---
title: "ForceChoice Model Theory and Selection Guide"
author: "Haijiang Qin & Lei Guo"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Model Theory and Selection Guide}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Overview

This vignette describes the theoretical foundations of the eight model
families in **ForceChoice** and provides guidance for model selection.
Core models support Stan (full Bayesian HMC) and iStEM (iterative Stochastic
EM); FCGDINA additionally supports deterministic EM.

The equations below use the parameterizations implemented in the package.
For several families, especially MGPCM and MGGUM, this means the notation is
not a verbatim copy of the original unidimensional source article. The cited
papers identify the model family and response-process assumptions; the
implementation details are those exposed by the corresponding `fit.*()`
documentation.

## Traditional (Single-Stimulus) Models

### MIRT: Multidimensional Item Response Theory

The MIRT model extends unidimensional IRT to multiple latent dimensions.
For binary response $Y_{ij} \in \{0, 1\}$, the item response function is:

$$
P(Y_{ij}=1 \mid \boldsymbol{\theta}_j) = c_i + (d_i - c_i) \times
\frac{1}{1 + \exp[-( \mathbf{a}_i' \boldsymbol{\theta}_j - b_i)]}
$$

where $\mathbf{a}_i = (a_{i1}, \ldots, a_{iD})'$ is the discrimination
vector, $b_i$ is the difficulty (intercept), $c_i$ is the lower asymptote
(pseudo-guessing), and $d_i$ is the upper asymptote.

**Model variants:**

| Model | Free Parameters | Fixed |
|---|---|---|
| M1PL (Rasch) | $b_i$ | $a_{id} = q_{id}$, $c_i=0$, $d_i=1$ |
| M2PL | $a_{id}, b_i$ | $c_i=0$, $d_i=1$ |
| M3PL | $a_{id}, b_i, c_i$ | $d_i=1$ |
| M4PL | $a_{id}, b_i, c_i, d_i$ | - |

**When to use:** Binary response data where items vary in discrimination
and difficulty. Dominance model: higher trait always increases endorsement
probability.

**Key reference:** Reckase, M. D. (2009). *Multidimensional Item Response
Theory*. Springer.

---

### MGPCM: Multidimensional Generalized Partial Credit Model

For polytomous responses $Y_{ij} \in \{0, 1, \ldots, K_i-1\}$,
**ForceChoice** implements a multidimensional softmax parameterization:

$$
P(Y_{ij} = k \mid \boldsymbol{\theta}_j) =
\frac{\exp\{k\eta_{ij} + d_{ik}\}}
{\sum_{r=0}^{K_i-1}\exp\{r\eta_{ij} + d_{ir}\}},
\qquad
\eta_{ij}=\sum_{d=1}^{D}a_{id}\theta_{jd}.
$$

The $d_{ik}$ values are category intercepts rather than cumulative step
difficulties; $d_{i0} \equiv 0$ is fixed for identification. This
implementation nests the generalized partial credit model of Muraki (1992)
in the unidimensional case.

**When to use:** Polytomous (ordered category) data such as Likert-type
responses. A dominance model: higher trait implies higher category.

**Key reference:** Muraki, E. (1992). A generalized partial credit model:
Application of an EM algorithm. *Applied Psychological Measurement*,
*16*(2), 159--176.

---

### MGGUM: Multidimensional Generalized Graded Unfolding Model

The MGGUM is an ideal-point (unfolding) model where endorsement is governed
by distance between the person and item locations. In **ForceChoice**, define

$$
r_{ij} =
\left[
  \sum_{d=1}^{D} a_{id}^2(\theta_{jd}-\delta_{id})^2
\right]^{1/2},
\qquad
S_i = \sum_{d=1}^{D} a_{id},
$$

and let $\tau_{i0}=0$, $\psi_{ik}=S_i\sum_{v=0}^{k}\tau_{iv}$, and
$M_i=2K_i-1$. Then

$$
P(Y_{ij}=k \mid \boldsymbol{\theta}_j)=
\frac{
  \exp\{kr_{ij}-\psi_{ik}\}+
  \exp\{(M_i-k)r_{ij}-\psi_{ik}\}
}{
  \sum_{r=0}^{K_i-1}
  [\exp\{rr_{ij}-\psi_{ir}\}+
   \exp\{(M_i-r)r_{ij}-\psi_{ir}\}]
}.
$$

The signed Q-matrix ($q_{id} \in \{-1, 0, 1\}$) indicates active
dimensions and the side of the latent continuum on which the item location
lies. The sign controls $\delta_{id}$, not the sign of the discrimination
parameter.

**When to use:** Attitude/ preference measurement where both ends of the
trait continuum can lead to disagreement (e.g., "I like moderate
temperatures": both extreme cold-lovers and heat-lovers disagree).

**Key references:**
- 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.
- Tu, N., Zhang, B., Angrave, L., Sun, T., & Neuman, M. (2023). Estimating
  the Multidimensional Generalized Graded Unfolding Model with covariates
  using a Bayesian approach. *Journal of Intelligence*, *11*, 163.

---

## Forced-Choice (Comparative) Models

Forced-choice (FC) formats ask respondents to compare items within blocks
rather than rating items in isolation. This can reduce response-style
biases such as acquiescence, social desirability, and extreme responding,
but it introduces comparative/ipsative information with model-specific
identification requirements.

### FCMIRT: Forced-Choice MIRT

Item-level endorsement follows the MIRT model (1PL--4PL). At the block
level, **ForceChoice** uses a sequential Luce--Plackett ranking kernel with
utilities equal to the logit of the item-level endorsement probability:

$$
u_i(\boldsymbol{\theta}_j) =
\log\left[\frac{P_i(\boldsymbol{\theta}_j)}
{1-P_i(\boldsymbol{\theta}_j)}\right].
$$

$$
P(i_{(1)} \succ \cdots \succ i_{(K_b)} \mid \boldsymbol{\theta}_j) =
\prod_{m=1}^{K_b-1}
\frac{\exp\{u_{i_{(m)}}(\boldsymbol{\theta}_j)\}}
{\sum_{r=m}^{K_b}\exp\{u_{i_{(r)}}(\boldsymbol{\theta}_j)\}}.
$$

**Identification:** Block difficulties sum to zero: $\sum_{i \in b} b_i = 0$.

**FC types:**
- **RANK**: Full ranking of all items within each block
- **MOLE**: Only most and least preferred items identified
- **PICK**: Only most preferred item identified

**When to use:** Forced-choice questionnaires with dominance items
(personality, vocational preferences).

**Key reference:** Zheng, C., Liu, J., Li, Y., et al. (2024). A 2PLM-RANK
multidimensional forced-choice model and its fast estimation algorithm.
*Behavior Research Methods*, *56*, 6363--6388.

**Ranking-model references:** Luce (1959) and Plackett (1975).

---

### FCGGUM: Forced-Choice GGUM

Combines the GGUM ideal-point model at the item level with the same
sequential Luce--Plackett ranking kernel used by FCMIRT. The block-level
ranking probability uses the logit of the GGUM endorsement probability as
the statement utility.

**Identification:** Same block-level sum-to-zero constraint as FCMIRT.
Signed Q-matrix indicates both dimension membership and statement direction.

**When to use:** Forced-choice questionnaires with ideal-point/unfolding
items (e.g., personality items where both extremes disagree).

**Key reference:** 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.

---

### TIRT: Thurstonian IRT for Forced-Choice

The TIRT model works directly with pairwise comparisons via a probit link.
For each pair $(i, k)$ in block $b$, the latent comparative judgment is:

$$
y_{j,ik}^* = -\gamma_{ik} + \lambda_i \mathbf{q}_i'\boldsymbol{\theta}_j -
\lambda_k \mathbf{q}_k'\boldsymbol{\theta}_j + \varepsilon_{j,i} - \varepsilon_{j,k}
$$

with $\varepsilon_{j,i} \sim N(0, \psi_i^2)$. The probability of preferring
$i$ over $k$ is:

$$
P(Y_{j,ik}=1 \mid \boldsymbol{\theta}_j) =
\Phi\left(\frac{-\gamma_{ik} + \lambda_i\mathbf{q}_i'\boldsymbol{\theta}_j -
\lambda_k\mathbf{q}_k'\boldsymbol{\theta}_j}
{\sqrt{\psi_i^2 + \psi_k^2}}\right)
$$

**Identification constraints:**
- **Case A** ($I_b=2, D>2$): All $\psi_i^2 = 0.5$
- **Case B** ($D=2, I_b=2$): First block $\lambda_i = 0.80$, all $\psi_i^2 = 0.5$
- **General case**: Last item per block has $\psi_i^2 = 1.0$

**When to use:** When you want a factor-analytic decomposition of
forced-choice data with factor loadings and uniquenesses, rather than the
item-level endorsement approach of FCMIRT.

**Key reference:** Brown, A., & Maydeu-Olivares, A. (2011). Item response
modeling of forced-choice questionnaires. *Educational and Psychological
Measurement*, *71*(3), 460--502.

---

### FCDCM: Forced-Choice Diagnostic Classification Model

A higher-order cognitive diagnostic model for paired-comparison FC data.
Each block contains exactly two statements. The discrete attribute mastery
profile $\boldsymbol{\alpha}_j$ is governed by a continuous higher-order
trait $\theta_j$:

$$
P(\alpha_{jd} = 1 \mid \theta_j) =
\frac{1}{1 + \exp[-(\delta_{1d}\theta_j - \delta_{1d}\delta_{0d})]}
$$

**Condensation rules:**
- **DINA (conjunctive):** $\zeta_{ij} = \prod_{d: q_{id}=1} \alpha_{jd}$
  (all required attributes must be mastered)
- **DINO (disjunctive):** $\zeta_{ij} = 1 - \prod_{d: q_{id}=1} (1-\alpha_{jd})$
  (at least one required attribute mastered)

Block response probabilities are parameterized by $\eta_{0b}$ (base rate)
and $\eta_{ABb}$ (joint-mastery bonus), with exact marginalization over
all $2^D$ attribute profiles.

**When to use:** Diagnostic classification with forced-choice format -
classifying respondents into discrete attribute mastery profiles while
avoiding response biases via the FC format.

**Key references:**
- Huang, H.-Y. (2022). Diagnostic classification model for forced-choice
  items and noncognitive tests. *Educational and Psychological Measurement*,
  *83*(1), 146--180.
- Zhu, Y.-A., Xu, J., Wang, D., Li, X., Cai, Y., & Tu, D. (2024). A
  ranking forced choice diagnostic classification model for psychological
  assessment. *British Journal of Mathematical and Statistical Psychology*,
  *78*, 617--646.

---

### FCGDINA: Forced-Choice GDINA

FCGDINA combines a statement-level cognitive diagnostic model with a
forced-choice block response model. The statement model may be GDINA, DINA,
DINO, or ACDM. For a person with attribute profile
$\boldsymbol{\alpha}_j \in \{0,1\}^D$, a statement-level endorsement
probability is computed from the selected CDM design matrix. The block
probabilities are then obtained by the same Luce--Plackett transformation
used by FCMIRT and FCGGUM.

The likelihood marginalizes over all $2^D$ attribute profiles:

$$
P(\mathbf{Y}_j) =
\sum_{c=1}^{2^D}\pi_c
\prod_{b=1}^{B}
P(Y_{jb}\mid \boldsymbol{\alpha}_c, \boldsymbol{\delta}, \text{block}_b).
$$

**Important identification note:** Forced-choice data identify within-block
utility contrasts, not absolute single-statement endorsement levels. In the
Stan implementation, the nonidentified null space of the delta effects is
removed and the reported coefficients use the package's identifying
convention. EM and iStEM provide probability-scale CDM estimates under the
same block response likelihood.

**When to use:** Diagnostic classification with ranking, most-least, or
best-only forced-choice blocks, especially when the statement-level CDM
needs GDINA/ACDM flexibility rather than the paired-comparison FCDCM
structure.

**Key references:** de la Torre (2011) for GDINA; Luce (1959) and Plackett
(1975) for the ranking kernel.

---

## Model Selection Guide

| Criterion | Recommended Model |
|---|---|
| Binary responses, dominance | `fit.MIRT()` |
| Polytomous (Likert), dominance | `fit.MGPCM()` |
| Polytomous, ideal-point | `fit.MGGUM()` |
| FC ranking, dominance items | `fit.FCMIRT()` |
| FC ranking, ideal-point items | `fit.FCGGUM()` |
| FC ranking, factor-analytic approach | `fit.TIRT()` |
| FC paired comparison, diagnostic classification | `fit.FCDCM()` |
| FC ranking/MOLE/PICK, diagnostic classification | `fit.FCGDINA()` |

## Estimation Method Selection

| Criterion | Recommended |
|---|---|
| Final inference, moderate N | Stan |
| Large datasets, exploratory analysis | iStEM |
| Need posterior SDs and R-hat | Stan |
| Need fast point estimates + SEs | iStEM |
| Complex model, multi-chain diagnostics | Stan |
| Very large N (> 5000) or many items | iStEM |

---

## Goodness-of-Fit Framework

All models use the limited-information M2 framework
(Maydeu-Olivares & Joe, 2005, 2006):

- **M2 statistic**: Tests the null of exact model fit using
  univariate and bivariate margins.
- **RMSEA**: Approximate fit index with 90% confidence interval.
- **CFI / TLI**: Comparative fit against an independence baseline.
- **SRMSR**: Standardized root mean square residual.
- **Yen's Q3**: Local dependence diagnostics from residual correlations.

For FC models, within-block pairwise residuals are excluded (by design,
items in the same block are locally dependent through the ranking
mechanism).

## 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>

Tu, N., Zhang, B., Angrave, L., Sun, T., & Neuman, M. (2023). Estimating the
multidimensional generalized graded unfolding model with covariates using a
Bayesian approach. *Journal of Intelligence*, *11*(8), 163.
<https://doi.org/10.3390/jintelligence11080163>

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>

Zhu, Y.-A., Xu, J., Wang, D., Li, X., Cai, Y., & Tu, D. (2024). A ranking
forced choice diagnostic classification model for psychological assessment
using forced choice questionnaires. *British Journal of Mathematical and
Statistical Psychology*, *78*(2), 617--646.
<https://doi.org/10.1111/bmsp.12376>
