Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results

Abstract

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

Appendices

Appendix A. Identifiability of the Scale Parameter σ by ω ₁₂,δ ₁,δ ₂,ω ₁, and ω ₂

To prove that the function ω ₁₂ given by (3.11) is a strictly increasing continuous function of σ, we need to study the sign of its derivative with respect to σ. This requires not only using the Implicit Function Theorem (Spivak 1965), but also assuming regularity conditions that allow performing derivatives under the integral sign. We accordingly assume that the cumulative distribution function F has a continuous density function f strictly positive on ℝ. Furthermore, to prove that ω ₁₂ is a strictly increasing continuous function of σ, we need to obtain the derivatives under the integral sign of the function p(σ,β) as defined in (3.7) with respect to σ and to β. Consequently, it is assumed that, $\forall\sigma\in\mathbb{R}_{0}^{+}$ and ∀β∈ℝ, there exist ϵ>0 and η>0, such that

Under these regularity conditions, the function p(σ,β) is continuously differentiable under the integral on $\mathbb {R}_{0}^{+}\times\mathbb{R}$ and, therefore,

$$ \everymath{\displaystyle} \begin{array}{r@{\quad }l} \mbox{(i)} & D_{2} p ( \sigma, \beta) \doteq \frac{\partial}{\partial\beta} p ( \sigma, \beta) = \int_{\mathbb{R}} f ( \sigma x - \beta) G ( dx ) \\[10pt] \mbox{(ii) } & D_{1} p ( \sigma, \beta) \doteq\frac{\partial}{\partial\sigma} p ( \sigma, \beta) = - \int_{\mathbb{R}} x f ( \sigma x - \beta) G ( dx ). \end{array} $$

(A.1)

Thus, $\overline{p} ( \sigma, \alpha)$ as defined by (3.9) is also continuously differentiable on $\mathbb{R}_{0}^{+} \times( 0 , 1 )$; and from (3.10), we obtain that

$$ \everymath{\displaystyle} \begin{array}{r@{\quad }lcl} \mbox{(i)}& 1 & = &\frac{\partial}{\partial\beta} \overline{p} \bigl[ \sigma, p ( \sigma, \beta) \bigr] \\[9pt] && = & D_2 \overline{p} \bigl[ \sigma, p ( \sigma, \beta) \bigr] \times D_2 p ( \sigma, \beta) \\[9pt] \mbox{(ii)}& 0 & = &\frac{\partial}{\partial\sigma} \overline{p} \bigl[ \sigma, p ( \sigma, \beta) \bigr] \\[9pt] & & = & D_1 \overline{p} \bigl[ \sigma, p ( \sigma, \beta) \bigr] + D_2 \overline{p} \bigl[ \sigma, p ( \sigma, \beta) \bigr] \times D_1 p ( \sigma, \beta), \end{array} $$

(A.2)

where

$$D_{1} \overline{p} ( \sigma, \omega) \doteq\frac{\partial }{\partial\sigma} \overline{p} ( \sigma, \omega), \qquad D_{2} \overline{p} ( \sigma, \omega) \doteq\frac{\partial }{\partial\omega} \overline{p} ( \sigma, \omega). $$

Combining (A.1) and (A.2), we obtain that

$$ \everymath{\displaystyle} \begin{array}{r@{\quad }lcl} \mbox{(i)}& D_{2} \overline{p} ( \sigma, \omega) & = & \frac{1}{ D_2 p [ \sigma, \overline{p} ( \sigma, \omega) ]} = \frac{1}{\int_{\mathbb{R}} f [ \sigma x - \overline{p} ( \sigma, \omega) ] G ( dx )} \\[12pt] \mbox{(ii)}& D_{1} \overline{p} ( \sigma, \omega) & = & - \frac{D_{1} p [ \sigma, \overline{p} ( \sigma, \omega) ]}{ D_{2} p [ \sigma, \overline{p} ( \sigma, \omega) ]}= \frac{\int_{\mathbb{R}} x f [ \sigma x - \overline{p} ( \sigma, \omega) ] G ( dx )}{\int_{\mathbb{R}} f [ \sigma x - \overline{p} ( \sigma, \omega) ] G ( dx )} \\[12pt] && \doteq& E_{\sigma, \omega} ( X ),\end{array} $$

(A.3)

where

$$ P_{\sigma, \omega} [ X \in dx ] \doteq G_{\sigma, \omega} ( dx ) \doteq \frac{f [ \sigma x - \overline{p} ( \sigma, \omega) ] G ( dx )}{\int_{\mathbb{R}} f [ \sigma x - \overline{p} ( \sigma, \omega) ] G ( dx )}. $$

Thanks to the regularity conditions allowing to perform derivatives of p(σ,β), and to the fact that F≤1, it can be shown that ω ₁₂ is continuously differentiable under the integral sign in σ, β ₁ and β ₂; therefore, the function φ(σ,δ ₁,δ ₂,ω ₁,ω ₂) is continuously differentiable under the integral sign with respect to σ. It remains to show that the derivative w.r.t. σ is strictly positive. Let us consider the sign of one of the two terms of the derivative of φ(σ,δ ₁,δ ₂,ω ₁,ω ₂). Using (A.3.ii), we obtain that

$$ \frac{\partial}{\partial\sigma} \bigl\{1-F \bigl[\sigma x - \overline{p} ( \sigma, \omega_1/ \delta_1 ) \bigr]\bigr\} = -f\bigl[ \sigma x- \overline{p} (\sigma,\omega_1/\delta_1) \bigr]\bigl( x - E_{\sigma, \omega_1/\delta_1} ( X ) \bigr). $$

Therefore, such a second term can be written as

Now, since $F [\sigma\theta- \overline{p} ( \sigma, \omega_{2}/\delta_{2} ) ]$ is a strictly increasing function of θ, the covariance between θ and $F [\sigma\theta- \overline{p} ( \sigma, \omega_{2}/\delta_{2} ) ]$ (with respect to $G_{\sigma, \omega_{1}/\delta_{1}}$) is strictly positive (if θ is not degenerate). Furthermore,

$$\int_{\mathbb{R}} f \bigl[ \sigma x - \overline{p} ( \sigma, \omega_1/\delta_1 ) \bigr] G ( dx ) $$

is clearly strictly positive. The two terms of the derivative of φ(σ,δ ₁,δ ₂,ω ₁,ω ₂) are, therefore, strictly positive.  □

Appendix B. Identification of the Random-Effects 2PL Model

Random-effects 2PL-type models are specified under the same hypotheses of the random-effects 1PL-G model (see Section 3.1), but the conditional distribution of Y _ij given the person specific ability θ _i is given by

$$ P[Y_{ij}=1\mid\theta_i,\alpha_j, \beta_j]=F(\alpha_j\theta_i- \beta_j), $$

(B.1)

where F is a strictly increasing distribution function with a continuous density function f strictly positive on ℝ. Let us suppose that the person specific abilities are distributed according to a known distribution G.

2.1 B.1 Identification of the Difficulty Parameters

Let

$$ \gamma_j\doteq\int_{\mathbb{R}}F( \alpha_j\theta-\beta_j) G(d\theta)\doteq p( \alpha_j,\beta_j), $$

which is a continuous function strictly decreasing in β _j . Define

$$\overline{p} ( \alpha,\gamma)=\inf\bigl\{\beta:p ( \alpha,\beta )<\gamma\bigr\}. $$

Since $\overline{p} [ \alpha,p(,\alpha,\beta)]=\beta$, it follows that $\beta_{j}=\overline{p} [ \alpha_{j},\gamma_{j} ]$ for each j=1,…,J. Thus, the item parameter β _j becomes identified once the discrimination parameter α _j becomes identified.

2.2 B.2 Monotonicity of P[Y _ij =1,Y _ik =1∣α _1:J ,β _1:J ]

In order to identify the discrimination parameters, we need to study the monotonicity of P[Y _ij =1,Y _ik =1∣α _1:J ,β _1:J ] as a function of the discrimination parameters. Using the equality $\overline{p} [ \alpha, p(\alpha,\beta) ]=\beta$, it follows that

$$\everymath{\displaystyle} \begin{array}{rcl} \frac{\partial}{\partial\zeta} \overline{p} \bigl[ \alpha,p( \alpha,\beta) \bigr] & = & -\frac{1}{\int_{\mathbb{R}}f(\alpha\theta -\beta) G(d\theta)};\\[12pt] \frac{\partial}{\partial\alpha} \overline{p} \bigl[ \alpha,p( \alpha,\beta) \bigr] & = & \frac{\int_{\mathbb{R}}\theta f(\alpha\theta -\beta) G(d\theta)}{\int_{\mathbb{R}}f(\alpha\theta-\beta) G(d\theta)} \doteq E_{\alpha,\beta}[ X ], \end{array} $$

where

$$P_{\alpha,\beta}[ X\in d\theta]\doteq G_{\alpha,\beta}(d\theta )= \frac{f(\alpha\theta-\beta)}{\int_{\mathbb{R}}f(\alpha\theta -\beta) G(d\theta)}. $$

Thus,

$$ \frac{\partial}{\partial\alpha}F\bigl[ \alpha\theta-\overline{p}( \alpha, \gamma) \bigr] = \biggl\{\theta-\frac{\partial}{\partial \alpha}\overline{p}( \alpha,\gamma) \biggr\} f\bigl[\alpha\theta\:-\: \overline{p}( \alpha,\gamma)\bigr], $$

(B.2)

where $\frac{\partial}{\partial\alpha} \overline{p} [ \alpha ,\gamma ]=E_{\alpha,\overline{p}[ \alpha,\gamma]}[ X ]$.

Let

(B.3)

Using (B.2), it follows that

(B.4)

provided α _k >0 since in this case $F[ \alpha_{k}X-\overline {p}(\alpha_{k},\gamma_{k}) ]$ is a strictly increasing function of X and, consequently, the covariance between X and $F[ \alpha_{k}X-\overline{p}(\alpha_{k},\gamma_{k}) ]$ is positive (if X is not degenerate). Thus, h(α _j ,γ _j ,α _k ,γ _k ) is a strictly increasing function in α _j . Similarly, it is concluded that h(α _j ,γ _j ,α _k ,γ _k ) is also a strictly increasing function in α _k provided α _j >0. The inverse function of h can, therefore, be defined as

$$\overline{h}(\alpha_j,\gamma_j,\zeta, \gamma_k)=\inf\bigl\{\alpha_k':h\bigl( \alpha_j,\gamma_j,\alpha_k', \gamma_k\bigr)>\zeta\bigr\} $$

and consequently,

$$ \everymath{\displaystyle} \begin{array}{r@{\quad }l} \mbox{(i)} &\overline{h} \bigl[ \alpha_j,\gamma_j,h(\alpha_j,\gamma_j,\alpha_k,\gamma_k),\gamma_k \bigr]=\alpha_k;\\[6pt] \mbox{(ii)}& h \bigl[ \alpha_j,\gamma_j, \overline{h}(\alpha_j,\gamma_j,\zeta_k,\gamma_k),\gamma_k \bigr]=\zeta. \end{array} $$

(B.5)

2.3 B.3 Identification of the Discrimination Parameters

Let J≥3. Using (B.3), we have that

$$\gamma_{12}=h(\alpha_1,\gamma_1, \alpha_2,\gamma_2),\qquad\gamma_{13}=h( \alpha_1,\gamma_1,\alpha_3, \gamma_3),\qquad\gamma_{23}=h(\alpha_2, \gamma_2,\alpha_3,\gamma_3). $$

Therefore, by (B.5.i) it follows that $\alpha_{2}=\overline{h}(\alpha_{1},\gamma_{1},\gamma_{12},\gamma_{2})$ and $\alpha_{3}=\overline{h}(\alpha_{1},\gamma_{1},\gamma_{13},\gamma_{3})$. Thus,

The identification of α ₁ follows because the function k is invertible. As a matter of fact,

But

$$\frac{\partial\overline{h}}{\partial\alpha_1}[ \alpha_1,\gamma_1, \gamma_{12}, \gamma_2 ] = -\frac{\frac{\partial h}{\partial \alpha_i}[ \alpha_1,\gamma_1,\overline{h}(\alpha_1,\gamma_1,\gamma_{12},\gamma_2),\gamma_2 ]}{\frac{\partial h}{\partial \alpha_2}[ \alpha_1,\gamma_1,\overline{h}(\alpha_1,\gamma_1,\gamma_{12},\gamma_2),\gamma_2 ]}. $$

Using (B.4), we conclude that $\frac{\partial }{\partial\alpha_{1}}k(\alpha_{1},\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{12},\gamma_{13})<0$ and, therefore, k is invertible. Finally, by (B.5.i), the identification of the remaining discrimination parameters then follows.

The previous arguments have been established assuming that the distribution generating the person specific abilities is known. If such a distribution is known up to the scale parameter σ, the previous arguments apply for $\widetilde{\alpha}_{j}=\alpha_{j}\sigma$. Thus, we obtain the following theorem.

Theorem B.1

Consider the statistical model induced by both the 2PL model (B.1) and the person specific abilities distributed according to a distribution G known up to the scale parameter σ, where the F is a strictly continuous increasing distribution function with a density function f strictly positive on ℝ. The parameters of interest (α _1:J ,β _1:J ,σ) are identified by one observation provided that

• 1. At least three items are available.

• 2. The discrimination parameter α ₁ is fixed at 1.

If the distribution G is fully known, then parameters of interest (α _1:J ,β _1:J ) are identified provided that at least three items are available.

It is relevant to remark that the positivity of the discrimination parameters is established by the identification analysis.

Appendix C. Proof of Theorem 5

The identification analysis of the parameters of interest (β _1:∞,c _1:∞,G) should be done in the asymptotic Bayesian model defined on (Y _i ,β _1:∞,c _1:∞,G), where Y _i ∈{0,1}^ℕ corresponds to the response pattern of person i. According to Definition 2, the corresponding minimal sufficient parameter is given by the following σ-field:

$$\mathcal{A}^*\doteq\sigma\bigl\{E(f\mid\boldsymbol{\beta}_{1:\infty }, \boldsymbol{c}_{1:\infty},G) : f\in\bigl[\sigma(\boldsymbol{Y}_1) \bigr]^+\bigr\}, $$

where [σ(Y ₁)]⁺ denotes the set of positive functions f such that f=g(Y ₁), with g a measurable function. The identification of the semi-parametric 1PL-G model leads to proving, under identification restrictions if necessary, that (β _1:∞,c _1:∞,G) is a measurable function of the parameter $\mathcal{A}^{*}$. By the Doob–Dynkin lemma, this is equivalent to proving that, under identification restrictions if necessary,

$$\sigma(\boldsymbol{\beta}_{1:\infty},\boldsymbol{c}_{1:\infty },G)=\mathcal{A}^*. $$

This equality relies on the following steps:

Step 1::

By the same arguments used to establish identity (4.4), it follows that

$$\sigma(\boldsymbol{\beta}_{2:\infty}, \boldsymbol{\delta}_{2:\infty}) \doteq\sigma(\beta_j :j\ge2)\vee \sigma(\delta_j:j\geq2) \subset\mathcal{A}^*, $$

where δ _j ≐1−c _j .

Step 2::

Hypotheses H1, H2, H3, and H4 jointly imply that {(u _j ,v _j ):2≤j<∞} are iid conditionally on (β ₁,δ ₁,K,H). By the Strong Law of Large Numbers, it follows that

for $B\in\mathcal{B}^{+}\times\mathcal{B}$. But Propositions 2 and 3 ensure that {u _j :2≤j<∞} and {v _j :2≤j<∞} are identified parameters. It follows that {(u _j ,v _j ):2≤j<∞} is measurable w.r.t. $\mathcal{A}^{*}$ and, consequently, $W^{\beta _{1},\delta_{1}}(B)$ is measurable w.r.t. $\overline{\mathcal{A}^{*}}$ for all $B\in\mathcal{B}^{+}\times\mathcal{B}$. The upper-bar denotes a σ-field completed with measurable sets; for a definition, see Chapter 2 in Florens et al. (1990).

Step 3::

Using (4.4), it follows that

and

Step 3.a::

By the law of large deviations (see Shiryaev 1995, Chapter IV, Section 5), it follows that

$$\overline{Y}_{iJ}-\overline{p}_{iJ}\longrightarrow0\quad \mbox {a.s. conditionally on $(\beta_{1},\delta_{1}, \theta_{i})$ as $J\to\infty$.} $$

But as J→∞

Therefore,

$$\overline{Y}_{iJ}\longrightarrow p(\beta_1, \delta_1,\theta_i)\quad \mbox{a.s. conditionally on}\ ( \beta_1,\delta_1,\theta_i)\ \mbox{as}\ J\to \infty. $$

Step 3.b::

It follows that, for all g∈C _b ([0,1]),

$$g (\overline{Y}_{iJ} )\longrightarrow g \bigl(p(\beta_1, \delta_1,\theta_i) \bigr)\quad\mbox{a.s. and in}\ L^1\ \mbox{conditionally on}\ (\beta_1,\delta_1, \theta_i)\ \mbox{as}\ J\to\infty. $$

Then for all g∈C _b ([0,1]),

By definition of conditional expectation, $E [g (\overline{Y}_{iJ} )\mid\boldsymbol{\beta}_{1:\infty },\boldsymbol{c}_{1:\infty},G ]$ is measurable w.r.t. $\mathcal{A}^{*}$. Thus,

$$\int_{\mathbb{R}_+} g \biggl\{ 1-E \biggl[\frac{u_j}{v_j+\frac {1}{\delta_1}+\frac{1}{\delta_1\exp(\beta_1)}\exp(\theta_i)}\Bigm| \beta_1,\delta_1,\theta_i \biggr] \biggr\}G(d\theta) $$

is measurable w.r.t. $\overline{\mathcal{A}^{*}}$; the bar is added because such an integral is the a.s. limit of the sequence $\{E [g (\overline{Y}_{iJ} )\mid\boldsymbol{\beta}_{1:\infty },\boldsymbol{c}_{1:\infty},G ]:J\in\mathbb{N}\}$.

Step 3.c::

Using the transformation (4.10), it is follows that

$$\int_{\mathbb{R}_+}g \bigl[L(x) \bigr]G_{\beta_1,\delta_1}(dx) \quad \mbox{is}\ \overline{\mathcal{A}^*}\hbox{-measurable}, $$

where

$$L(x)=\int_{\mathbb{R}_+\times\mathbb{R}} \biggl(1-\frac {u}{v+x} \biggr) W^{\beta_1,\delta_1}(du,dv). $$

The function L(⋅) is a strictly continuous function from (δ ⁻¹,∞) to (0,1) that is known because it is measurable w.r.t. $\sigma(W^{\beta_{1},\delta_{1}})$. By Step 2, $\sigma(W^{\beta_{1},\delta_{1}})\subset\overline{\mathcal{A}^{*}}$. In particular, for every function f∈C _b (ℝ⁺), take $g(y)=f [\overline{L}(y) ]$, where $\overline{L}(\alpha )=\inf\{x:L(x)\geq\alpha\}$. It follows that

$$\int_{\mathbb{R}_+}f(x)G_{\beta_1,\delta_1}(dx) $$

is measurable w.r.t. $\overline{\mathcal{A}^{*}}$. Considering

as n→∞, the monotone convergence theorem implies that, for every x∈ℝ⁺, $G_{\beta_{1},\delta_{1}}((0,x])$, and so $G_{\beta_{1},\delta_{1}}$, is measurable w.r.t. $\overline{\mathcal{A}^{*}}$.

Step 4::

From Steps 1 and 3c, it follows that $(\boldsymbol{\beta}_{2:\infty},\boldsymbol{\delta}_{2:\infty },G_{\beta_{1}, \delta_{1}})$ is measurable w.r.t. $\overline{\mathcal{A}^{*}}$. By the Doob–Dynkin lemma, this is equivalent to

$$\sigma(\boldsymbol{\beta}_{2:\infty},\boldsymbol{\delta }_{2:\infty}) \vee\sigma(G_{\beta_1,\delta_1})\subset\overline {\mathcal{A}^*}. $$

However, $\sigma(G_{\beta_{1},\delta_{1}})\subset\sigma(G)\vee\sigma (\beta_{1},\delta_{1})$. Therefore, two restrictions should be introduced in order to obtain the equality $\sigma(G_{\beta_{1},\delta_{1}})= \sigma(G)\vee\sigma(\beta_{1},\delta_{1})$. Two possibilities can be considered:

• 1. The first possibility consists in fixing two q-quantiles of G. In fact, let

  $$x_1=\inf\bigl\{x:G_{\beta_1,\delta_1}(x)>q_1\bigr\},\qquad x_2=\inf\bigl\{ x:G_{\beta_1,\delta_1}(x)>q_2\bigr\}. $$

  Using (4.10), this is equivalent to

  

  It follows that

  $$\beta_1=\ln \biggl[\frac{x_1e^{y_2}-x_2e^{y_1}}{x_2-x_1} \biggr], \qquad \delta_1=\frac{e^{y_2}-e^{y_1}}{x_1e^{y_2}-x_2e^{y_1}}; $$

  that is, β ₁ and δ ₁ are identified since x ₁,x ₂,y ₁,y ₂ depends on G that it is identified.

• 2. The second possibility consists in fixing the mean and the variance of the distribution of exp(θ), namely

  $$E_G\bigl(e^{\theta}\bigr)=\mu,\qquad V_G \bigl(e^{\theta}\bigr)=\sigma^2. $$

  Using (4.10), this is equivalent to

  $$m=E_{G_{\beta_1,\delta_1}}(X)=\frac{1}{\delta_1}+\frac{\mu }{\delta_1e^{\beta_1}},\qquad v^2=V_{G_{\beta_1,\delta_1}}(X)=\frac {\sigma^2}{\delta_1^2e^{2\beta_1}}. $$

  It follows that

  $$\beta_1=\ln \biggl(\frac{m\sigma}{v}-\mu \biggr),\qquad \delta_1^{-1}=m-\frac{\mu v}{\sigma}. $$

  For instance, if μ=0 and σ=1, then

  $$\beta_1=\ln \biggl(\frac{m}{v} \biggr),\qquad \delta_1=\frac{1}{m}; $$

  that is, β ₁ and δ ₁ are identified since m and v depend on G which is identified.