Unifying Differential Item Functioning in Factor Analysis for Categorical Data Under a Discretization of a Normal Variant

Abstract

The multiple-group categorical factor analysis (FA) model and the graded response model (GRM) are commonly used to examine polytomous items for differential item functioning to detect possible measurement bias in educational testing. In this study, the multiple-group categorical factor analysis model (MC-FA) and multiple-group normal-ogive GRM models are unified under the common framework of discretization of a normal variant. We rigorously justify a set of identified parameters and determine possible identifiability constraints necessary to make the parameters just-identified and estimable in the common framework of MC-FA. By doing so, the difference between categorical FA model and normal-ogive GRM is simply the use of two different sets of identifiability constraints, rather than the seeming distinction between categorical FA and GRM. Thus, we compare the performance on DIF assessment between the categorical FA and GRM approaches through simulation studies on the MC-FA models with their corresponding particular sets of identifiability constraints. Our results show that, under the scenarios with varying degrees of DIF for examinees of different ability levels, models with the GRM type of identifiability constraints generally perform better on DIF detection with a higher testing power. General guidelines regarding the choice of just-identified parameterization are also provided for practical use.

Appendix

1.1 A.1 Proof for Theorem 1

First, define $\mathcal{A}^{J}\equiv \{0,1,\cdots ,C-1\}^{J}$ be a J-dimensional space, which consists of $C^J$ unique patterns and forms the support for the observed response vector $\varvec{y}_i^{(g)}=(y_{i1}^{(g)},y_{i2}^{(g)},\dots ,y_{iJ}^{(g)})'$ for examinee i in group g. For each pattern $\varvec{s}\in {\mathcal {A}}^J$, let

$$\begin{aligned} x_{\varvec{s}}^{(g)}\equiv & {} \sum _{i=1}^{N_g} 1(\varvec{y}_i^{(g)}=\varvec{s}), \quad \hbox {satisfying}~ \sum _{\varvec{s}\in {\mathcal {A}}^J}x_{\varvec{s}}^{(g)}=N_g, \end{aligned}$$

(23)

which counts the number of examinees in group g having the response pattern $\varvec{s}$. We further define

$$\begin{aligned} p_{\varvec{s}}^{(g)}(\varvec{\zeta })\equiv & {} \int ^{\delta _{1,\left( y_{i1}^{(g)}+1\right) }^{(g)}}_{\delta _{1,y_{i1}^{(g)}}^{(g)}}\cdots \int ^{\delta _{J,\left( y_{iJ}^{(g)}+1\right) }^{(g)}}_{\delta _{J,y_{iJ}^{(g)}}^{(g)}} \phi \left( \varvec{z}_{i}^{(g)}|\varvec{0},\varvec{\Gamma }^{(g)}\right) dz_{iJ}^{(g)} \cdots dz_{i1}^{(g)}, \end{aligned}$$

(24)

where $(y_{i1}^{(g)},y_{i2}^{(g)},\cdots ,y_{iJ}^{(g)})'=\varvec{s}$ and $\phi (\cdot |\varvec{\mu },\varvec{\Sigma })$ denotes the multivariate normal density with mean $\varvec{\mu }$ and variance $\varvec{\Sigma }$. The quantities $\{p_{\varvec{s}}^{(g)}(\varvec{\zeta }):\varvec{s}\in {\mathcal {A}}^J\}$ divide the integration term in (6) into $C^J$ partitions subject to different response pattern $\varvec{s}$. That is, under (23) and (24), the likelihood in (6) can be written as

$$\begin{aligned} L(\varvec{\zeta }|\varvec{Y})= & {} \prod _g \prod _{i=1}^{N_g} \left[ \prod _{\varvec{s}\in {\mathcal {A}}^J} p_{\varvec{s}}^{(g)}(\varvec{\zeta })^{1(\varvec{y}_i^{(g)}=\varvec{s})}\right] \nonumber \\= & {} \prod _g \prod _{\varvec{s}\in {\mathcal {A}}^J} \left[ p_{\varvec{s}}^{(g)}(\varvec{\zeta })\right] ^{\sum _{i}1(\varvec{y}_i^{(g)}=\varvec{s})} = \prod _g \prod _{\varvec{s}\in {\mathcal {A}}^J} \left[ p_{\varvec{s}}^{(g)}(\varvec{\zeta })\right] ^{x_{\varvec{s}}^{(g)}}. \end{aligned}$$

(25)

Note that the functional form in (25) is identical to the probability structure of a multinomial distribution with $C^J$ possible categories, see similar arguments in San Martín and Rolin (2013) and San Martín et al. (2013). In particular, $p_{\varvec{s}}^{(g)}(\varvec{\zeta })$’s are functions of parameter $\varvec{\zeta }$, which do not depend on the observed response data, while $x_{\varvec{s}}^{(g)}$’s are sufficient statistics which are functions of the response data and do not depend on $\varvec{\zeta }$. As a result, that $L(\varvec{\zeta })=L(\varvec{\zeta }')$ for each of all possible $\{\varvec{y}_i^{(g)}\}$ leads to, for each of all possible $\{x_{\varvec{s}}^{(g)}: \varvec{s}\in {\mathcal {A}}^J, g=r,f\}$,

$$\begin{aligned} \log L(\varvec{\zeta }|\varvec{Y})- \log L(\varvec{\zeta }'|\varvec{Y}) = \sum _g\sum _{\varvec{s}\in {\mathcal {A}}^J} {x_{\varvec{s}}^{(g)}}\left( \log p_{\varvec{s}}^{(g)}(\varvec{\zeta })-\log p_{\varvec{s}}^{(g)}(\varvec{\zeta }')\right) = 0, \end{aligned}$$

which further implies

$$\begin{aligned} p_{\varvec{s}}^{(g)}(\varvec{\zeta })=p_{\varvec{s}}^{(g)}(\varvec{\zeta }'), \quad \hbox {for all}~\varvec{s}\in {\mathcal {A}}^J. \end{aligned}$$

(26)

Therefore, to prove Theorem 1, it suffices to check that (26) implies

$$\begin{aligned} (\varvec{\delta }(\varvec{\zeta }),\varvec{\alpha }(\varvec{\zeta }))=(\varvec{\delta }(\varvec{\zeta }'),\varvec{\alpha }(\varvec{\zeta }')) \end{aligned}$$

Step I: The equivalence of the (marginal) distributions of each item and each two-item pair:

For $J=2$, (24) involves two-dimensional integrals, i.e.,

$$\begin{aligned} p_{\varvec{s}}^{(g)}(\varvec{\zeta }) =\int _{{\delta _{1,s_1}^{(g)}}}^{{\delta _{1,s_1+1}^{(g)}}} \int _{{\delta _{2,s_2}^{(g)}}}^{{\delta _{2,s_2+1}^{(g)}}} \phi (\varvec{z}_{i}^{(g)}|\varvec{0},\varvec{\Gamma }^{(g)}) d z_{i2}^{(g)}d z_{i1}^{(g)}, \quad \hbox {where}~\varvec{s}=(s_1,s_2)\in {\mathcal {A}}^2, \end{aligned}$$

(27)

where $\varvec{\Gamma }^{(g)}$ is the correlation matrix. Clearly,

$$\begin{aligned} \sum _{s_2=0}^{C-1} p_{\varvec{s}}^{(g)}(\varvec{\zeta })= & {} \int _{{\delta _{1,s_1}^{(g)}}}^{{\delta _{1,s_1+1}^{(g)}}} \left\{ \sum _{s_2=0}^{C-1}\int _{{\delta _{2,s_2}^{(g)}}}^{{\delta _{2,s_2+1}^{(g)}}} \phi (\varvec{z}_{i}^{(g)}|\varvec{0},\varvec{\Gamma }^{(g)}) d z_{i2}^{(g)}\right\} d z_{i1}^{(g)}\\= & {} \int _{{\delta _{1,s_1}^{(g)}}}^{{\delta _{1,s_1+1}^{(g)}}} \left\{ \int _{-\infty }^{\infty } \phi (\varvec{z}_{i}^{(g)}|\varvec{0},\varvec{\Gamma }^{(g)}) d z_{i2}^{(g)}\right\} d z_{i1}^{(g)} \\= & {} \int _{{\delta _{1,s_1}^{(g)}}}^{{\delta _{1,s_1+1}^{(g)}}} \phi (z_{i1}^{(g)}|0,1) d z_{i1}^{(g)} \equiv p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta }), \quad s_1\in {\mathcal {A}}^1, \end{aligned}$$

where $p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta })$ reduces to the multinomial probability for a single item case by summing up the response in the second item. Therefore, given the condition (26), we have

$$\begin{aligned}&\quad p_{\varvec{s}}^{(g)}(\varvec{\zeta }) = p_{\varvec{s}}^{(g)}(\varvec{\zeta }'), \,\forall \varvec{s}=(s_1,s_2)'\in {\mathcal {A}}^2 \quad \Longrightarrow \quad \sum _{s_i=0}^{C-1}p_{\varvec{s}}^{(g)}(\varvec{\zeta })=\sum _{s_i=0}^{C-1}p_{\varvec{s}}^{(g)}(\varvec{\zeta }'), \quad i=1,2, \\&\quad \Longrightarrow \quad p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta }) = p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta }'), \quad p_{(\bullet ,s_2)}^{(g)}(\varvec{\zeta }) = p_{(\bullet ,s_2)}^{(g)}(\varvec{\zeta }'),\quad s_1,s_2\in {\mathcal {A}}^1. \end{aligned}$$

Therefore, for the general case of $J \ge 3$, the above result implies that the marginal distributions of each item and each two-item pair under $\varvec{\zeta }$ and $\varvec{\zeta }'$ are the same through summing up all other items.

Step II: For each item, the equivalence of its marginal distributions under $\varvec{\zeta }$ and $\varvec{\zeta }'$ leads to $\varvec{\delta }(\varvec{\zeta })=\varvec{\delta }(\varvec{\zeta }')$:

For any given item (without loss of generality, may assume the item to be item 1), we start from the equivalence of its marginal distributions under $\varvec{\zeta }$ and $\varvec{\zeta }'$, i.e.,

$$\begin{aligned} p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta }) = p_{(s_1,\bullet )}^{(g)}(\varvec{\zeta }'), \quad s_1\in {\mathcal {A}}^1. \end{aligned}$$

For item 1, its marginal distribution can be formulated using one-dimensional integral only, i.e.,

$$\begin{aligned} p_s^{(g)}(\varvec{\zeta }) =\int _{{\delta _{1,s}^{(g)}}}^{{\delta _{1,s+1}^{(g)}}} \phi (z_{i1}^{(g)}|0,1) d z_{i1}^{(g)} = \Phi (\delta _{1,s+1}^{(g)})-\Phi (\delta _{1,s}^{(g)}), \quad s\in {\mathcal {A}}^1, \end{aligned}$$

where $\Phi (\cdot )$ is the cumulative distribution function of standard normal distribution. Starting with $s=0$ in ${\mathcal {A}}^1$, we have

$$\begin{aligned} p_0^{(g)}(\varvec{\zeta })=p_0^{(g)}(\varvec{\zeta }') \quad \Longrightarrow \quad \Phi (\delta _{1,1}^{(g)})=\Phi ({\delta \,'}^{(g)}_{1,1}) \quad \Longrightarrow \quad \delta _{1,1}^{(g)}={\delta \,'}^{(g)}_{1,1}, \end{aligned}$$

since $\Phi (\cdot )$ is a monotone function, and $\Phi (\delta _{1,0}^{(g)})=\Phi ({\delta \,'}^{(g)}_{1,0})=\Phi (-\infty )=0$. Given $\delta _{1,1}^{(g)}={\delta \,'}^{(g)}_{1,1}$ and considering $p_s^{(g)}(\varvec{\zeta })$ for $s=1$, we have

$$\begin{aligned} p_1^{(g)}(\varvec{\zeta })=p_1^{(g)}(\varvec{\zeta }') \quad \Longrightarrow \quad \Phi (\delta _{1,2}^{(g)})-\Phi (\delta _{1,1}^{(g)})=\Phi ({\delta \,'}^{(g)}_{1,2})-\Phi ({\delta \,'}^{(g)}_{1,1}) \quad \Longrightarrow \quad \delta _{1,2}^{(g)}={\delta \,'}^{(g)}_{1,2}. \end{aligned}$$

Similar arguments can be sequentially applied for $s=2,3,\ldots ,C-1$, leading to the results that $\delta _{1,s}^{(g)}={\delta \,'}^{(g)}_{1,s}$ for $\forall ~s\in {\mathcal {A}}^1$, i.e., $\varvec{\delta }(\varvec{\zeta })=\varvec{\delta }(\varvec{\zeta }')$.

Step III: For each two-item pair, the equivalence of its marginal distributions under $\varvec{\zeta }$ and $\varvec{\zeta }'$ leads to $\varvec{\gamma }(\varvec{\zeta }) = \varvec{\gamma }(\varvec{\zeta }')$:

For any given two-item pair (without loss of generality, may assume the pair of items 1 and 2), we start from the equivalence of its marginal distributions under $\varvec{\zeta }$ and $\varvec{\zeta }'$, i.e.,

$$\begin{aligned} p_{(\varvec{s},\bullet )}^{(g)}(\varvec{\zeta }) = p_{(\varvec{s},\bullet )}^{(g)}(\varvec{\zeta }'), \quad \forall \varvec{s}=(s_1,s_2)' \in {\mathcal {A}}^2. \end{aligned}$$

Based on the results in Step II, we have $\varvec{\delta }(\varvec{\zeta }) = \varvec{\delta }(\varvec{\zeta }')$. We only need to check that $\varvec{\gamma }(\varvec{\zeta }) = \varvec{\gamma }(\varvec{\zeta }')$, i.e., $\gamma _{12}^{(g)}={\gamma '}^{(g)}_{12}$ for the pair of items 1 and 2. This part is proved by first assuming $\gamma _{12}^{(g)}=\gamma \ne \gamma '={\gamma '}^{\,(g)}_{12}$ but leading to a contradiction of (26). Hereafter, the superscript (g) is omitted if no confusion occurs. Without loss of generality, we assume $\gamma <\gamma '$. Rewrite (27) for $\varvec{s}=(0,0)$ as

$$\begin{aligned} p_{(0,0)}^{(g)}(\varvec{\zeta })= & {} \int _{-\infty }^{\delta _{1,1}} \int _{-\infty }^{\delta _{2,1}} \phi (z_{i2}^{(g)}|\gamma z_{i1}^{(g)},1-\gamma ^2) \phi (z_{i1}^{(g)}|,0,1) dz_{i2}^{(g)} dz_{i1}^{(g)}\\= & {} \int _{-\infty }^{\delta _{1,1}} \Phi \left( \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}\right) \phi (u|0,1)du, \end{aligned}$$

which implies

$$\begin{aligned} p_{(0,0)}^{(g)}(\varvec{\zeta })-p_{(0,0)}^{(g)}(\varvec{\zeta }')= & {} \int _{-\infty }^{\delta _{1,1}} \Phi \left( \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}\right) \phi (u|0,1)du \nonumber \\&-\int _{-\infty }^{\delta '_{1,1}} \Phi \left( \frac{\delta '_{2,1}-\gamma ' u}{\sqrt{1-\gamma '^2}}\right) \phi (u|0,1)du \nonumber \\= & {} \int _{-\infty }^{\delta _{1,1}} \left\{ \Phi \left( \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}\right) - \Phi \left( \frac{\delta _{2,1}-\gamma ' u}{\sqrt{1-\gamma '^2}}\right) \right\} \phi (u|0,1)du, \end{aligned}$$

(28)

$$\begin{aligned}= & {} \int ^{\infty }_{\delta _{1,1}} \left\{ \Phi \left( \frac{\delta _{2,1}-\gamma ' u}{\sqrt{1-\gamma '^2}}\right) - \Phi \left( \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}\right) \right\} \phi (u|0,1)du, \end{aligned}$$

(29)

due to $\delta _{1,1}=\delta '_{1,1}$ and $\delta _{2,1}=\delta '_{2,1}$. Let

$$\begin{aligned} b \equiv \frac{ \delta _{2,1} / \sqrt{1-\gamma ^2} - \delta _{2,1} / \sqrt{1-\gamma '^2} }{ \gamma / \sqrt{1-\gamma ^2} - \gamma ' / \sqrt{1-\gamma '^2} }. \end{aligned}$$

We derive the contradiction under two possible scenarios: (i) $\delta _{1,1} \le b$; (ii) $\delta _{1,1} > b$:

• (i) When $\delta _{1,1}\le b$,

  $$\begin{aligned} \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}<\frac{\delta _{2,1}-\gamma ' u}{\sqrt{1-\gamma '^2}}, \,\forall u<\delta _{1,1} \quad \begin{array}{c}(28) \\ \Longrightarrow \\ ~~\end{array} \quad p_{(0,0)}^{(g)}(\varvec{\zeta })-p_{(0,0)}^{(g)}(\varvec{\zeta }')<0. \end{aligned}$$

• (ii) When $\delta _{1,1} > b$,

  $$\begin{aligned} \frac{\delta _{2,1}-\gamma u}{\sqrt{1-\gamma ^2}}>\frac{\delta _{2,1}-\gamma ' u}{\sqrt{1-\gamma '^2}}, \,\forall u>\delta _{1,1} \quad \begin{array}{c}(29) \\ \Longrightarrow \\ ~~\end{array} \quad p_{(0,0)}^{(g)}(\varvec{\zeta })-p_{(0,0)}^{(g)}(\varvec{\zeta }')<0. \end{aligned}$$

Both scenarios lead to a contradiction of (26). Therefore, $\varvec{\gamma }(\varvec{\zeta })$ must be equal to $\varvec{\gamma }(\varvec{\zeta }')$ to avoid the contradiction, which completes the proof.

Step IV: $\varvec{\gamma }(\varvec{\zeta }) = \varvec{\gamma }(\varvec{\zeta }') \Longleftrightarrow \varvec{\alpha }(\varvec{\zeta }) = \varvec{\alpha }(\varvec{\zeta }')$

When $J \ge 3$, take three distinct items arbitrarily, say $j,k,\ell \in \{1,2,\ldots ,J\}$, $\varvec{\gamma }(\varvec{\zeta }) = \varvec{\gamma }(\varvec{\zeta }')$ implies

$$\begin{aligned} \left( \alpha _j^{(g)}\right) ^2= & {} \frac{\left( \alpha _j^{(g)}\alpha _k^{(g)}\right) \left( \alpha _j^{(g)}\alpha _{\ell }^{(g)}\right) }{\alpha _k^{(g)}\alpha _{\ell }^{(g)}} = \frac{\gamma _{jk}^{(g)}\gamma _{j\ell }^{(g)}}{\gamma _{k\ell }^{(g)}} = \frac{{\gamma '}^{(g)}_{jk}{\gamma '}^{(g)}_{j\ell }}{{\gamma '}^{(g)}_{k\ell }} = \frac{\left( {\alpha '}^{(g)}_j{\alpha '}^{(g)}_k\right) \left( {\alpha '}^{(g)}_j{\alpha '}^{(g)}_{\ell }\right) }{{\alpha '}^{(g)}_k{\alpha '}^{(g)}_{\ell }} = \left( {\alpha '}^{(g)}_j\right) ^2, \end{aligned}$$

Therefore, either $\alpha _j^{(g)}={\alpha '}^{(g)}_j$ or $\alpha _j^{(g)}=-{\alpha '}^{(g)}_j$ holds for all j. From (7), the sign of $\alpha _1^{(r)}$ must coincide with the sign of $\lambda _1^{(r)}$ under either parameterization $\varvec{\zeta }$ or $\varvec{\zeta }'$. Moreover, the sign of $\lambda _1^{(r)}$ (or ${\lambda '}^{(r)}_1$) characterizes the positive or negative association between item 1 (the reference item) and the latent variable. The sign is determined once the meaning of the latent variable is specified. Thus, the sign should be invariant regardless of using $\varvec{\zeta }$ or $\varvec{\zeta }'$. Therefore, the case $\alpha _j^{(g)}=-{\alpha '}^{(g)}_j$ is further eliminated, and the proof for Theorem 1 is completed.

1.2 A.2 Proof for Theorem 2

Adopting similar techniques considered in Millsap and Tein (2004) and San Martín et al. (2013), our proof is carried out by showing the one-to-one correspondence between the identified parameters $(\varvec{\delta }(\varvec{\zeta }),\varvec{\alpha }(\varvec{\zeta }))$ and the parameters to be freely estimated in $\varvec{\zeta }$ under the given constraints, detailed in Part I and Part II.

Part I: Parameters for the reference group $g=r$

• Parameters associated with item 1: Since $\lambda _1^{(r)}$ and ${\sigma _1^2}^{(r)}$ are constrained in (10) and (11), $\varphi ^{(r)}$ is the only parameter to be freely estimated in $\alpha _1^{(r)}$. Therefore, from (7), represent $\varphi ^{(r)}$ in terms of the identified parameter $\alpha _1^{(r)}$ via

  $$\begin{aligned} \varphi ^{(r)}=\frac{{\sigma _1^2}^{(r)}}{{\lambda _1^{(r)}}^2} \frac{\alpha _1^{(r) 2}}{1-\alpha _1^{(r) 2}} =\frac{\alpha _1^{(r) 2}}{1-\alpha _1^{(r) 2}}, \quad \hbox {provided}\quad \alpha _1^{(r)} \ne 1. \end{aligned}$$

  (30)

  Then, from (5), represent $\tau _{1,c}^{(r)}$ in terms of $(\delta _{1,c}^{(r)}, \alpha _1^{(r)})$:

  $$\begin{aligned}&\tau _{1,c}^{(r)} = \lambda _1^{(r)} \kappa ^{(r)} + \delta _{1,c}^{(r)}\left[ \left( \lambda _1^{(r)}\right) ^2\varphi ^{(r)} + {\sigma _1^2}^{(r)}\right] ^{1/2}= \frac{\delta _{1,c}^{(r)}}{\sqrt{1-\alpha _1^{(r) 2}}},~c=1,\cdots ,C-1, \end{aligned}$$

  (31)

  in which $\varphi ^{(r)}$ is substituted as a function of $\alpha _1^{(r)}$ using (30).

• Parameters associated with items $j\ne 1$: First, for all $j\ne 1$, (7) implies

  $$\begin{aligned} \lambda _j^{(r)} = \frac{\alpha _j^{(r)} \sigma _j^{(r)}}{\left[ \varphi ^{(r)}(1-\alpha _j^{(r)\,2})\right] ^{1/2}} \quad \begin{array}{c} (10) \\ = \\ ~~\end{array} \frac{\alpha _j^{(r)}}{\left[ \varphi ^{(r)}(1-\alpha _j^{(r)\,2})\right] ^{1/2}} \quad \begin{array}{c} (30) \\ = \\ ~~\end{array} \frac{\alpha _j^{(r)}}{|\alpha _1^{(r)}|} \left[ \frac{1-\alpha _1^{(r)\,2}}{1-\alpha _j^{(r)\,2}}\right] ^{1/2}. \end{aligned}$$

  (32)

  Second, (5) implies

  $$\begin{aligned} \tau _{j,c}^{(r)}&= \lambda _j^{(r)} \kappa ^{(r)} + \delta _{j,c}^{(r)} \cdot \left[ \left( \lambda _j^{(r)}\right) ^2\varphi ^{(r)} + {\sigma _j^2}^{(r)}\right] ^{1/2} \nonumber \\&= \delta _{j,c}^{(r)} \cdot \left[ \left( \lambda _j^{(r)}\right) ^2\varphi ^{(r)} +1 \right] ^{1/2} = \frac{\delta _{j,c}^{(r)}}{\sqrt{1-\alpha _j^{(r)\,2}}}, \end{aligned}$$

  in which the second equality holds given (10) and (11), and the third equality holds due to (30) and (32).

Part II: Parameters for the focal group $g=f$

Following similar steps in Part I, the parameters to be freely estimated for the focal group can also be represented in terms of the identified parameters:

$$\begin{aligned} \varphi ^{(f)}&\begin{array}{c} (7) \\ = \\ ~~\end{array} \frac{{\sigma _1^2}^{(f)}}{{\lambda _1^{(f)}}^2} \frac{\alpha _1^{(f) 2}}{1-\alpha _1^{(f) 2}} \begin{array}{c} (12) \\ = \\ ~~\end{array} \frac{\alpha _1^{(f) 2}}{1-\alpha _1^{(f) 2}}, \end{aligned}$$

(33)

$$\begin{aligned} \kappa ^{(f)}&\begin{array}{c} (5) \\ = \\ ~~\end{array} \frac{1}{\lambda _1^{(f)}} \left[ \tau _{1,c}^{(f)} - \delta _{1,c}^{(f)} \left( (\lambda _1^{(f)})^2 \varphi ^{(f)} + {\sigma _1^2}^{(f)}\right) ^{1/2}\right] \begin{array}{c} (12) \\ = \\ ~~\end{array} \tau _{1,c}^{(r)} - \delta _{1,c}^{(f)} \left( \varphi ^{(f)} + 1\right) ^{1/2} \nonumber \\&\begin{array}{c} (31) \\ = \\ ~~ \end{array} \frac{\delta _{1,c}^{(r)}}{\sqrt{1-\alpha _1^{(r) 2}}} - \delta _{1,c}^{(f)} \left( \varphi ^{(f)} + 1\right) ^{1/2} \begin{array}{c} (33) \\ = \\ ~~ \end{array} \frac{\delta _{1,c}^{(r)}}{\sqrt{1-\alpha _1^{(r) 2}}} - \frac{\delta _{1,c}^{(f)}}{\sqrt{1-\alpha _1^{(f) 2}}},\end{aligned}$$

(34)

$$ \begin{aligned} \lambda _j^{(f)}&\begin{array}{c} (7) \\ = \\ ~~\end{array} \frac{\alpha _j^{(f)} \sigma _j^{(f)}}{\left[ \varphi ^{(f)}(1-\alpha _j^{(f) 2})\right] ^{1/2}} \begin{array}{c} (10) \& (13) \\ = \\ ~~\end{array} \frac{\alpha _j^{(f)} }{\left[ \varphi ^{(f)}(1-\alpha _j^{(f) 2})\right] ^{1/2}} \quad \begin{array}{c} (33) \\ = \\ ~~\end{array} \frac{\alpha _j^{(f)}}{|\alpha _1^{(f)}|} \left[ \frac{1-\alpha _1^{(f)\,2}}{1-\alpha _j^{(f)\,2}}\right] ^{1/2},\end{aligned}$$

(35)

$$ \begin{aligned} \tau _{j,c'}^{(f)}&\begin{array}{c} (5) \& (13) \\ = \\ ~~\end{array} \lambda _j^{(f)} \kappa ^{(f)} + \delta _{j,c'}^{(f)} \left[ \left( \lambda _j^{(f)}\right) ^2\varphi ^{(f)} + 1\right] ^{1/2} \begin{array}{c} (33) \& (34) \\ = \\ ~~ \end{array} \lambda _j^{(f)} \kappa ^{(f)} + \frac{\delta _{j,c'}^{(f)}}{\sqrt{1-\alpha _j^{(f)\,2}}} \nonumber \\&\begin{array}{c} (34) \& (35) \\ = \\ ~~ \end{array} \frac{1}{\sqrt{1-\alpha _j^{(f)\,2}}} \left\{ \delta _{j,c'}^{(f)} - \frac{\alpha _j^{(f)}}{|\alpha _1^{(f)}|} \left( \delta _{1,c}^{(f)}-\delta _{1,c}^{(r)} \frac{\sqrt{1-\alpha _1^{(f)\,2}}}{\sqrt{1-\alpha _1^{(r)\,2}}}\right) \right\} , \forall c' \in \{1, \cdots , C-1\}, \end{aligned}$$

(36)

where the equations used for deriving each step are directly labeled at the corresponding step, and c is the category chosen for setting identifiability constraints on threshold in (12).

To summarize, the transformations of (30)–(36) illustrate the mapping from the identified parameters $\{\delta _{j,c}^{(g)}$, $\alpha _j^{(g)},\forall ~j,c,g\}$ to the model parameters to be freely estimated

$\left( \{\varphi ^{(g)}: \, g=r,f\},\kappa ^{(f)},\{\lambda _j^{(g)}: \, j \ne 1,~g=r,f\}, \{\tau _{j,c}^{(g)}: \, \forall ~j,c,g\}\setminus \{\tau _{1,c}^{(g)}: \, g=r,f\}\right)$. Both sets, having the dimension 2JC, are clearly of a one-to-one correspondence to each other, and therefore, the whole model is just-identified under the set of constraints (10)–(13).

1.3 A.3 Derivations for Section 2.2.3

According to Section 2.2.3, a set of identifiability constraints is composed of three parts: (I) constraints for the reference group; (II) relating parameters/linking the metric between groups via item 1 (the anchor); and (III) determining the scale for the focal group. In the following, we check that those choices listed in Section 2.2.3 cover the possible forms for MC-FA with the Theta parameterization through representing parameters to be estimated in each part as a function of identified parameters.

Part (I): constraints for the reference group

Constraint (10) is the prototype for the Theta parameterization. Together with (11), a natural and the most common setting for fixing the scale of the reference group, there are $(J+2)$ constraints imposed for the reference group.

Part (II): relating parameters/linking the metric between groups via item 1

By counting the degrees of freedom for parameters, there are $JC+C$ identified parameters, $\{\delta _{j,c}^{(r)},\alpha _j^{(r)},\,\forall j,c\}\cup \{\delta _{1,c}^{(f)},\alpha _1^{(f)},\,\forall c\}$, related to item 1 for both groups and the nonanchor items for reference group only. Accordingly, the overall parameters involved to the same part of the model, $\{\varphi ^{(r)},\kappa ^{(r)},\lambda _j^{(r)},\sigma _j^{(r)},\tau _{j,c}^{(r)},\,\forall j,c\}\cup \{\varphi ^{(f)},\kappa ^{(f)},\lambda _1^{(f)},\sigma _1^{(f)},\tau _{1,c}^{(f)},\,\forall c\}$, have the dimension $JC+C+J+5$. Focusing on this part of the model, $J+5$ constraints have to be imposed for achieving the just-identifiability. Besides $(J+2)$ constraints in Part (I), we need exactly 3 extra constraints involving at least one term in $\{\varphi ^{(f)},\kappa ^{(f)},\lambda _1^{(f)},\sigma _1^{(f)},\tau _{1,c}^{(f)},\,\forall c\}$ for each constraint. Since $(\varphi ^{(f)},\kappa ^{(f)})$ is of interest, the constraints for item 1 are further limited to $\{\lambda _1^{(f)},\sigma _1^{(f)},\tau _{1,c}^{(f)},\,\forall c\}$, from which 3 quantities are chosen to form four types of combinations: $(\lambda _1^{(f)},\sigma _1^{(f)},\tau _{1,c}^{(f)})$, $(\lambda _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$, $(\sigma _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$, and $(\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)},\tau _{1,c_3}^{(f)})$. The first and second types, respectively, correspond to (12) in the GRM-type parameterization and (16) in M&T parameterization. In the following, the other two types are checked to be invalid for just-identifiability.

First, suppose $(\sigma _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$ is used to set 3 constraints for relating parameters and linking the metric between groups via item 1. From (5), we have

$$\begin{aligned} \tau _{1,c_i}^{(f)}&= \lambda _1^{(f)}\kappa ^{(f)}+\delta _{1,c_i}^{(f)} \left[ (\lambda _1^{(f)})^2\varphi ^{(f)}+\sigma _1^{2(f)}\right] ^{1/2} \\&= \lambda _1^{(f)}\kappa ^{(f)}+\delta _{1,c_i}^{(f)} \,\sigma _1^{(f)} \left[ 1-\alpha _1^{(f) 2}\right] ^{-1/2}, \quad i=1,2, \end{aligned}$$

where (7) with $j=1$ and $g=f$ is used to obtain the second equality. Consequently, the parameters satisfy

$$\begin{aligned} \lambda _1^{(f)}\kappa ^{(f)} =\tau _{1,c_i}^{(f)}-\delta _{1,c_i}^{(f)} \sigma _1^{(f)} \left[ 1-\alpha _1^{(f) 2}\right] ^{-1/2}, \quad i=1,2, \end{aligned}$$

which resulted in an extra constraint on $(\sigma _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$ such that

$$\begin{aligned} \frac{\tau _{1,c_1}^{(f)}-\tau _{1,c_2}^{(f)}}{\sigma _1^{(f)}} = \frac{\delta _{1,c_1}^{(f)}-\delta _{1,c_2}^{(f)}}{\sqrt{1-\alpha _1^{(f)\,2}}}. \end{aligned}$$

(37)

However, for general values of identified parameters, (37) is impossible to hold simultaneously with the constraints given on $(\sigma _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$ for relating groups.

Second, suppose $(\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)},\tau _{1,c_3}^{(f)})$ is used to set 3 constraints for relating parameters and linking the metric between groups via item 1. From (5), we have

$$\begin{aligned} \tau _{1,c_i}^{(f)} = \lambda _1^{(f)}\kappa ^{(f)}+\delta _{1,c_i}^{(f)} \left[ (\lambda _1^{(f)})^2 \varphi ^{(f)}+\sigma _1^{2(f)}\right] ^{1/2}, \quad i=1,2,3, \end{aligned}$$

which implies

$$\begin{aligned} \frac{\tau _{1,c_1}^{(f)}-\lambda _1^{(f)}\kappa ^{(f)}}{\tau _{1,c_2}^{(f)}-\lambda _1^{(f)}\kappa ^{(f)}} = \frac{\delta _{1,c_1}^{(f)}}{\delta _{1,c_2}^{(f)}} \quad \hbox {and}\quad \frac{\tau _{1,c_1}^{(f)}-\lambda _1^{(f)}\kappa ^{(f)}}{\tau _{1,c_3}^{(f)}-\lambda _1^{(f)}\kappa ^{(f)}} = \frac{\delta _{1,c_1}^{(f)}}{\delta _{1,c_3}^{(f)}}, \end{aligned}$$

leading to two inconsistent solutions for $\lambda _1^{(f)}\kappa ^{(f)}$, i.e.,

$$\begin{aligned} \lambda _1^{(f)}\kappa ^{(f)} =\frac{\delta _{1,c_1}^{(f)}\tau _{1,c_2}^{(f)}-\delta _{1,c_2}^{(f)}\tau _{1,c_1}^{(f)}}{\delta _{1,c_1}^{(f)}-\delta _{1,c_2}^{(f)}} \ne \frac{\delta _{1,c_1}^{(f)}\tau _{1,c_3}^{(f)}-\delta _{1,c_3}^{(f)}\tau _{1,c_1}^{(f)}}{\delta _{1,c_1}^{(f)}-\delta _{1,c_3}^{(f)}}=\lambda _1^{(f)}\kappa ^{(f)}, \end{aligned}$$

where the inequality holds for general values of identified parameters. Therefore, using $(\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)},\tau _{1,c_3}^{(f)})$ fails to construct the one-to-one correspondence of parameters to be estimated and identified parameters.

To summarize, setting constraints based on $(\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)},\tau _{1,c_3}^{(f)})$ or $(\sigma _1^{(f)},\tau _{1,c_1}^{(f)},\tau _{1,c_2}^{(f)})$ cannot achieve the goal of the one-to-one correspondence. The setting (12) in the GRM-type parameterization and (16) in M&T parameterization are the only two valid choices to achieve the goal in this part.

Part (III): determining the scale for the focal group

For each item other than the anchor, the parameters associated with the focal group in that item include $\{\lambda _j^{(f)},\sigma _j^{(f)},\tau _{j,c}^{(f)},\,\forall c\}$ with dimension $C+1$. Compared with the relevant identified parameters $\{\alpha _j^{(f)},\delta _{j,c}^{(f)},\,\forall c\}$ with dimension C, each item requires one extra constraint to make the parameter in that particular item just-identified. There are three possibilities to choose from $\{\lambda _j^{(f)},\sigma _j^{(f)},\tau _{j,c}^{(f)},\,\forall c\}$. The GRM-type parameterization picks $\sigma _j^{(f)}$ to set the constraint; M&T parameterization uses $\tau _{j,c}^{(f)}$ instead. Meanwhile, the third possibility is using the factor loading $\lambda _j^{(f)}$. Here, we show that

$$\begin{aligned} \lambda _j^{(f)}=\lambda _j^{(r)}, \hbox { for } j\ne 1, \end{aligned}$$

(38)

is also a valid constraint. Again, from (5) and (7), we have

$$\begin{aligned} \sigma _j^{2 (f)} = \lambda _j^{(f) 2} \varphi ^{(f)} \frac{1-\alpha _j^{(f) 2}}{\alpha _j^{(f) 2}}, \quad \tau _{j,c}^{(f)} = \lambda _j^{(f)}\kappa ^{(f)} + \delta _{j,c}^{(f)} \frac{\lambda _j^{(f)}\sqrt{\varphi ^{(f)}}}{\alpha _j^{(f)}}, \,\forall c. \end{aligned}$$

Since both $\kappa ^{(f)}$ and $\varphi ^{(f)}$ are represented as functions of the identified parameters in Part (I) and (II), together with (38), $\{\sigma _j^{(f)},\tau _{j,c}^{(f)},\forall j,c\}$ can be uniquely determined by the identified parameters. Therefore, (38) is also a valid choice to achieve the one-to-one correspondence.