2.1. General Notions

The Rasch model (RM), also known as 1-Parameter Logistic (1PL) model, is nested within the 2-Parameter Logistic (2PL) model, which is in turn nested within the 3-Parameter Logistic (3PL) and 4-Parameter Logistic (4PL) models. The form of the latter is here given by:

(1) $\begin{array}{cc}P(X_i=1|\theta ,{\Gamma }_i^4)& =c_i+(1-d_i-c_i)\frac{e^{a_i(\theta -b_i)}}{1+e^{a_i(\theta -b_i)}}=\frac{(1-d_i)e^{a_i\theta }+c_i{e}^{a_i{b}_i}}{e^{a_i\theta }+e^{a_i{b}_i}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X_i = 1|\theta , \Gamma ^4_i)&= c_i+ (1-d_i -c_i)\frac{e^{a_i(\theta -b_i)}}{1+e^{a_i(\theta -b_i)}} = \frac{(1-d_i)e^{a_i\theta }+c_ie^{a_ib_i}}{e^{a_i\theta }+e^{a_ib_i}} \end{aligned}$$\end{document}

where ${\Gamma }_i^4=\{b_i,a_i,c_i,d_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^4_i=\{b_i,a_i,c_i,d_i\}$$\end{document} contains the difficulty parameter $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} of the RM, the discrimination parameter $a_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i$$\end{document} of the 2PL, the guessing parameter $c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} of the 3PL (Birnbaum, Reference Birnbaum, Lord and Novick1968), and the slipping parameter $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} of the 4PL (Barton and Lord, Reference Barton and Lord1981). A common formulation of the IRF (1) replaces $1-d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-d_i$$\end{document} with $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} so that $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} captures the upper asymptote of the IRF. In the present work, since we are interested in relating KST and IRT left-side added parameters, the notation of Equation (1) is more convenient.

Equation (1) is written using a random-effects notation, in which the ability is treated as a latent variable. If instead a fixed-effects notation is considered, abilities are incidental parameters ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} for $j\in \{1,\dots ,N\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \{1,\ldots , N\}$$\end{document} , with N the number of persons. The IRF (1) can be given notation $P(X_{\mathrm{ji}}=1|{\theta }_j,{\Gamma }_i^n)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_{ji} = 1|\theta _j, \Gamma ^n_i)$$\end{document} with $X_{\mathrm{ji}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{ji}$$\end{document} the response of the j-th individual to the i-th item. Random- and fixed-effects perspectives are, respectively, grounded within a ‘random sampling’ view and a ‘stochastic subject’ view of the IRT process (see, e.g., Holland, Reference Holland1990). In the former, an IRF like (1) represents the proportion of individuals with a level of ability $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} that provides a correct answer to the i-th item. In the latter, the IRF represents the probability of the j-th individual to answer correctly the i-th item. Typical estimation methods for the fixed-effects case are joint maximum likelihood (JML) and conditional maximum likelihood (CML). CML is, however, restricted to the Rasch-family of models, while JML is well known to provide inconsistent estimates of $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} due to the incidental parameters problem (see, e.g., Haberman, Reference Haberman1977; Andersen, Reference Andersen1980; Ghosh, Reference Ghosh1995). For these reasons, the fixed-effects perspective is often used for didactic purposes, while a random-effect perspective is preferred in practice. With a random-effect specification, ability can be integrated out when using marginal maximum likelihood (MML). A final remark on the ${\Gamma }_i^n$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^n_i$$\end{document} notation, in which n is the number of item parameters. The RM or 1PL model is associated with ${\Gamma }_i^1=\left\{b_i\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^1_i=\{b_i\}$$\end{document} , the 2PL model to ${\Gamma }_i^2=\{b_i,a_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^2_i=\{b_i,a_i\}$$\end{document} , the 3PL model to ${\Gamma }_i^3=\{b_i,a_i,c_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^3_i=\{b_i,a_i,c_i\}$$\end{document} , and the 4PL model to ${\Gamma }_i^4=\{b_i,a_i,c_i,d_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^4_i=\{b_i,a_i,c_i,d_i\}$$\end{document} . ${\Gamma }_i^{1,3}=\{b_i,c_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{1,3}_i=\{b_i,c_i\}$$\end{document} is used for the 1-Parameter model plus Guessing (1PL-G), ${\Gamma }_i^{1,4}=\{b_i,d_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{1,4}_i=\{b_i,d_i\}$$\end{document} is used for the 1-Parameter model plus Slipping (1PL-S), and ${\Gamma }_i^{-2}=\{b_i,c_i,d_i\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{-2}_i=\{b_i,c_i,d_i\}$$\end{document} is used for the 4PL without discrimination, which is labeled as 1PL-GS.

2.2. Unidentifiability of IRT Models

It is well known that the parameters of the RM are not unique since a uniform translation of both ability $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} and difficulty $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} for the same constant yields the same response probability. This is the only form of local unidentifiability for the RM/1PL. Additionally, in the 2PL model, one can dilate person and difficulty parameters by the same constant while dilating the discrimination parameter $a_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i$$\end{document} by the reciprocal. As to the guessing parameter $c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} , it has been a source of debate in the literature. Some problematic features pertaining the global unidentifiability of the 3PL model had already been discussed by Samejima (Reference Samejima1973), i.e., the nonuniqueness of the maximum for the likelihood w.r.t. the latent trait $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} (see also Yen et al., Reference Yen, Burket and Sykes1991). Instability of the estimates has also been long known in the literature. Thissen and Wainer (Reference Thissen and Wainer1982) highlighted that estimation of the c-parameters can “wreak havoc” with the estimation of the b-parameters since easy items have few observations low enough to provide enough information. A similar problem can occur for very difficult items (see, e.g., Drasgow and Parsons, Reference Drasgow and Parsons1983; Hulin et al., Reference Hulin, Lissak and Drasgow1982). Instability also does not concern only the difficulty parameters; van Der Linden and Hambleton (Reference van Der Linden, Hambleton, van der Linden, Updike and Hambleton1997) stated that small changes in the guessing parameter can be compensated by small changes in the slope of the curve. More in general, Mislevy (Reference Mislevy1986) highlighted that the instability of the ML estimates in the 3PL model stems from the fact that different triples ${\Gamma }_i^3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _i^3$$\end{document} “can trace similar IRFs in the region of the ability scale where the sample of examinees is to be found” often resulting in nearly flat likelihood surfaces. Trade-offs seem to occur between c and all the other parameters. Lack of stability of the estimates of the 3PL model has also been discussed by Patz and Junker (Reference Patz and Junker1999), DeMars (Reference DeMars2001), and Pelton (Reference Pelton2002). As a consequence of the difficulty in estimating the lower asymptote of the 3PL, the 4PL model is often considered even more problematic to estimate (see, e.g., Embretson and Reise, Reference Embretson and Reise2000; Baker and Kim, Reference Baker and Kim2004). Nonetheless, it is worth mentioning that there has recently been a renewed interest in 4-Parameter models (see, e.g., Hessen, Reference Hessen2005; Loken and Rulison, Reference Loken and Rulison2010; Ogasawara, Reference Ogasawara2012; Culpepper, Reference Culpepper2016).

Local identifiability of IRT models is typically assessed by establishing an injective mapping between the parameters of interest of the model and the identified parametrization associated with the experimental outcomes. The choice of both parameters of interest and identified parametrization depends on whether the model is given a fixed-effects, a random-effects, or a semi-parametric specification (see, e.g., San Martín et al., Reference San Martín, González and Tuerlinckx2009; San Martín and Rolin, Reference San Martín and Rolin2013; San Martín, Reference San Martín and van Der Linden2016). Interpretation of the parameters based on the identification analysis was provided by Fariña et al. (Reference Fariña, González and San Martín2019). In the fixed-effects case, the parameters of interest consist of the item parameters in ${\Gamma }_i^n$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _i^n$$\end{document} and the latent abilities ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} for $j\in \{1,\dots ,N\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \{1,\ldots , N\}$$\end{document} , while the identified parametrization is given by the parameters of mutually independent Bernoulli distributions. In the random-effects case, the parameters of interest are the item parameters in ${\Gamma }_i^n$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _i^n$$\end{document} and the scale $\sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} and location $\mu$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} parameters of some distribution $f(\theta ;\mu ,\sigma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta ; \mu , \sigma )$$\end{document} of the individual’s ability, while the identified parametrization is given by the parameters of a Multinomial distribution associated with the response patterns. Finally, the semi-parametric case follows the same reasoning as the random-effects case, but the distribution of the ability itself is treated as a parameter. In what follows, we are interested in the results for fixed- and random-effects models.

Although the 3PL model was initially believed to be always identifiable (see, e.g., Lord, Reference Lord1980), Maris (Reference Maris2002) and Maris and Bechger (Reference Maris and Bechger2009) showed that trade-offs can occur between the parameters of the 1PL-G model in a non-trivial way. Indeed, the model of Eq. (1), in the case $d_i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i=0$$\end{document} and when all discrimination parameters are equal (i.e., 1PL-G model, $a_i=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_i=1$$\end{document} ), is unidentifiable since the transformations

(2) $\begin{array}{c}\left(\begin{array}{c}{c}_i^{\ast }=\frac{c_i{e}^{b_i}-\ell }{e^{b_i}-\ell }\\ e^{{\theta }^{\ast }}=e^{\theta }+\ell \\ e^{b_i^{\ast }}=e^{b_i}-\ell \end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} c_i^* = \frac{c_ie^{b_i}-\ell }{e^{b_i}-\ell }\\ e^{\theta ^*}= e^{\theta }+\ell \\ e^{b_i^*}= e^{b_i}-\ell \end{array}\right. } \end{aligned}$$\end{document}

yield equivalent response probabilities

$\begin{array}{cc}P(X_i=1|{\theta }^{\ast },{\Gamma }_i^{1,3\ast })& =\frac{e^{{\theta }^{\ast }}+c_i^{\ast }{e}^{b_i^{\ast }}}{e^{{\theta }^{\ast }}+e^{b_i^{\ast }}}=\frac{e^{\theta }+\ell +c_i{e}^{b_i}-\ell }{e^{\theta }+l+e^{b_i}-\ell }=\frac{e^{\theta }+c_i{e}^{b_i}}{e^{\theta }+e^{b_i}}=P(X_i=1|\theta ,{\Gamma }_i^{1,3}).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X_i=1|\theta ^*, \Gamma ^{1,3*}_i)&= \frac{e^{\theta ^*}+c_i^*e^{b_i^*}}{e^{\theta ^*}+e^{b_i^*}} = \frac{e^{\theta }+\ell +c_ie^{b_i}-\ell }{e^{\theta }+l+e^{b_i}-\ell } = \frac{e^{\theta }+c_ie^{b_i}}{e^{\theta }+e^{b_i}} = P(X_i=1|\theta , \Gamma ^{1,3}_i). \end{aligned}$$\end{document}

Notice that by transformations (2) it follows that $\ell \in [-e^{min\theta },e^{{min}_i{b}_i})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in [-e^{\min {\theta }},e^{\min _i{b_i}})$$\end{document} . If a fixed-effects interpretation of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is considered, there is a minimal value ${\theta }_m={min}_j{\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _m = \min _{j}{\theta _j}$$\end{document} such that $e^{min\theta }=e^{{\theta }_m}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{\min {\theta }}=e^{\theta _m}$$\end{document} . If instead a random-effects interpretation is considered, one has that $\underset{\theta \to -\infty }{lim}{e}^{min\theta }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \lim _{\theta \rightarrow -\infty } e^{\min {\theta }}=0$$\end{document} so that $\ell \in [0,e^{{min}_i{b}_i})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in [0,e^{\min _i{b_i}})$$\end{document} . It should be stressed that transformations (2) provide an unidentifiability result in the fixed-effects case, but in the random-effects case they are an example of empirically indistinguishable statistical models. Indeed, in spite of the fact that the mathematical forms of the IRFs stay the same, the mathematical form of the distribution of the latent variable changes under the transformations. As a matter of fact, the second part of the work of Maris and Bechger (Reference Maris and Bechger2009) provides an example of empirically indistinguishable alternative models.

As the results of Maris and Bechger (Reference Maris and Bechger2009) cast doubts on the identifiability of the 3PL model, or of at least some degenerate case of it, several authors investigated the local identifiability of both fixed- and random-effects IRT models. A general discussion about identifiability conditions for 1PL, 2PL, and 1PL-G models for both fixed- and random-effects models can be found in San Martín (Reference San Martín and van Der Linden2016). Results for the random-effects 1PL, 2PL, and 1PL-G models have been discussed by San Martín and Rolin (Reference San Martín and Rolin2013), San Martín et al. (Reference San Martín, Rolin and Castro2013). Specifically, the parameters $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} , $c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} , $\mu$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} , and $\sigma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} of the random-effects 1PL-G model are identified if (1) at least three items are available, (2) the guessing parameter $c_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1$$\end{document} is set to zero, and (3) the traditional linear restrictions (e.g., $b_1=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1=0$$\end{document} or ${\sum }_i{b}_i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _ib_i =0$$\end{document} ) are imposed to remove the unidentifiability of the 1PL model. Up to our knowledge, there are no results for the random-effects 3PL and 4PL models.

More results are instead available for the fixed-effects models. van der Linden and Barrett (Reference van der Linden and Barrett2016) showed that identifiability issues arise also in the degenerate case in which all individuals have the same ability values. It has been shown that the fixed-effects 1PL-G model is identifiable if difficulty and guessing parameters of one item are set to known constants, as in the random-effects case, and at least two persons with different abilities are available (San Martín et al., Reference San Martín, Rolin and Castro2013). An alternative set of restrictions (two items with equal guessing but distinct difficulties) was later provided by Ogasawara (Reference Ogasawara2020). San Martín et al. (Reference San Martín, González and Tuerlinckx2015) argued that the 3PL is unidentified, but this latter result was rectified by Wu (Reference Wu2016), which showed that as long as four persons with distinct abilities and two items with distinct discrimination parameters are available, the fixed-effects 3PL model is identifiable up to the usual permissible transformations of the 2PL model. These sufficient conditions were also shown to be necessary by Ogasawara (Reference Ogasawara2017; Reference Ogasawara2020). Hence, in the case of the fixed-effects 3PL model, the belief of the locally identified 3PL model as found in Lord (Reference Lord1980) is still generally correct in that both the cases considered in Maris and Bechger (Reference Maris and Bechger2009) and van der Linden and Barrett (Reference van der Linden and Barrett2016) are special degenerate cases of the 3PL model with added restrictions to their parameters. As Wu (Reference Wu2016) pointed out, although it is not unusual for the parameter space of an identified model to contain sub-spaces in which the model is unidentified, these situations are still relevant since they might impact procedures like likelihood-ratio testing. Cases of identifiability and unidentifiability for the fixed-effects 3PL (and 4PL) were also recently summarized, discussed, and extended by Ogasawara (Reference Ogasawara2017; Reference Ogasawara2020). More in detail, Ogasawara (Reference Ogasawara2017) systematized the previous results and provided a wider set of unidentified models, extended the analysis to non-logistic fixed-effects 3-parameter models by briefly discussing the probit model, and by exploring the unidentifiability of the general family of IRFs of the form

(3) $\begin{array}{c}P(X_i=1|\theta ,{\Gamma }_i^{1,3})=c_i+(1-c_i)\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X_i = 1|\theta , \Gamma _i^{1,3}) = c_i+(1-c_i)\frac{G(\theta )}{G(\theta )+H(b_i)} \end{aligned}$$\end{document}

for arbitrary strictly increasing functions G and H. Ogasawara (Reference Ogasawara2017, Proposition 2) showed that the 4PL is identified under the same conditions of Wu (Reference Wu2016) plus either the $c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} ’s or the $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} ’s are given and that in the 1PL-GS model there exist further trade-offs of parameters in addition to those highlighted by transformations (2). Specifically, the following transformations of the left-side added parameters

(4) $\begin{array}{c}\left(\begin{array}{c}(1-d_i^{\ast })=(1-d_i)+k^{\ast }\\ c_i^{\ast }=c_i-\frac{k^{\ast }P(X_{i,j}=1|{\theta }_j,{\Gamma }_i^1)}{1-P(X_i=1|\theta ,{\Gamma }_i^1)}=c_i-k^{\ast }{e}^{{\theta }_j-b_i}\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} (1-d_i^*) = (1-d_i) + k^*\\ c_i^* = c_i - \frac{k^*P(X_{i,j}=1|\theta _j, \Gamma ^{1}_i)}{1-P(X_i=1|\theta , \Gamma ^{1}_i)} = c_i- k^*e^{\theta _j-b_i} \end{array}\right. }, \end{aligned}$$\end{document}

for $k^{\ast }\in (-(1-c_i-d_i)(1-P(X_i=1|\theta ,{\Gamma }_i^1)),{min}_{i,j}(d_i,c_i{e}^{{\theta }_j-b_i})]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^*\in (-(1-c_i-d_i)(1-P(X_i=1|\theta , \Gamma ^{1}_i)), \min _{i,j}{(d_i, c_ie^{\theta _j-b_i})}]$$\end{document} leave the IRF of both the 1PL-GS and the 1PL models unchanged (Ogasawara, Reference Ogasawara2017, Theorem 5). Transformations (2) can also be further extended to the 1PL-GS model by considering an additional condition

(5) $\begin{array}{cc}(1-d_i^{\ast })& =\frac{(1-d_i)e^{\theta }+\ell }{e^{\theta }+\ell }\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (1-d_i^*)&= \frac{(1-d_i)e^{\theta }+\ell }{e^{\theta }+\ell } \end{aligned}$$\end{document}

which actually transforms the slipping parameter $d_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i^*$$\end{document} into a function $d_i^{\ast }\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i^*(\theta )$$\end{document} of the ability (Ogasawara, Reference Ogasawara2017). Equation (5) might be of interest if the slipping parameter $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} is assumed to be person-dependent, i.e., of the form $d_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_j$$\end{document} . Transformations (4) and (5) are, in the present context, not considered unidentifiability results but rather empirically indistinguishable alternative models since, in both cases, some left-side added parameters have become ability-dependent and are not constant anymore. Finally, it is worth mentioning that Ogasawara (Reference Ogasawara2020) discussed additional identifiability restrictions for the 1PL-G model based on the minimization of the absolute value of the skewness of the ${\theta }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^*$$\end{document} abilities. More in general, Ogasawara (Reference Ogasawara2021) discussed the maximization of Fisher information, total score information, and total quasi-information to impose optimal restrictions that allow to resolve the identifiability of the 1PL-G model and of the general family of models given by Equation (3). As it will be shown in Sect. 4, the transformations (2) introduced by Maris and Bechger (Reference Maris and Bechger2009), as well as the general case (3) and the special cases (4) and (5) introduced by Ogasawara (Reference Ogasawara2017), or the other results on unidentifiability provided by Wu (Reference Wu2016), van der Linden and Barrett (Reference van der Linden and Barrett2016), and San Martín (Reference San Martín and van Der Linden2016) can be derived from the general transformations describing the trade-off between the parameters $c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i$$\end{document} , $d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} , and the entire $P(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta ,\Gamma ^2_i)$$\end{document} as a consequence of the forward- and backward-gradedness of the power set describing the local stochastic independence assumption in IRT between the different items. In order to show this, we first need to introduce KST.

3.1. A Brief Introduction to KST

KST is a combinatorial and set-theoretical approach that classifies individuals by means of the collections of items that they can master in a given domain of knowledge (see, e.g., Falmagne and Doignon, Reference Falmagne and Doignon2011). Let Q be a nonempty set of items q, then a knowledge state $K\subseteq Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\subseteq Q$$\end{document} is a collection of problems $q\in Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in Q$$\end{document} that an individual is capable of mastering. A knowledge structure is a pair $(Q,K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q,\mathcal {K})$$\end{document} where $K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} is a family of subsets of Q that always includes the full domain Q and the empty set $\varnothing$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\emptyset $$\end{document} . As an example, for the domain $Q=\{q_1,q_2,q_3\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=\{q_1,q_2,q_3\}$$\end{document} , a possible knowledge structure is defined by the collection

(6) $\begin{array}{cc}K& =\{\varnothing ,\left\{q_1\right\},\left\{q_2\right\},\{q_1,q_2\},\{q_1,q_3\},Q\}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {K}&=\{\emptyset , \{q_1\}, \{q_2\},\{q_1, q_2\},\{q_1, q_3\}, Q\}. \end{aligned}$$\end{document}

and represents a situation in which item $q_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_1$$\end{document} and $q_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_2$$\end{document} can be mastered independently of each other, while item $q_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_3$$\end{document} requires item $q_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_1$$\end{document} to be mastered before it can be mastered. Knowledge structures are graphically displayed by Hasse diagrams as shown in Fig. 1, where each node is a different knowledge state.

Figure 1 Example of a knowledge structure for a set of five items $Q=\{q_1,q_2,q_3\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q=\{q_1, q_2,q_3\}$$\end{document} .

Different properties of and relations between the knowledge states characterize different families of structures. For instance, if the structure is closed under union (i.e., $K,L\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K,L\in \mathcal {K}$$\end{document} implies $K\cup L\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\cup L \in \mathcal {K}$$\end{document} ), the knowledge structure is called a knowledge space. Such a property is a natural requirement for substantive applications in learning as it implies that any item can be mastered at any time (provided, of course, that the prerequisite conditions for learning it are fulfilled). Another characteristic which is deemed a necessary requirement to learning is that items can be learned one at a time, that is, the structure is well-graded (for every state $K\in K\backslash Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}\setminus Q$$\end{document} there is an item $q\in Q\backslash K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in Q\setminus K$$\end{document} such that $K\cup \left\{q\right\}\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\cup \{q\}\in \mathcal {K}$$\end{document} ). If the structure is closed under union and well-graded, it is called a learning space. The knowledge structure shown in Fig. 1 can be easily verified to satisfy the requirements for a learning space (moreover, it is also closed under intersection).

Deterministic structures like (6) are made probabilistic by considering a probabilistic knowledge structure (PKS), i.e., a triple $(Q,K,\pi )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q,\mathcal {K},\pi )$$\end{document} in which $\pi :K\to [0,1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :\mathcal {K}\rightarrow [0,1] $$\end{document} is a probability distribution over the knowledge states $K\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}$$\end{document} . Although usually in KST, the response pattern is denoted by $R\in 2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\in 2^Q$$\end{document} , we will use here the notation $X\in 2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\in 2^Q$$\end{document} to uniform the notation with the IRT one. The probability of a given response pattern $X\in 2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\in 2^Q$$\end{document} in the data (i.e., solved and non-solved items) is given by:

(7) $\begin{array}{cc}P\left(X\right)& =\sum _{K\in K}P\left(X\right|K)\pi \left(K\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X)&=\sum _{K\in \mathcal {K}}P(X|K)\pi (K), \end{aligned}$$\end{document}

where one only needs to set the conditional probabilities P(X|K). The most widely used model is the Basic Local Independence Model (BLIM), in which the conditional probabilities P(X|K) are written as

(8) $\begin{array}{cc}P\left(X\right|K)& =\prod _{q\in X}{\varphi }_{q,K}\prod _{q^{\prime }\in Q\backslash X}(1-{\varphi }_{q^{\prime },K}),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X|K)&= \prod _{q\in X} \phi _{q,K} \prod _{q'\in Q\setminus X} (1-\phi _{q',K}), \end{aligned}$$\end{document}

in which

(9) $\begin{array}{cc}{\varphi }_{q,K}& =\left(\begin{array}{cc}1-{\beta }_q& \text{if}q\in K\\ {\eta }_q& \text{if}q\notin K.\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \phi _{q,K}&= {\left\{ \begin{array}{ll} 1-\beta _q &{} \text {if}\,q\in K\\ \eta _q &{} \text {if}\,q\notin K. \end{array}\right. } \end{aligned}$$\end{document}

where lucky guesses $\eta =\left\{{\eta }_q\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta = \{\eta _{q}\}$$\end{document} model correct observed responses even though an item is not in the knowledge state of an individual, whereas careless errors $\beta =\left\{{\beta }_q\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = \{\beta _{q}\}$$\end{document} model incorrect observed responses even though an item is in the knowledge state of an individual.

Unidentifiability of the BLIM

Given an arbitrary knowledge structure $K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} , some of the parameters in the set ${\Gamma }_K=\{\pi ,\beta ,\eta \}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\mathcal {K}=\{\pi ,\beta ,\eta \}$$\end{document} used in the BLIM (8) can turn out to be unidentified as a consequence of a specific nature of the knowledge structure $K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} . For more detailed discussions, the reader is referred, for instance, to Spoto, Stefanutti, and Vidotto (Reference Spoto, Stefanutti and Vidotto2012; Reference Spoto, Stefanutti and Vidotto2013), Heller (Reference Heller2017), Stefanutti, Spoto and Vidotto (Reference Stefanutti, Spoto and Vidotto2018), and Stefanutti and Spoto (Reference Stefanutti and Spoto2020). For the purpose of the present manuscript, it is sufficient to introduce the notions of forward- and backward-graded knowledge structure. In more details, a knowledge structure $(Q,K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q,\mathcal {K})$$\end{document} is said to be forward-graded (FG) in an item q if $K\cup \left\{q\right\}\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\cup \{q\}\in \mathcal {K}$$\end{document} for every $K\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}$$\end{document} . Conversely, a knowledge structure $(Q,K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q,\mathcal {K})$$\end{document} is said to be backward-graded (BG) in an item q if $K\backslash \left\{q\right\}\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K{\setminus } \{q\}\in \mathcal {K}$$\end{document} for every $K\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}$$\end{document} . In the FG (BG) case, joining (removing) an item q to (from) any state of the structure always yields another state of the structure. For instance, consider the structure given in example (6). It is FG in $q_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_1$$\end{document} , but not BG in that item (e.g., $\{q_1,q_3\}\backslash \left\{q_1\right\}=\left\{q_3\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{q_1,q_3\}\setminus \{q_1\}=\{q_3\}$$\end{document} is not a state). The structure is also both FG and BG in $q_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_2$$\end{document} . It is BG in $q_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_3$$\end{document} , but it is not FG in this item ( $\varnothing \cup \left\{q_3\right\}=\left\{q_3\right\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\emptyset \cup \{q_3\}=\{q_3\}$$\end{document} is not a state). Finally, notice that the power set $K=2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}=2^Q$$\end{document} is both FG and BG in all items $q\in Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in Q$$\end{document} since any item can be both added to or removed from any state.

In order to describe the relation between FG and BG structures and identifiability, it is convenient to introduce the following collections of states:

(10) $\begin{array}{cc}{K}_q& =\{K\in K|q\in K\}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {K}_q&= \{K\in \mathcal {K}|\, q\in K\} \end{aligned}$$\end{document}

(11) $\begin{array}{cc}{\overline{K}}_q& =\{K\in K|q\notin K\}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \overline{\mathcal {K}}_q&= \{K\in \mathcal {K}|\, q\not \in K\} \end{aligned}$$\end{document}

(12) $\begin{array}{cc}{K}_q^{-}& =\{K\backslash \left\{q\right\}|K\in K_q\}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {K}_{q}^-&= \{K\setminus \{q\}|\,K\in \mathcal {K}_q\} \end{aligned}$$\end{document}

(13) $\begin{array}{cc}{\overline{K}}_q^{+}& =\{K\cup \left\{q\right\}|K\in {\overline{K}}_q\}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \overline{\mathcal {K}}_{q}^+&= \{K\cup \{q\}|\,K\in \overline{\mathcal {K}}_q\} \end{aligned}$$\end{document}

Collection (10) is the collection of all states that contain a given item. Collection (11) is the complement in $K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}$$\end{document} of the former collection. Collection (12) is the collection resulting from removing the item q from all states in $K_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_q$$\end{document} . Collection (13) is the collection resulting from adding the item q to all states in the complement of $K_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_q$$\end{document} . Let m be the dimension of the parameter space of ${\Gamma }_K=\{\eta ,\beta ,\pi \}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\mathcal {K}=\{\eta ,\beta ,\pi \}$$\end{document} , in the KST literature (see, e.g., Stefanutti and Spoto, Reference Stefanutti and Spoto2020) it has been established that, if the structure is FG in an item q, then for every $t\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \mathbb {R}$$\end{document} there exists a transformation $f_q^t:R^m\to R^m$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^t_q:\mathbb {R}^{m}\rightarrow \mathbb {R}^{m}$$\end{document} that yields a new set of parameter values ${\Gamma }_K^t=\{{\eta }^{\prime },{\beta }^{\prime },{\pi }^{\prime }\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_\mathcal {K}=\{\eta ',\beta ',\pi '\}$$\end{document} given by ${\Gamma }_K^t=f_q^t\left({\Gamma }_K\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_\mathcal {K}=f^t_q(\Gamma _{\mathcal {K}})$$\end{document} , that is

(14) $\begin{array}{c}\left(\begin{array}{c}{\beta }_p^{\prime }={\beta }_p\text{for all}p\in Q\\ {\eta }_p^{\prime }={\eta }_p\text{for all}p\in Q,p\ne q\\ {\eta }_q^{\prime }={\eta }_q+(1-{\eta }_q-{\beta }_q)(1-e^t)\\ {\pi }^{\prime }\left(K\right)=\left(\begin{array}{cc}\pi \left(K\right)+(1-e^{-t})\pi (K\backslash \left\{q\right\})& \text{for all}K\in {\overline{K}}_q^{+}\\ e^{-t}\pi \left(K\right)& \text{for all}K\in {\overline{K}}_q\\ \pi \left(K\right)& \text{for all}K\in K\backslash ({\overline{K}}_q\cup {\overline{K}}_q^{+})\end{array}\right)\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \beta '_p = \beta _p\quad \text {for all } p\in Q \\ \eta '_p = \eta _p\quad \text {for all } p\in Q, p\ne q \\ \eta '_q = \eta _q +(1-\eta _q-\beta _q)(1-e^t)\\ \pi '(K) = {\left\{ \begin{array}{ll} \pi (K)+(1-e^{-t})\pi (K\setminus \{q\}) &{} \text {for all } K\in \overline{\mathcal {K}}^+_q \\ e^{-t}\pi (K) &{} \text {for all } K\in \overline{\mathcal {K}}_q \\ \pi (K) &{}\text {for all } K\in \mathcal {K}\setminus (\overline{\mathcal {K}}_q\cup \overline{\mathcal {K}}^+_q ) \end{array}\right. } \end{array}\right. } \end{aligned}$$\end{document}

such that both sets of parameters ${\Gamma }_K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathcal {K}}$$\end{document} and ${\Gamma }_K^t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_{\mathcal {K}}$$\end{document} result in the same P(X) as given by Equation (7), that is $P\left(X\right)=P\left(X\right|{\Gamma }_K^t)=P\left(X\right|f_q^t\left({\Gamma }_K\right))=P\left(X\right|{\Gamma }_K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X)=P(X|\Gamma ^t_{\mathcal {K}}) = P(X|f_q^t(\Gamma _{\mathcal {K}})) = P(X|\Gamma _{\mathcal {K}})$$\end{document} . If instead the structure is BG in an item q, then for every $t\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \mathbb {R}$$\end{document} there exists a transformation $b_q^t:R^m\to R^m$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^t_q:\mathbb {R}^{m}\rightarrow \mathbb {R}^{m}$$\end{document} that yields a new set of parameter values ${\Gamma }_K^t=\{{\eta }^{\prime },{\beta }^{\prime },{\pi }^{\prime }\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_\mathcal {K}=\{\eta ',\beta ',\pi '\}$$\end{document} given by ${\Gamma }_K^t=b_q^t\left({\Gamma }_K\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_\mathcal {K}=b^t_q(\Gamma _{\mathcal {K}})$$\end{document} , that is

(15) $\begin{array}{c}\left(\begin{array}{c}{\eta }_p^{\prime }={\eta }_p\text{for all}p\in Q\\ {\beta }_p^{\prime }={\beta }_p\text{for all}p\in Q,p\ne q\\ {\beta }_q^{\prime }={\beta }_q+(1-{\eta }_q-{\beta }_q)(1-e^t)\\ {\pi }^{\prime }\left(K\right)=\left(\begin{array}{cc}\pi \left(K\right)+(1-e^{-t})\pi (K\cup \left\{q\right\})& \text{for all}K\in K_q^{-}\\ e^{-t}\pi \left(K\right)& \text{for all}K\in K_q\\ \pi \left(K\right)& \text{for all}K\in K\backslash (K_q\cup K_q^{-})\end{array}\right)\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \eta '_p = \eta _p\quad \text {for all } p\in Q \\ \beta '_p = \beta _p\quad \text {for all } p\in Q, p\ne q \\ \beta '_q = \beta _q +(1-\eta _q-\beta _q)(1-e^t)\\ \pi '(K) = {\left\{ \begin{array}{ll} \pi (K)+(1-e^{-t})\pi (K\cup \{q\}) &{} \text {for all } K\in \mathcal {K}^-_q \\ e^{-t}\pi (K) &{} \text {for all } K\in \mathcal {K}_q \\ \pi (K) &{}\text {for all } K\in \mathcal {K}\setminus (\mathcal {K}_q\cup \mathcal {K}^-_q ) \end{array}\right. } \end{array}\right. } \end{aligned}$$\end{document}

such that both sets of parameters ${\Gamma }_K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\mathcal {K}}$$\end{document} and ${\Gamma }_K^t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^t_{\mathcal {K}}$$\end{document} result in the same P(X) as given by Equation (7), that is $P\left(X\right)=P\left(X\right|{\Gamma }_K^t)=P\left(X\right|b_q^t\left({\Gamma }_K\right))=P\left(X\right|{\Gamma }_K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X)=P(X|\Gamma ^t_{\mathcal {K}}) = P(X|b_q^t(\Gamma _{\mathcal {K}})) = P(X|\Gamma _{\mathcal {K}})$$\end{document} . As it will be shown in what follows, the set of transformations (14) and (15) allow to derive the IRT transformations (2) and as such they allow to describe the unidentifiability of 3PL and 4PL models (or more, in general, any IRF to which guessing and slipping parameters are left-side added) as a consequence of the forward- and backward- gradedness of the power set structure in all its items. In order to do so, one needs to first better detail the relation between the IRT and the KST frameworks. It is important to stress that forward- and backward-gradedness might, however, not be the only sources of unidentifiability in KST models. Trade-offs among $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} and $\beta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} parameters can indeed also occur both in the presence and absence of forward- and backward-gradedness (see, e.g., Heller, Reference Heller2017). A complete description of the unidentifiability problem in KST for an arbitrary structure has yet to be achieved. Nonetheless, as it will be discussed in the next subsection, the KST structure underlying IRT models is the power set, and such a structure is both FG and BG in all items. As it will be shown, the IRT results on unidentifiability can be fully traced back to these transformations even when a trade-off between the left-side added parameters occurs, without affecting the IRF, as in Equation (4). Finally, it is worthy to remark that the transformations (14) and (15) have been given for arbitrary values of a parameter $t\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \mathbb {R}$$\end{document} . In what follows, in order to distinguish FG and BG transformations, we will denote their parameters as $t_F$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F$$\end{document} and $t_B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B$$\end{document} , respectively. Although these parameters are defined over the entire $R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} , their actual domains are restricted by the domain of the parameters. When transformations (14) and (15) are restricted in such a way, they are called inner transformations, and their domains have been given by Stefanutti, Spoto and Vidotto (Reference Stefanutti, Spoto and Vidotto2018). For the backward case, it must hold ${\beta }_q^{\prime }\in (0,1-{\eta }_q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \beta '_q \in (0,1-\eta _q)$$\end{document} and ${\pi }^{\prime }\left(K\right)\in (0,1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi '(K)\in (0,1)$$\end{document} , which implies $t_B\in (max[log\frac{\pi (K\cup \{q\left\}\right)}{\pi \left(K\right)+\pi (K\cup \{q\left\}\right)}],log\frac{1-{\eta }_q}{1-{\beta }_q-{\eta }_q})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B\in (\max {[\log {\frac{\pi (K\cup \{q\})}{\pi (K)+\pi (K\cup \{q\})}}]}, \log {\frac{1-\eta _q}{1-\beta _q-\eta _q}})$$\end{document} . For the forward case, it must hold ${\eta }_q^{\prime }\in (0,1-{\beta }_q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \eta '_q \in (0,1-\beta _q)$$\end{document} and ${\pi }^{\prime }\left(K\right)\in (0,1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi '(K)\in (0,1)$$\end{document} , which implies $t_F\in (max[log\frac{\pi (K\backslash \{q\left\}\right)}{\pi \left(K\right)+\pi (K\backslash \{q\left\}\right)}],log\frac{1-{\beta }_q}{1-{\beta }_q-{\eta }_q})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F\in (\max {[\log {\frac{\pi (K{\setminus }\{q\})}{\pi (K)+\pi (K{\setminus }\{q\})}}]}, \log {\frac{1-\beta _q}{1-\beta _q-\eta _q}})$$\end{document} .

3.3. The Simple Learning Model and the IRT-KST Relation

The present subsection summarizes results reported in Noventa et al. (Reference Noventa, Spoto, Heller and Kelava2019). The interested reader is referred to the original source for a more comprehensive treatise. Large domains of items imply large numbers of parameters in the set ${\Gamma }_K=\{\pi ,\beta ,\eta \}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\mathcal {K}=\{\pi ,\beta ,\eta \}$$\end{document} . A possible approach for reducing the number of parameters involves constraining the distribution of knowledge-state probabilities $\pi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} . An example of such an approach is provided by the Simple Learning Model (SLM, see, e.g., Falmagne and Doignon, Reference Falmagne and Doignon2011, p. 199) for learning spaces, that is

(16) $\begin{array}{c}\pi \left(K\right)=\prod _{q\in K}{g}_q\prod _{q^{\prime }\in K^O}(1-g_{q^{\prime }})\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi (K) = \prod \limits _{q\in K} g_q \prod \limits _{q'\in K^{\mathcal {O}}} (1- g_{q'}) \end{aligned}$$\end{document}

where $g_q\in (0,1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q\in (0,1)$$\end{document} is the probability of mastering item q and where $K^O$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{\mathcal {O}}$$\end{document} is the outer fringe of the knowledge state K, which is given by

(17) $\begin{array}{c}{K}^O:=\{q\in Q\backslash K:K\cup \left\{q\right\}\in K\}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K^{\mathcal {O}}:=\{q\in Q\setminus K : K \cup \{q\}\in \mathcal {K}\}. \end{aligned}$$\end{document}

Intuitively, the outer fringe is the set of items that can be learned next when moving from the knowledge state K. Hence, the SLM factorizes the probability of each state into the product of the probabilities of the items that have already been learned (i.e., those in K) and the complementary probabilities of the items that can be learned next (i.e., items in $K^O$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{\mathcal {O}}$$\end{document} ). Generalized versions of the SLM have been explored by Noventa, Heller and Stefanutti (Reference Noventa, Heller and Stefanutti2021). The relevance of the SLM (16) for the present manuscript is that it provides a generalized version of local stochastic independence in IRT so that, when the power set $K=2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}=2^Q$$\end{document} is considered, the combination of SLM and BLIM (8) yields exactly the likelihood of a 4-parameter IRT model in the presence of local independence. As formally shown by Noventa et al. (Reference Noventa, Spoto, Heller and Kelava2019), both the state probability $\pi \left(K\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K)$$\end{document} and the item probability $g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q$$\end{document} can be extended to encompass some latent variable $\theta \in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {R}$$\end{document} (e.g., the ability of individuals), thus yielding, respectively, some functions $\pi \left(K\right|\theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K|\theta )$$\end{document} and $g_q\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q}(\theta )$$\end{document} . While the latter is the probability of a correct response to a given item, conditional to the value of a latent variable, and as such, it captures an IRF, the former provides a state probability conditional to the value of a latent variable, and it is called a state response function (SRF). It is worth mentioning that both IRFs and SRFs can be given both a fixed-effects or a random-effects interpretation. In the former case, the latent variable $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is replaced with an incidental parameter ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} as discussed in Sect. 2, and the SRF $\pi \left(K\right|{\theta }_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K|\theta _j)$$\end{document} can be interpreted compatibly with a stochastic subject view, as the probability of an individual with a certain ability ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} to be in the knowledge state $K\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}$$\end{document} . If instead a random-effects interpretation is chosen, the SRF $\pi \left(K\right|\theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K|\theta )$$\end{document} can be interpreted compatibly with a random sampling view, as the proportion of individuals with ability $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} that are in the knowledge state $K\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \mathcal {K}$$\end{document} . In the random-effects interpretation, the state probability $\pi \left(K\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K)$$\end{document} can then be obtained by marginalizing out the ability in the SRF, that is $\pi \left(K\right)=\int d\theta \pi \left(K\right|\theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi (K) = \int d\theta \pi (K|\theta )$$\end{document} .

Given then the previously defined IRFs and SRFs, one can reformulate the SLM (16) as a generalization of the IRT notion of (strong) local stochastic independence, that is,

(18) $\begin{array}{cc}\pi \left(K\right|\theta )& =\prod _{q\in K}{g}_q\left(\theta \right)\prod _{q^{\prime }\in K^O}(1-g_{q^{\prime }}\left(\theta \right)).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi (K|\theta )&= \prod \limits _{q \in K} g_{q}(\theta )\prod \limits _{q' \in K^{\mathcal {O}}}(1- g_{q'}(\theta )). \end{aligned}$$\end{document}

Intuitively, (18) generalizes local independence in that it allows for the presence of items that cannot be mastered from a given state because their prerequisites are not satisfied (i.e., they are not in the outer fringe). The traditional formulation of local independence is retrieved only if the knowledge structure is a power set, $K=2^Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K} = 2^Q$$\end{document} . Indeed, in such case, one can master or fail any of the items in $K^O=Q\backslash K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{\mathcal {O}}=Q\setminus K$$\end{document} as required by local independence. Let $X_q={\chi }_K\left(q\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_q=\chi _{K}(q)$$\end{document} the indicator function that takes value one if $q\in K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in K$$\end{document} and zero if $q\notin K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\notin K$$\end{document} , then one has

(19) $\begin{array}{cc}\pi \left(K\right|\theta )& =\prod _{q\in K}{g}_q\left(\theta \right)\prod _{q^{\prime }\in Q\backslash K}(1-g_{q^{\prime }}\left(\theta \right))=\prod _{q\in Q}{g}_q{\left(\theta \right)}^{X_q}{(1-g_q\left(\theta \right))}^{1-X_q}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \pi (K|\theta )&= \prod \limits _{q\in K} g_q(\theta ) \prod \limits _{q'\in Q\setminus K} (1- g_{q'}(\theta ))= \prod \limits _{q\in Q} g_q(\theta )^{X_q}(1- g_{q}(\theta ))^{1-X_q} \end{aligned}$$\end{document}

which for $g_q\left(\theta \right)=P(X_q=1|\theta )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta )=P(X_q=1|\theta )$$\end{document} yields the traditional IRT definition of (strong) local stochastic independence. As a further consequence, the combination of SLM plus BLIM in the KST-IRT framework yields the likelihood of the 4-parameter IRT model. Indeed, let ${\Gamma }_K=\{\eta ,\beta ,g\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\mathcal {K}=\{\eta , \beta , g\}$$\end{document} be the KST set of parameters, then Equation (7) can be rewritten, by substitution of Equations (8) and (16), as

$\begin{array}{cc}P\left(X\right|\theta ,{\Gamma }_K)& =\sum _{K\in 2^Q}P\left(X\right|K)\pi \left(K\right|\theta )\\ & =\sum _{K\in 2^Q}\prod _{q\in X}{\varphi }_{q,K}\prod _{q^{\prime }\in Q\backslash X}(1-{\varphi }_{q^{\prime },K})\prod _{q\in K}{g}_q\left(\theta \right)\prod _{q^{\prime }\in Q\backslash K}(1-g_{q^{\prime }}\left(\theta \right))\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X|\theta , \Gamma _\mathcal {K})&= \sum _{K\in 2^Q}P(X|K)\pi (K|\theta )\\&= \sum _{K\in 2^Q}\prod _{q\in X} \phi _{q,K} \prod _{q'\in Q\setminus X} (1-\phi _{q',K}) \prod \limits _{q\in K} g_q(\theta ) \prod \limits _{q'\in Q\setminus K} (1- g_{q'}(\theta )) \end{aligned}$$\end{document}

which can be shown (see, Noventa et al., Reference Noventa, Spoto, Heller and Kelava2019, Theorem 3 for a proof of the result) to be equivalent to

(20) $\begin{array}{cc}P\left(X\right|\theta ,{\Gamma }_K)& =\prod _{X_q}P{(X_q=1|\theta ,{\Gamma }_K)}^{X_q}{(1-P(X_q=1|\theta ,{\Gamma }_K))}^{1-X_q}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X|\theta , \Gamma _\mathcal {K})&= \prod \limits _{X_q} P(X_q=1|\theta , \Gamma _\mathcal {K})^{X_q}(1- P(X_q=1|\theta , \Gamma _\mathcal {K}))^{1-X_q} \end{aligned}$$\end{document}

where

(21) $\begin{array}{cc}P(X_q=1|\theta ,{\Gamma }_K)& ={\eta }_q+(1-{\beta }_q-{\eta }_q)g_q\left(\theta \right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X_q=1|\theta , \Gamma _\mathcal {K})&= \eta _q+(1-\beta _q-\eta _q)g_q(\theta ) \end{aligned}$$\end{document}

which is clearly equivalent to the likelihood of a 4-parameter IRT model, e.g., the 4PL in Equation (1), as soon as one identifies ${\eta }_q:=c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _q:= c_i$$\end{document} , ${\beta }_q:=d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _q:= d_i$$\end{document} , and $g_q\left(\theta \right):=P(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta ): =P(X_i=1|\theta ,\Gamma ^2_i)$$\end{document} so that ${\Gamma }_K:={\Gamma }_i^4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _\mathcal {K}:=\Gamma ^4_i$$\end{document} . It is important to remark that, as a consequence of the results summarized in the present section, traditional IRT models, which assume local stochastic independence, are, therefore, the power set case of a KST-IRT approach. Assuming any other structure except the power set amounts to not assume local stochastic independence, and it has been suggested to provide an alternative way of modeling local dependence between items (see, e.g., Noventa et al., Reference Noventa, Spoto, Heller and Kelava2019; Ye et al., Reference Ye, Kelava and Noventa2023). Since the present work aims to recover traditional IRT unidentifiability results, the only knowledge structure considered is the power set, as it is the only structure equivalent to the traditional IRT models that assume local independence. Further considerations on the use of other structures are briefly given in the discussion section.

4.1. Unidentifiability of the SLM Model in the Power Set Case

Since the power set is FG and BG in any item q, the following transformations of the parameters ${\Gamma }_K=\{\eta ,\beta ,g\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Gamma _{\mathcal {K}}=\{\eta ,\beta , g\}$$\end{document} of the SLM can be obtained

(22) $\begin{array}{c}\left(\begin{array}{cc}{\beta }_q^{\prime }={\beta }_q+(1-{\eta }_q-{\beta }_q)(1-e^{t_B})& \\ {\eta }_q^{\prime }={\eta }_q+(1-{\eta }_q-{\beta }_q)(1-e^{t_F})& \\ g_q^{\prime }=\frac{g_q+e^{t_F}-1}{e^{t_F}+e^{t_B}-1}& \\ g_p^{\prime }=g_p,{\eta }_p^{\prime }={\eta }_p,{\beta }_p^{\prime }={\beta }_p& \text{for all}p\ne q,\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \beta '_q = \beta _q+(1-\eta _q -\beta _q)(1-e^{t_B})&{}\\ \eta '_q = \eta _q+(1-\eta _q -\beta _q)(1-e^{t_F})&{} \\ g'_q = \frac{g_q+e^{t_F}-1}{e^{t_F}+e^{t_B}-1} &{} \\ g'_p = g_p, \quad \eta '_p = \eta _p, \quad \beta '_p = \beta _p &{}\text {for all } p\ne q, \end{array}\right. } \end{aligned}$$\end{document}

where $t_B\in (log{g}_q,log\frac{1-{\eta }_q}{1-{\beta }_q-{\eta }_q})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B\in (\log {g_q}, \log {\frac{1-\eta _q}{1-\beta _q-\eta _q}})$$\end{document} and $t_F\in (log(1-g_q),log\frac{1-{\beta }_q}{1-{\beta }_q-{\eta }_q})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F\in (\log {(1-g_q)}, \log {\frac{1-\beta _q}{1-\beta _q-\eta _q}})$$\end{document} with the additional condition that $e^{t_F}+e^{t_B}>1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{t_F}+e^{t_B}>1$$\end{document} . A proof of system (22) and the associated ranges for $t_F$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F$$\end{document} and $t_B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B$$\end{document} are given in Appendix A. Since all items follow Equations (22) there are as many trade-off parameters $t_B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B$$\end{document} and $t_F$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F$$\end{document} as there are items. Hence, it is convenient to denote them as $t_F^q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_F^q$$\end{document} and $t_B^q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_B^q$$\end{document} to highlight such a dependence. It is shown in the next subsection that the proliferation of trade-off parameters can be reduced in IRT if the mathematical forms of the 2- and 4-parameter IRFs are preserved. Before moving to the IRT case, further considerations on the KST case are provided. For the purposes of what follows, it is convenient to redefine the trade-off parameters as $r_T^q=1-e^{t_T^q}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_T = 1-e^{t^q_T}$$\end{document} for $T\in \{B,F\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in \{B,F\}$$\end{document} . In such a way, the transformations (22) can be rewritten as:

where $r_B^q\in (-\frac{{\beta }_q}{1-{\beta }_q-{\eta }_q},1-g_q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_B\in (-\frac{\beta _q}{1-\beta _q-\eta _q}, 1-g_q)$$\end{document} and $r_F^q\in (-\frac{{\eta }_q}{1-{\beta }_q-{\eta }_q},g_q)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_F\in (-\frac{\eta _q}{1-\beta _q-\eta _q}, g_q)$$\end{document} with the additional condition that $r_F^q+r_B^q<1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_F+r^q_B < 1$$\end{document} . Equations (23a) and (23b) state that forward and backward gradedness respectively affect the lucky guess ${\eta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _q$$\end{document} and the careless error ${\beta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _q$$\end{document} by means of the trade-off parameters, $r_F^q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_F$$\end{document} and $r_B^q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_B$$\end{document} . By considering the differences $\Delta {\eta }_q={\eta }_q^{\prime }-{\eta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \eta _q = \eta '_q-\eta _q$$\end{document} and $\Delta {\beta }_q={\beta }_q^{\prime }-{\beta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \beta _q = \beta '_q-\beta _q$$\end{document} , these trade-off parameters can be interpreted as re-scaled differences of left-side added parameters, that is $r_F^q=\frac{\Delta {\eta }_q}{1-{\eta }_q-{\beta }_q}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_F = \frac{\Delta \eta _q}{1-\eta _q-\beta _q}$$\end{document} and $r_B^q=\frac{\Delta {\beta }_q}{1-{\eta }_q-{\beta }_q}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^q_B = \frac{\Delta \beta _q}{1-\eta _q-\beta _q}$$\end{document} . Equation (23c) states that both forward- and backward-gradedness affect the item probability $g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q$$\end{document} in the SLM by means of the same trade-off parameters. In particular, the difference $\Delta g_q=g_q^{\prime }-g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta g_q = g'_q-g_q$$\end{document} can be rewritten as:

(24) $\begin{array}{cc}\Delta g_q& =\frac{r_B^q{g}_q-r_F^q(1-g_q)}{1-r_B^q-r_F^q},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta g_q&= \frac{r^q_Bg_q-r^q_F(1-g_q) }{1-r^q_B-r^q_F}, \end{aligned}$$\end{document}

which highlights how the change in the item probability parameter is an exchange between the probability of failing and that of succeeding, mediated by the trade-off parameters capturing the associated changes in the left-side added parameters. Hence, the KST transformations highlight a trade-off between the left-side added parameters and the g-parameters, which in the IRT case transfers into a trade-off between the left-side added parameters and the 2-parameter IRF. Equations (23d) state that all the parameters associated with any item $p\ne q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ne q$$\end{document} are not affected by the forward- and backward-gradedness of the structure in q. In order to better understand the implications of transformations (22) for the IRT models, it is convenient to stress that they provide the solution to the following equation

(25) $\begin{array}{c}{\eta }_q^{\prime }+(1-{\eta }_q^{\prime }-{\beta }_q^{\prime })g_q^{\prime }={\eta }_q+(1-{\eta }_q-{\beta }_q)g_q,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta '_q +(1-\eta '_q-\beta '_q)g'_q = \eta _q +(1-\eta _q-\beta _q)g_q, \end{aligned}$$\end{document}

which is clearly (minus the dependence on $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} ) the unidentifiability condition of a 4-parameter model as in Sect. 2. By considering the differences $\Delta {\eta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \eta _q$$\end{document} , $\Delta {\beta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \beta _q$$\end{document} , and $\Delta g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta g_q$$\end{document} , Equation (25) can be re-written as:

(26) $\begin{array}{c}(1-g_q)\Delta {\eta }_q-g_q\Delta {\beta }_q+(1-{\eta }_q-{\beta }_q)\Delta g_q-(\Delta {\eta }_q+\Delta {\beta }_q)\Delta g_q=0\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (1-g_q)\Delta \eta _q- g_q\Delta \beta _q + (1- \eta _q-\beta _q)\Delta g_q -(\Delta \eta _q+\Delta \beta _q)\Delta g_q = 0 \end{aligned}$$\end{document}

and it is straightforward to verify that, by substitution of Equations (23a) and (23b), Equation (26) becomes Equation (24), which is indeed a consequence of Equation (23c). Although such a result is expected as transformations (23) are the solutions to Equation (25), this rewriting highlights that other solutions, which have been identified in the literature, are sub-cases of System (23). Although there are several potential sub-cases, up to our knowledge, only two have been considered in the IRT literature:

1. 1. If $\Delta g_q=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta g_q =0$$\end{document} , then the g-parameter is constant under transformations (23), and one obtains

   (27) $\begin{array}{c}(1-g_q)\Delta {\eta }_q-g_q\Delta {\beta }_q=0\iff \Delta {\eta }_q=\frac{g_q}{1-g_q}\Delta {\beta }_q\iff r_F^q=\frac{g_q}{1-g_q}{r}_B^q\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (1-g_q)\Delta \eta _q - g_q \Delta \beta _q = 0 \quad \Leftrightarrow \quad \Delta \eta _q = \frac{g_q}{1-g_q}\Delta \beta _q \quad \Leftrightarrow \quad r^q_F = \frac{g_q}{1-g_q} r^q_B \end{aligned}$$\end{document}
   that represents an exclusive trade-off between the conditional error parameters in KST, which was described by Heller (Reference Heller2017). There, the trade-off was obtained by assuming that $\pi \left(K\right)=\pi (K\cup \{q\left\}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (K)=\pi (K\cup \{q\})$$\end{document} , which corresponds to assume $1-g_q=g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-g_q = g_q$$\end{document} in the SLM. In the KST context, Equation (27) can be generalized to $\Delta {\eta }_q=\frac{\pi (K\cup \{q\left\}\right)}{\pi \left(K\right)}\Delta {\beta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \eta _q = \frac{\pi (K\cup \{q\})}{\pi (K)}\Delta \beta _q$$\end{document} for all $K\in {\overline{K}}_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in \overline{\mathcal {K}}_q$$\end{document} and is therefore a sub-case of a structure which is both BG and FG in an item. This case is shown in Sect. 4.3 to correspond to the IRT case described by Ogasawara (Reference Ogasawara2017, Theorem 5), and given in Sect. 2 by System (4).

2. 2. Sub-cases of transformations (23) can be obtained by splitting Equation (26) into a system of more equations. For instance, one might consider the following split with associated solutions

   (28) $\begin{array}{c}\left(\begin{array}{c}(1-g_q-\Delta g_q)\Delta {\eta }_q+(1-{\eta }_q)\Delta g_q=0\\ (g_q+\Delta g_q)\Delta \beta +{\beta }_q\Delta g_q=0\end{array}\right)\iff \left(\begin{array}{c}\Delta {\eta }_q=-(1-{\eta }_q)\frac{\Delta g_q}{(1-g_q-\Delta g_q)}\\ \Delta {\beta }_q=-{\beta }_q\frac{\Delta g_q}{g_q+\Delta g_q}\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} (1-g_q -\Delta g_q)\Delta \eta _q + (1- \eta _q)\Delta g_q = 0\\ (g_q+\Delta g_q)\Delta \beta + \beta _q\Delta g_q = 0 \end{array}\right. } \quad \Leftrightarrow {\left\{ \begin{array}{ll} \Delta \eta _q = - (1- \eta _q)\frac{\Delta g_q}{(1-g_q -\Delta g_q)}\\ \Delta \beta _q = -\beta _q\frac{\Delta g_q}{g_q +\Delta g_q} \end{array}\right. }, \end{aligned}$$\end{document}
   which will be shown to correspond to the sub-case given in Sect. 2 by Equation (5).

4.2. Application of the KST Transformations to the KST-IRT Case

According to the KST transformations (23), the 2-parameter IRF $g_q\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta )$$\end{document} is expected to trade-off with the left-side added parameters. However, such a trade-off can occur in two distinct situations: the first one, in which only the KST transformations (23) are applied; and the second one, in which additional assumptions are imposed such that the mathematical form of the 2-parameter IRFs and/or the 4-parameter IRFs is preserved. The former case yields empirically indistinguishable models, in which different values of the left-side added parameters are associated with different 2-parameter IRFs, but the parameters within the 2-parameter IRFs $g_q\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta )$$\end{document} are unaffected. The latter case yields instead a more nuanced situation in which the parameters $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} and ${\Gamma }_i^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{2}_i$$\end{document} within the 2-parameter IRFs $g_q\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta )$$\end{document} are affected by the transformations, and both empirical indistinguishability and unidentifiability can manifest themselves, depending on whether or not the transformations also preserve the mathematical form of the 4-parameter IRF.

4.2.1. Empirically Indistinguishable IRFs

Application of transformations (23) to a 4-parameter IRF yields the following system of equations

in which a new set of 4-parameter IRFs is provided that yields the same probability of the responses. System (29) is a case of empirical indistinguishability as the mathematical forms of the 2-parameter IRFs $g_q\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_q(\theta )$$\end{document} and $g_q^{\prime }\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g'_{q}(\theta )$$\end{document} are different. In order to discuss the IRT case, it is convenient to rewrite System (29) in a notation that is more consistent with the IRT notation used in Sect. 2 and to invert it to match the mathematical form of the results that can be found in the IRT literature on unidentifiability. Given then the previous system, it is convenient to invert Equations (29a) and (29b) to obtain the transformations

$\begin{array}{c}\left(\begin{array}{c}{\eta }_q=\frac{1}{1-r_B^q-r_F^q}[(1-r_B^q){\eta }_q^{\prime }-(1-{\beta }_q^{\prime })r_F^q]\\ {\beta }_q=\frac{1}{1-r_B^q-r_F^q}[(1-r_F^q){\beta }_q^{\prime }-(1-{\eta }_q^{\prime })r_B^q]\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \eta _q = \frac{1}{1-r^q_B-r^q_F}[ (1-r^q_B)\eta '_q-(1-\beta '_q)r^q_F]\\ \beta _q = \frac{1}{1-r^q_B-r^q_F}[ (1-r^q_F)\beta '_q-(1-\eta '_q)r^q_B] \end{array}\right. } \end{aligned}$$\end{document}

so that, once set $c_i:={\eta }_{q_i}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_i:= \eta '_{q_i}$$\end{document} , $c_i^{\ast }:={\eta }_{q_i}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c^*_i:= \eta _{q_i}$$\end{document} , $d_i:={\beta }_{q_i}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i:= \beta '_{q_i} $$\end{document} , $d_i^{\ast }:={\beta }_{q_i}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i:= \beta _{q_i}$$\end{document} , $r_B^i:=r_B^{q_i}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B:= r^{q_i}_B$$\end{document} , $r_F^i:=r_F^{q_i}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F:= r^{q_i}_F$$\end{document} , and once identified the IRFs $P^{\ast }(X_i=1|\theta ,{\Gamma }_i^2):=g_{q_i}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P^*(X_i=1|\theta , \Gamma ^{2}_i):= g_{q_i}(\theta )$$\end{document} and $P(X_i=1|\theta ,{\Gamma }_i^2):=g_{q_i}^{\prime }\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , \Gamma ^{2}_i):= g'_{q_i}(\theta )$$\end{document} we can then set the general transformations for empirical indistinguishability in the IRT 4-parameter case as

and where $P^{\ast }(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^*(X_i=1|\theta , \Gamma ^{2}_i)$$\end{document} and $P(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , \Gamma ^{2}_i)$$\end{document} depend on the same set of parameters $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} and ${\Gamma }_i^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{2}_i$$\end{document} but have different mathematical form. Let ${\Gamma }_i^{4\ast }=\{a_i,b_i,c_i^{\ast },d_i^{\ast }\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{4*}_i=\{a_i,b_i,c^*_i,d^*_i\}$$\end{document} , the KST transformations yield the 4-parameter IRFs $P^{\ast }(X_i=1|\theta ,{\Gamma }_i^{4\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^*(X_i=1|\theta , \Gamma ^{4*}_i)$$\end{document} that are empirically indistinguishable from the 4-parameter IRFs $P(X_i=1|\theta ,{\Gamma }_i^4)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , \Gamma ^{4}_i)$$\end{document} , that is $P(X_i=1|\theta ,{\Gamma }_i^4)=P^{\ast }(X_i=1|\theta ,{\Gamma }_i^{4\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , \Gamma ^{4}_i) = P^*(X_i=1|\theta , \Gamma ^{4*}_i)$$\end{document} . Like in the KST case, there are as many trade-off parameters $r_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F$$\end{document} and $r_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B$$\end{document} as there are items. In passing, one might read the newly obtained $P^{\ast }(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^*(X_i=1|\theta , \Gamma ^{2}_i)$$\end{document} in Equation (30c) themselves as 4-parameter IRFs in which the trade-off parameters act similarly to left-side added parameters but can attain negative values. System (30) holds in both fixed- and random-effects cases. In the fixed-effects case, the empirical indistinguishability concerns the 4-parameter IRFs $P^{\ast }(X_{\mathrm{ji}}=1|{\theta }_j,{\Gamma }_i^{4\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^*(X_{ji}=1|\theta _j, \Gamma ^{4*}_i)$$\end{document} . In the random-effects case, the empirical indistinguishability can be interpreted in terms of either the likelihood of a response pattern X or of the marginal probability of a correct response to a given item. In the former case, it indeed holds that

$\begin{array}{cc}P\left(X\right|{\Gamma }_i^4)& =\int f(\theta ;\mu ,\sigma )\prod _{i=1}^{\left|Q\right|}P\left(X_i\right|\theta ,{\Gamma }_i^4)d\theta \\ & =\int f(\theta ;\mu ,\sigma )\prod _{i=1}^{\left|Q\right|}{P}^{\ast }\left(X_i\right|\theta ,{\Gamma }_i^{4\ast })d\theta =P^{\ast }\left(X\right|{\Gamma }_i^{4\ast })\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X|\Gamma ^{4}_i)&= \int f(\theta ; \mu , \sigma ) \prod _{i = 1}^{|Q|}P(X_i|\theta , \Gamma ^{4}_i)d\theta \\&= \int f(\theta ; \mu , \sigma ) \prod _{i = 1}^{|Q|}P^*(X_i|\theta , \Gamma ^{4*}_i)d\theta = P^*(X|\Gamma ^{4*}_i) \end{aligned}$$\end{document}

so that both sets of IRFs return the same likelihood of a pattern of responses. In the latter case, by local independence, the marginal probability $P(X_i=1|{\Gamma }_i^4)={\sum }_{X\in 2^Q,X_i=1}P\left(X\right|{\Gamma }_i^4)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\Gamma ^{4}_i) = \sum _{X \in 2^Q, X_i =1} P(X|\Gamma ^{4}_i)$$\end{document} of a correct response to the i-th item coincides with the marginalization of the 4-parameter IRFs, that is

$\begin{array}{cc}P(X_i=1|{\Gamma }_i^4)& =c_i+(1-c_i-d_i)\int f(\theta ;\mu ,\sigma )P\left(X_i\right|\theta ,{\Gamma }_i^2)d\theta \\ & =c_i^{\ast }+(1-c_i^{\ast }-d_i^{\ast })\int f(\theta ;\mu ,\sigma )P^{\ast }\left(X_i\right|\theta ,{\Gamma }_i^2)d\theta =P^{\ast }(X_i=1|{\Gamma }_i^{4\ast }),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X_i=1|\Gamma ^{4}_i)&= c_i+(1-c_i-d_i)\int f(\theta ; \mu ,\sigma )P(X_i|\theta , \Gamma ^{2}_i)d\theta \\&= c^*_i+(1-c^*_i-d^*_i)\int f(\theta ; \mu ,\sigma )P^*(X_i|\theta , \Gamma ^{2}_i)d\theta = P^*(X_i=1|\Gamma ^{4*}_i), \end{aligned}$$\end{document}

and one can interpret Equation (30c) as a transformation of the marginal probability $P(X_i=1|{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\Gamma ^{2}_i)$$\end{document} , that is

$\begin{array}{cc}{P}^{\ast }(X_i=1|{\Gamma }_i^2)& =\int f(\theta ;\mu ,\sigma )P^{\ast }(X_i=1|\theta ,{\Gamma }_i^2)d\theta =r_F^i+[1-r_B^i-r_F^i]P(X_i=1|{\Gamma }_i^2).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P^*(X_i=1|\Gamma ^{2}_i)&= \int f(\theta ; \mu ,\sigma ) P^*(X_i=1|\theta , \Gamma ^{2}_i)d\theta = r^i_F+[1-r^i_B-r^i_F]P(X_i=1|\Gamma ^{2}_i). \end{aligned}$$\end{document}

Finally, it is worth mentioning that in the random-effects case, the distribution of the latent trait $f(\theta ;\mu ,\sigma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta ;\mu ,\sigma )$$\end{document} is unaffected by the transformations (30), so the empirical indistinguishability only concerns the IRFs.

4.2.2. Preserving the Mathematical Form of the IRFs

The IRT results summarized in Sect. 2 can be obtained if System (30) is supplemented with the assumption that the mathematical forms of the IRFs must be preserved. Let us assume that only the mathematical form of the 2-parameter IRFs is preserved. The transformation (30c) must then imply changes in the parameters of the 2-parameter IRF. Namely, there exist ${\theta }^{\ast }\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^*\in \mathbb {R}$$\end{document} and ${\Gamma }_i^{2\ast }=\{b_i^{\ast },a_i^{\ast }\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ^{2*}_i=\{b^*_i, a^*_i\}$$\end{document} such that one can replace $P^{\ast }(X_i=1|\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^*(X_i=1|\theta , \Gamma ^{2}_i)$$\end{document} with $P(X_i=1|{\theta }^{\ast },{\Gamma }_i^{2\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i =1|\theta ^*, \Gamma ^{2*}_i)$$\end{document} in Equation (30c), thus yielding the system

such that $P(X_i=1|\theta ,{\Gamma }_i^4)=P(X_i=1|{\theta }^{\ast },{\Gamma }_i^{4\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , \Gamma ^{4}_i) = P(X_i=1|\theta ^*, \Gamma ^{4*}_i)$$\end{document} . Equation (30d) has been removed since changes propagate to all IRFs via the $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} parameter. It follows that the trade-off between the left-side added parameters and the 2-parameter IRF, which was occurring in the system of transformations (30) for each item independently, is not anymore independent. That is, the trade-off parameters $r_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F$$\end{document} and $r_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B$$\end{document} must lose some if not all, their independence. In order to explore these implications, it is convenient to rewrite the 2-parameter IRF $g_{q_i}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta )$$\end{document} as $g_{q_i}(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta , \Gamma ^2_i)$$\end{document} to underline the item parameters. Therefore, Equation (31c) can be rewritten as

(32) $\begin{array}{cc}{g}_{q_i}({\theta }^{\ast },{\Gamma }_i^{2\ast })& =r_F^i+[1-r_B^i-r_F^i]g_{q_i}(\theta ,{\Gamma }_i^2),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_{q_i}(\theta ^*,\Gamma ^{2*}_i)&= r^i_F+[1-r^i_B-r^i_F]g_{q_i}(\theta , \Gamma ^2_i), \end{aligned}$$\end{document}

which establishes the general relation that the sets of parameters $({\theta }^{\ast },{\Gamma }_i^{2\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta ^*,\Gamma ^{2*}_i)$$\end{document} and $(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \Gamma ^{2}_i)$$\end{document} must obey in order for the mathematical form of the 2-parameter IRFs to be preserved. Any transformation of the parameters that satisfy Equation (32) provides either empirically indistinguishable or unidentifiable transformations for the 4-parameter IRFs in System (31). However, not all possible transformations that satisfy (32) are also meaningful in IRT. For instance, one might want to restrict the set of all possible transformations to those in which the ability parameter ${\theta }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^*$$\end{document} is independent of the difficulty parameter $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} (same for $b_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^*_i$$\end{document} and $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} ). For this parameter independence to hold, one needs to transform the trade-off parameters $r_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F$$\end{document} and $r_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B$$\end{document} into trade-off functions $r_F^i(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F(\theta , \Gamma ^2_i)$$\end{document} and $r_B^i(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B(\theta , \Gamma ^2_i)$$\end{document} that cancel out the dependence on the undesired parameters. However, the functional forms of $r_F^i(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F(\theta , \Gamma ^2_i)$$\end{document} and $r_B^i(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B(\theta , \Gamma ^2_i)$$\end{document} must be set on a case-by-case basis as they depend on the specific choice of the 2-parameter IRF $g_{q_i}(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta ,\Gamma ^{2}_i)$$\end{document} . As an example, let us consider the 1-parameter IRF $g_{q_i}(\theta ,b_i)=\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g_{q_i}(\theta , b_i) = \frac{G(\theta )}{G(\theta )+H(b_i)}$$\end{document} of Equation (3), which was suggested by Ogasawara (Reference Ogasawara2017) to generalize the 1PL-G model (for which $H=G=exp$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=G=\exp $$\end{document} ). By substitution of the IRF (3), Equation (32) becomes

(33) $\begin{array}{c}\frac{G\left({\theta }^{\ast }\right)}{G\left({\theta }^{\ast }\right)+H\left(b_i^{\ast }\right)}=r_F^i+[1-r_B^i-r_F^i]\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}=\frac{r_F^{i}H\left(b_i\right)+(1-r_B^i)G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{G(\theta ^*)}{G(\theta ^*)+H(b^*_i)} = r^i_F+[1-r^i_B-r^i_F]\frac{G(\theta )}{G(\theta )+H(b_i)} = \frac{r^i_FH(b_i)+(1-r^i_B)G(\theta )}{G(\theta )+H(b_i)} \end{aligned}$$\end{document}

that, after a little algebra, can be rewritten as

(34) $\begin{array}{c}\frac{G\left({\theta }^{\ast }\right)}{H\left(b_i^{\ast }\right)}=\frac{r_F^{i}H\left(b_i\right)+(1-r_B^i)G\left(\theta \right)}{r_B^{i}G\left(\theta \right)+(1-r_F^i)H\left(b_i\right)}=\frac{G\left(\theta \right)+r_F^{i}H\left(b_i\right)-r_B^{i}G\left(\theta \right)}{H\left(b_i\right)-(r_F^{i}H\left(b_i\right)-r_B^{i}G\left(\theta \right))}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{G(\theta ^*)}{H(b^*_i)} = \frac{r^i_FH(b_i)+(1-r^i_B)G(\theta )}{r^i_BG(\theta )+(1-r^i_F)H(b_i)} = \frac{G(\theta )+r^i_FH(b_i)-r^i_BG(\theta )}{H(b_i)-(r^i_FH(b_i)-r^i_BG(\theta ))}. \end{aligned}$$\end{document}

Any solution to Equation (34) satisfies system (31). However, if one wants the transformations of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} and $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} to be independent on the other parameter, as is the case in IRT, one needs to consider the solution

(35) $\begin{array}{c}\left(\begin{array}{c}G\left({\theta }^{\ast }\right)=kG\left(\theta \right)+k(r_F^{i}H\left(b_i\right)-r_B^{i}G\left(\theta \right))\\ H\left(b_i^{\ast }\right)=kH\left(b_i\right)-k(r_F^{i}H\left(b_i\right)-r_B^{i}G\left(\theta \right))\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} G(\theta ^*) = kG(\theta )+k(r^i_FH(b_i)-r^i_BG(\theta ))\\ H(b^*_i) = kH(b_i)-k(r^i_FH(b_i)-r^i_BG(\theta )) \end{array}\right. } \end{aligned}$$\end{document}

for some $k\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {R}$$\end{document} , and require that the term $\ell :=r_F^{i}H\left(b_i\right)-r_B^{i}G\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell := r^i_FH(b_i)-r^i_BG(\theta )$$\end{document} be a constant $\ell \in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in \mathbb {R}$$\end{document} . This requires the trade-off parameters $r_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F$$\end{document} and $r_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B$$\end{document} to become the trade-off functions

(36) $\begin{array}{c}\left(\begin{array}{c}{r}_F^i\left(b_i\right)={\ell }_F^{i}H{\left(b_i\right)}^{-1}\\ r_B^i\left(\theta \right)=-{\ell }_B^{i}G{\left(\theta \right)}^{-1}\\ \ell ={\ell }_F^i+{\ell }_B^i\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} r^i_F(b_i) = \ell ^i_F H(b_i)^{-1}\\ r^i_B(\theta ) = -\ell ^i_B G(\theta )^{-1}\\ \ell = \ell ^i_F+\ell ^i_B \end{array}\right. } \end{aligned}$$\end{document}

for some newly defined trade-off parameters ${\ell }_F^i,{\ell }_B^i\in R$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_F,\ell ^i_B\in \mathbb {R}$$\end{document} . The trade-off functions $r_F^i\left(b_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F(b_i)$$\end{document} and $r_B^i\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_B(\theta )$$\end{document} remove the dependence of ${\theta }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^*$$\end{document} and $b_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i^*$$\end{document} on, respectively, $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} and $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} , thus yielding the system

(37) $\begin{array}{c}\left(\begin{array}{c}{\theta }^{\ast }=G^{-1}(kG\left(\theta \right)+k\ell )\\ b_i^{\ast }=H^{-1}(kH\left(b_i\right)-k\ell )\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \theta ^* = G^{-1}(kG(\theta )+k\ell )\\ b^*_i = H^{-1}(kH(b_i)-k\ell ) \end{array}\right. } \end{aligned}$$\end{document}

that provides the transformations discussed by Ogasawara (Reference Ogasawara2017; Reference Ogasawara2020) for the IRF (3). By combining together systems (31), (36), and (37), and by setting $k=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} for practical purposes, one obtains for the 1-parameter model $P(X_i=1|\theta ,b_i)=\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta , b_i) = \frac{G(\theta )}{G(\theta )+H(b_i)}$$\end{document} with guessing and slipping parameters, the system

(38) $\begin{array}{c}\left(\begin{array}{c}{c}_i^{\ast }=\frac{1}{1-r_B^i\left(\theta \right)-r_F^i\left(b_i\right)}[(1-r_B^i\left(\theta \right))c_i-(1-d_i)r_F^i\left(b_i\right)]\\ d_i^{\ast }=\frac{1}{1-r_B^i\left(\theta \right)-r_F^i\left(b_i\right)}[(1-r_F^i\left(b_i\right))d_i-(1-c_i)r_B^i\left(\theta \right)]\\ P(X_i=1|{\theta }^{\ast },b_i^{\ast })=r_F^i\left(b_i\right)+[1-r_B^i\left(\theta \right)-r_F^i\left(b_i\right)]P(X_i=1|\theta ,b_i)\\ G\left({\theta }^{\ast }\right)=G\left(\theta \right)+\ell ,H\left(b_i^{\ast }\right)=H\left(b_i\right)-\ell \\ r_F^i\left(b_i\right)={\ell }_F^{i}H{\left(b_i\right)}^{-1},r_B^i\left(\theta \right)=-{\ell }_B^{i}G{\left(\theta \right)}^{-1},\ell ={\ell }_F^i+{\ell }_B^i\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} c^*_i = \frac{1}{1-r^i_B(\theta )-r^i_F(b_i)}[ (1-r^i_B(\theta ))c_i-(1-d_i)r^i_F(b_i)]\\ d^*_i = \frac{1}{1-r^i_B(\theta )-r^i_F(b_i)}[ (1-r^i_F(b_i))d_i-(1-c_i)r^i_B(\theta )]\\ P(X_i=1|\theta ^*, b^*_i) = r^i_F(b_i)+[1-r^i_B(\theta )-r^i_F(b_i)]P(X_i=1|\theta , b_i) \\ G(\theta ^*) = G(\theta )+\ell , \quad H(b^*_i) = H(b_i)-\ell \\ r^i_F(b_i) = \ell ^i_F H(b_i)^{-1}, \quad r^i_B(\theta ) = -\ell ^i_B G(\theta )^{-1},\quad \ell = \ell ^i_F+\ell ^i_B \end{array}\right. } \end{aligned}$$\end{document}

such that $P(X_i=1|\theta ,{\Gamma }_i^{1,3})=P(X_i=1|{\theta }^{\ast },{\Gamma }_i^{1,3\ast })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X_i=1|\theta ,\Gamma ^{1,3}_i)=P(X_i=1|\theta ^*,\Gamma ^{1,3*}_i)$$\end{document} . Before moving to the logistic case, which yields the traditional IRT results, several general remarks are in order.

1. 1. The trade-off parameters ${\ell }_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B$$\end{document} and ${\ell }_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_F$$\end{document} are item-dependent: This is evident in the $\ell =0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =0$$\end{document} case, in which for each item a trade-off occurs exclusively between the left-side added parameters without involving the 2-parameter IRF. Indeed, for $\ell =0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =0$$\end{document} it holds ${\theta }^{\ast }=\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^* = \theta $$\end{document} and $b_i^{\ast }=b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^*_i = b_i$$\end{document} so that $g_{q_i}({\theta }^{\ast },b_i^{\ast })=g_{q_i}(\theta ,b_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta ^*, b^*_i)=g_{q_i}(\theta , b_i)$$\end{document} , but one can still have ${\ell }_B^i+{\ell }_F^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B+\ell ^i_F=0$$\end{document} , that is $r_F^i\left(b_i\right)H\left(b_i\right)=r_B^i\left(\theta \right)G\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ r^i_F(b_i)H(b_i)=r^i_B(\theta )G(\theta )$$\end{document} . This corresponds to the KST case in which $\Delta g_q=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta g_q = 0$$\end{document} and $\Delta {\eta }_q=\frac{g_q}{1-g_q}\Delta {\beta }_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \eta _q = \frac{g_q}{1-g_q}\Delta \beta _q$$\end{document} considered in Equation (27) of Sect. 4.1 and in the present case yields the trade-off $\Delta c_i=\frac{G\left(\theta \right)}{H\left(b_i\right)}\Delta d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta c_i = \frac{G(\theta )}{H(b_i)}\Delta d_i$$\end{document} . Further details are given in Sect. 4.3 for the logistic case.

2. 2. Alternative sets of transformations: The mathematical form of the trade-off functions does not only depend on the mathematical form of the IRF but also on how the IRF is written. For instance, in System (38), the dependence on $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} and $b_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_i$$\end{document} of the trade-off functions can be reversed by rewriting the 1-parameter IRF as $g_{q_i}(\theta ,b_i)=\frac{H{\left(b_i\right)}^{-1}}{G{\left(\theta \right)}^{-1}+H{\left(b_i\right)}^{-1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g_{q_i}(\theta , b_i) = \frac{H(b_i)^{-1}}{G(\theta )^{-1}+H(b_i)^{-1}}$$\end{document} and adapting the derivation accordingly to obtain

   (39) $\begin{array}{c}\left(\begin{array}{c}{c}_i^{\ast }=\frac{1}{1-r_B^i\left(b_i\right)-r_F^i\left(\theta \right)}[(1-r_B^i\left(b_i\right))c_i-(1-d_i)r_F^i\left(\theta \right)]\\ d_i^{\ast }=\frac{1}{1-r_B^i\left(b_i\right)-r_F^i\left(\theta \right)}[(1-r_F^i\left(\theta \right))d_i-(1-c_i)r_B^i\left(b_i\right)]\\ P(X_i=1|{\theta }^{\ast },b_i^{\ast })=r_F^i\left(\theta \right)+[1-r_B^i\left(b_i\right)-r_F^i\left(\theta \right)]P(X_i=1|\theta ,b_i)\\ G{\left({\theta }^{\ast }\right)}^{-1}=G{\left(\theta \right)}^{-1}-\ell ,H{\left(b_i^{\ast }\right)}^{-1}=H{\left(b_i\right)}^{-1}+\ell \\ r_F^i\left(\theta \right)={\ell }_F^{i}G\left(\theta \right),r_B^i\left(b_i\right)=-{\ell }_B^{i}H\left(b_i\right),\ell ={\ell }_F^i+{\ell }_B^i\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} c^*_i = \frac{1}{1-r^i_B(b_i)-r^i_F(\theta )}[ (1-r^i_B(b_i))c_i-(1-d_i)r^i_F(\theta )]\\ d^*_i = \frac{1}{1-r^i_B(b_i)-r^i_F(\theta )}[ (1-r^i_F(\theta ))d_i-(1-c_i)r^i_B(b_i)]\\ P(X_i=1|\theta ^*, b^*_i) = r^i_F(\theta )+[1-r^i_B(b_i)-r^i_F(\theta )]P(X_i=1|\theta , b_i) \\ G(\theta ^*)^{-1} = G(\theta )^{-1}-\ell , \quad H(b^*_i)^{-1} = H(b_i)^{-1}+\ell \\ r^i_F(\theta ) = \ell ^i_F G(\theta ), \quad r^i_B(b_i) = -\ell ^i_B H(b_i),\quad \ell = \ell ^i_F+\ell ^i_B \end{array}\right. } \end{aligned}$$\end{document}
   that provides an alternative version of transformations (38) based on the same trade-off parameters ${\ell }_F^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_F$$\end{document} and ${\ell }_B^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B$$\end{document} but with different trade-off functions and transformations. Identifiability conditions need to constrain these alternative sets of transformations. Although in line of principle, there might be many ways of rewriting an IRF, in practice, there are but a few of them if parameter independence is required as in System (35). For instance, the form $g_{q_i}(\theta ,b_i)=\frac{G\left(\theta \right)H{\left(b_i\right)}^{-1}}{1+G\left(\theta \right)H{\left(b_i\right)}^{-1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta , b_i) =\frac{G(\theta )H(b_i)^{-1}}{1+G(\theta )H(b_i)^{-1}}$$\end{document} would defy such an independence. Most of all, there are no alternative versions of the IRF that allow the trade-off functions to be both ability-independent; hence, the transformed left-side added parameters are not constants (see Remark 3 below). The only exception occurs if all individuals have the same value of ability (see Remark 4).

3. 3. Empirical indistinguishability vs. unidentifiability: As long as any of the trade-off functions is ability-dependent, the transformations (38) and (39) yield empirically indistinguishable 4-parameter IRFs with ability-dependent left-side added parameters $c_i^{\ast }(\theta ,b_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*_i(\theta , b_i)$$\end{document} and $d_i^{\ast }(\theta ,b_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i(\theta , b_i)$$\end{document} . The only exception occurs if all individuals have the same value of ability so that $c_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*_i$$\end{document} and $d_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i$$\end{document} can be treated as constants. Let us assume that this is not the case (see Remark 4) and that there are at least two individuals with different abilities (the random-effects case is implicitly considered to have infinite values for the ability). If one assumes that not only the mathematical form of the 2-parameter IRF but also the mathematical form of the 4-parameter IRF must be preserved, then $c_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*_i$$\end{document} and $d_i^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i$$\end{document} must be constant values and the trade-off parameters ${\ell }_T^i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_T$$\end{document} with $T\in \{B,F\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in \{B,F\}$$\end{document} associated with the ability-dependent trade-off functions $r_T^i\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_T(\theta )$$\end{document} must be equal to zero. In the case of System (38), this yields a restriction ${\ell }_B^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B=0$$\end{document} , which implies $d_i^{\ast }=d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i=d_i$$\end{document} and $c_i^{\ast }=\frac{c_i-(1-d_i)r_F^i\left(b_i\right)}{1-r_F^i\left(b_i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*_i = \frac{c_i-(1-d_i)r^i_F(b_i)}{1-r^i_F(b_i)}$$\end{document} , for all items. In other words, according to transformations (38), either $d_i^{\ast }=d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i=d_i$$\end{document} or the left-side added parameters depend on the ability. The empirical indistinguishability is thus removed, and one is left with the unidentifiability of the guessing parameters. In addition, if ${\ell }_B^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B=0$$\end{document} then ${\ell }_F^i=\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_F = \ell $$\end{document} for all items. The trade-off functions $r_F^i\left(b_i\right)=\ell H^{-1}\left(b_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^i_F(b_i) = \ell H^{-1}(b_i)$$\end{document} still differ from item to item but have a common parameter $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} that needs to be constrained to solve unidentifiability (e.g., set $c_1=c_1^{\ast }=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1=c^*_1=0$$\end{document} in a reference item). A similar reasoning holds for System (39). It thus appears that the identification issue of the 4-parameter models is one of empirical indistinguishability. Once the left-side added parameters are required to be constant in the presence of at least two individuals with different abilities, one is left with the unidentifiability issue of one of these parameters. Section 4.3 shows how these results relate to those already discussed in the literature for the logistic family of models.

4. 4. The equal abilities case: This sub-case was first highlighted by van der Linden and Barrett (Reference van der Linden and Barrett2016) in the context of the 3PL, but is of general relevance. When all individuals in a fixed-effects 4-parameter model have exactly the same value of ability, any set of transformations like (38) or (39) provides a constant reparameterization of the left-side added parameters. Similarly, the exclusive trade-off discussed in Remark 1 between the left-side added parameters becomes a constant trade-off. In the equal abilities case, the identification problem can thus be considered one of unidentifiability and not one of empirical indistinguishability. Hence, to enforce a restriction like ${\ell }_T^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_T=0$$\end{document} for $T\in \{B,F\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in \{B,F\}$$\end{document} , it is not sufficient to require the left-side added parameters to always be constant values, as discussed in Remark 3 above, and one needs at least two individuals with different abilities to be available.

5. 5. There are two independent sources of non-identification: The first source of non-identification is the well-known unidentifiability of the 2-parameter IRF $g_{q_i}(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta , \Gamma ^2_i)$$\end{document} . The second source is the trade-off between the left-side added parameters and the 2-parameter IRF $g_{q_i}(\theta ,{\Gamma }_i^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}(\theta , \Gamma ^2_i)$$\end{document} , which occurs independently of the mathematical form of the latter and is captured by System (31). As an example, let us consider the generic 1-parameter IRF $g_{q_i}(\theta ,b_i)=\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g_{q_i}(\theta , b_i) = \frac{G(\theta )}{G(\theta )+H(b_i)}$$\end{document} and let us rewrite Equation (33) as

   (40) $\begin{array}{c}{g}_{q_i}({\theta }^{\ast },b_i^{\ast })=g_{q_i}(\theta ,b_i)+\frac{r_F^i\left(b_i\right)H\left(b_i\right)-r_B^i\left(\theta \right)G\left(\theta \right)}{H\left(b_i\right)+G\left(\theta \right)}=g_{q_i}(\theta ,b_i)+\frac{\ell }{H\left(b_i\right)+G\left(\theta \right)}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_{q_i}(\theta ^*, b^*_i) = g_{q_i}(\theta , b_i)+ \frac{r^i_F(b_i)H(b_i)-r^i_B(\theta )G(\theta )}{H(b_i)+G(\theta )} = g_{q_i}(\theta , b_i)+\frac{\ell }{H(b_i)+G(\theta )} \end{aligned}$$\end{document}
   where the term $\frac{\ell }{H\left(b_i\right)+G\left(\theta \right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\ell }{H(b_i)+G(\theta )}$$\end{document} captures the change $\Delta g_q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta g_{q}$$\end{document} as it was discussed in Equation (24) for the KST case in Sect. 4.1. The first source of non-identification is that one can multiply both numerator and denominator of the generic 1-parameter IRF $g_{q_i}(\theta ,b_i)=\frac{G\left(\theta \right)}{G\left(\theta \right)+H\left(b_i\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g_{q_i}(\theta , b_i) = \frac{G(\theta )}{G(\theta )+H(b_i)}$$\end{document} by the same constant k without affecting the IRF (if $H=G=exp$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=G=\exp $$\end{document} this is the unidentifiability of the 1PL model). The second source of non-identification is the trade-off between the 1-parameter IRF and the left-side added parameters, which captures the change in the whole IRF and depends on $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} , $H\left(b_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(b_i)$$\end{document} , and $G\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\theta )$$\end{document} . The second source is independent of the first one since multiplying both numerator and denominator by the same constant k on both sides of Equation (40) leaves all terms unaffected as $\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} scales to $k\ell$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ell $$\end{document} as in System (37).

6. 6. Fixed- and random-effects: The previous results hold for both fixed- and random-effects specifications. In the fixed-effects case, the incidental parameter ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _j$$\end{document} replaces $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} , and System (31) provides a general result for the non-identification of the 4-parameter IRFs. In the random-effects case, the statistical model consists of both the IRF and the distribution of the ability, and as such, the identification problem is always one of empirical indistinguishability. Indeed, although both the mathematical forms of the 2- and 4-parameter IRFs can be constrained, thus removing, as discussed above, the problem of empirical indistinguishability and leaving only a problem of unidentifiability for the 4-parameter IRFs, the distribution $f^{\ast }\left({\theta }^{\ast }\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^*(\theta ^*)$$\end{document} of the transformed values ${\theta }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta ^*$$\end{document} in general does not coincide with $f(\theta ;\mu ,\sigma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta ; \mu , \sigma )$$\end{document} . Rather, as long as the IRF $g_{q_i}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{q_i}$$\end{document} is differentiable, the change of variables formula yields the distribution

   (41) $\begin{array}{cc}{f}^{\ast }\left({\theta }^{\ast }\right)& =f\left(g_{q_i}^{-1},\left(\frac{g_{q_i}\left({\theta }^{\ast }\right)-r_F^i}{1-r_F^i-r_B^i}\right),;,\sigma ,,,\mu \right)\frac{d}{d{\theta }^{\ast }}{g}_{q_i}^{-1}\left(\frac{g_{q_i}\left({\theta }^{\ast }\right)-r_F^i}{1-r_F^i-r_B^i}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^*(\theta ^*)&= f\left( g_{q_i}^{-1}\left( \frac{g_{q_i}(\theta ^*)-r^i_F}{1-r^i_F-r^i_B}\right) ;\sigma , \mu \right) \frac{d}{d\theta ^*}g_{q_i}^{-1}\left( \frac{g_{q_i}(\theta ^*)-r^i_F}{1-r^i_F-r^i_B} \right) , \end{aligned}$$\end{document}
   such that the different statistical models are observationally equivalent, that is
   $\begin{array}{cc}P\left(X\right|{\Gamma }_i^4)& =\int f(\theta ;\mu ,\sigma )\prod _{i=1}^{\left|Q\right|}P\left(X_i\right|\theta ,{\Gamma }_i^4)d\theta \\ & =\int f^{\ast }\left({\theta }^{\ast }\right)\prod _{i=1}^{\left|Q\right|}P\left(X_i\right|{\theta }^{\ast },{\gamma }_i^{4\ast })d{\theta }^{\ast }=P\left(X\right|{\Gamma }_i^{4\ast }).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(X|\Gamma ^{4}_i)&= \int f(\theta ; \mu , \sigma ) \prod _{i = 1}^{|Q|}P(X_i|\theta , \Gamma ^{4}_i)d\theta \\&= \int f^*(\theta ^*) \prod _{i = 1}^{|Q|}P(X_i|\theta ^*, \gamma ^{4*}_i)d\theta ^* = P(X|\Gamma ^{4*}_i). \end{aligned}$$\end{document}
   Hence, in the random-effects case, even if the identification problem of the 4-parameter IRFs is reduced to one of unidentifiability, the statistical models are empirically indistinguishable. However, resolving the unidentifiability of the 4-parameter IRFs also resolves the empirical indistinguishability.

7. 7. Sufficient and necessary conditions: For all items, the condition ${\ell }_B^i={\ell }_F^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B=\ell ^i_F=0$$\end{document} occurs if and only if it holds that $c_i^{\ast }=c_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^*_i=c_i$$\end{document} and $d_i^{\ast }=d_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*_i=d_i$$\end{document} . Hence the condition is both necessary and sufficient to eliminate the trade-off between the left-side added parameters and the 2-parameter IRF for both fixed- and random-effects specifications. Alternative versions of System (31), based on alternative versions of the same IRF, are solved by the same condition. Therefore, any identifiability constraint that sets ${\ell }_B^i={\ell }_F^i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^i_B=\ell ^i_F=0$$\end{document} for all alternative systems erases this source of non-identification. However, full resolution of the unidentifiability of an IRT model requires to also provide a) the identifiability conditions for the 2-parameter IRF and b) the conditions on the minimal numbers of individuals and/or items that are required to provide enough equations to identify the parameter values. These latter depend on the identified parametrization associated with the observed outcomes. Let N be the number of individuals, |Q| the number of items, I the number of constraints imposed to reduce unidentifiability, and n the number of parameters per item. In the fixed-effects case, the identified parametrization consists of mutually independent Bernoulli distributions capturing the N|Q| response probabilities; hence, one needs to satisfy the condition $N\left|Q\right|\ge n\left|Q\right|+N-I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N|Q|\ge n|Q|+N-I$$\end{document} . In the random-effects case, the identified parametrization consists of a Multinomial distribution capturing the $2^{\left|Q\right|}-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{|Q|}-1$$\end{document} independent patterns of responses, hence one needs to satisfy the condition $2^{\left|Q\right|}-1\ge n\left|Q\right|+2-I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{|Q|}-1\ge n|Q|+2-I$$\end{document} , where the additional term 2 comes from the location and scale parameters of the distribution $f(\theta ;\mu ,\sigma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta ; \mu , \sigma )$$\end{document} of the abilities.