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Extended Asymptotic Identifiability of Nonparametric Item Response Models

Yinqiu He

TL;DR

The paper addresses identifiability in nonparametric IRT by extending the model class to include widely used parametric IRFs and proving asymptotic identifiability as the number of items grows, under relaxed conditions. It employs a triangular sequence framework with a fixed latent density and derives new techniques to handle end-point and interior behavior, demonstrating that two IRF sequences producing the same manifest distributions are asymptotically indistinguishable. The main contributions are (i) relaxing key assumptions from Douglas (2001) to accommodate normal ogive and 4PL models, (ii) proving global identifiability on (0,1) in the large-item limit, and (iii) providing constructive examples and technical lemmas to support the result. The findings offer a solid theoretical basis for using nonparametric IRTs in large-scale assessments and for comparing parametric vs. nonparametric models in practice.

Abstract

Nonparametric item response models provide a flexible framework in psychological and educational measurements. Douglas (2001) established asymptotic identifiability for a class of models with nonparametric response functions for long assessments. Nevertheless, the model class examined in Douglas (2001) excludes several popular parametric item response models. This limitation can hinder the applications in which nonparametric and parametric models are compared, such as evaluating model goodness-of-fit. To address this issue, We consider an extended nonparametric model class that encompasses most parametric models and establish asymptotic identifiability. The results bridge the parametric and nonparametric item response models and provide a solid theoretical foundation for the applications of nonparametric item response models for assessments with many items.

Extended Asymptotic Identifiability of Nonparametric Item Response Models

TL;DR

The paper addresses identifiability in nonparametric IRT by extending the model class to include widely used parametric IRFs and proving asymptotic identifiability as the number of items grows, under relaxed conditions. It employs a triangular sequence framework with a fixed latent density and derives new techniques to handle end-point and interior behavior, demonstrating that two IRF sequences producing the same manifest distributions are asymptotically indistinguishable. The main contributions are (i) relaxing key assumptions from Douglas (2001) to accommodate normal ogive and 4PL models, (ii) proving global identifiability on (0,1) in the large-item limit, and (iii) providing constructive examples and technical lemmas to support the result. The findings offer a solid theoretical basis for using nonparametric IRTs in large-scale assessments and for comparing parametric vs. nonparametric models in practice.

Abstract

Nonparametric item response models provide a flexible framework in psychological and educational measurements. Douglas (2001) established asymptotic identifiability for a class of models with nonparametric response functions for long assessments. Nevertheless, the model class examined in Douglas (2001) excludes several popular parametric item response models. This limitation can hinder the applications in which nonparametric and parametric models are compared, such as evaluating model goodness-of-fit. To address this issue, We consider an extended nonparametric model class that encompasses most parametric models and establish asymptotic identifiability. The results bridge the parametric and nonparametric item response models and provide a solid theoretical foundation for the applications of nonparametric item response models for assessments with many items.
Paper Structure (10 sections, 7 theorems, 56 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set-up
  3. Conditions
  4. Examples
  5. Example 1. Two-Parameter Normal Ogive Model.
  6. Example 2. Four-Parameter Logistic Model.
  7. Results
  8. Step 1.
  9. Step 2.
  10. Discussions

Key Result

Proposition 1

Suppose IRFs in the sequence $\{Q_{n,i}(\lambda)\}$ follow the normal ogive models with parameters $(a_{n,i}, b_{n,i})$ and the latent trait $\Lambda\sim N(0,1)$. Let $\{P_{n,i}(\theta)\}$ denote the corresponding transformed IRFs following eq:normalogivemodel with the latent trait $\Theta=\Phi(\Lam

Figures (2)

  • Figure 1: $P_{n,i}(\theta)$ and $P'_{n,i}(\theta)$ for the normal ogive model in \ref{['eq:normalogivemodel']} with $a_{n,i}=b_{n,i}=1$.
  • Figure 2: $P_{n,i}(\theta)$ and $P'_{n,i}(\theta)$ for the 4PL in \ref{['eq:transicc4pl']} with $a_{n,i}=b_{n,i}=1$, $c_{n,i}=0.2$, and $d_{n,i}=0.8$.

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3: Hoeffding inequality of bounded variables