement, 83(1), 146–180. https://doi.org/10.1177/00131644211069906
Lee, P., Joo, S.-H., Stark, S., & Chernyshenko, O. S. (2018). GGUM-RANK statement and person parameter estimation with multidimensional forced choice triplets. Applied Psychological Measurement, 43(3), 226–240. https://doi.org/10.1177/0146621618768294
Luce, R. D. (1959). Individual choice behavior: A theoretical analysis. Wiley.
Maydeu-Olivares, A., & Joe, H. (2005). Limited- and full-information estimation and goodness-of-fit testing in 2^n contingency tables: A unified framework. Journal of the American Statistical Association, 100(471), 1009–1020. https://doi.org/10.1198/016214504000002069
Maydeu-Olivares, A., & Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71(4), 713–732. https://doi.org/10.1007/s11336-005-1295-9
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159–176. https://doi.org/10.1177/014662169201600206
Plackett, R. L. (1975). The analysis of permutations. Journal of the Royal Statistical Society: Series C (Applied Statistics), 24(2), 193–202. https://doi.org/10.2307/2346567
Reckase, M. D. (2009). Multidimensional Item Response Theory. Springer. https://doi.org/10.1007/978-0-387-89976-3
Roberts, J. S., Donoghue, J. R., & Laughlin, J. E. (2000). A general item response theory model for unfolding unidimensional polytomous responses. Applied Psychological Measurement, 24(1), 3–32. https://doi.org/10.1177/01466216000241001
Zheng, C., Liu, J., Li, Y., Xu, P., Zhang, B., Wei, R., Zhang, W., Liu, B., & Huang, J. (2024). A 2PLM-RANK multidimensional forced-choice model and its fast estimation algorithm. Behavior Research Methods, 56(6), 6363–6388. https://doi.org/10.3758/s13428-023-02315-x