# Smoothness-constrained model for nonparametric item response theory

## TL;DR

This paper introduces a smoothness-constrained nonparametric item response theory model that improves estimation accuracy of item characteristic curves and latent abilities, especially with limited data, by preventing overfitting and maintaining shape restrictions.

## Contribution

The paper proposes a novel NIRT model with smoothness constraints and an efficient EM algorithm, enhancing estimation stability and accuracy over existing models.

## Key findings

- Outperforms the two-parameter logistic model with bimodal ability distributions.
- Outperforms the monotone-homogeneity model due to smoothness constraints.
- Provides more accurate estimates with limited data.

## Abstract

This paper is concerned with the nonparametric item response theory (NIRT) for estimating item characteristic curves (ICCs) and latent abilities of examinees on educational and psychological tests. In contrast to parametric models, NIRT models can estimate various forms of ICCs under mild shape restrictions, such as the constraints of monotone homogeneity and double monotonicity. However, NIRT models frequently suffer from estimation instability because of the great flexibility of nonparametric ICCs, especially when there is only a small amount of item-response data. To improve the estimation accuracy, we propose a novel NIRT model constrained by monotone homogeneity and smoothness based on ordered latent classes. Our smoothness constraints avoid overfitting of nonparametric ICCs by keeping them close to logistic curves. We also implement a tailored expectation--maximization algorithm to calibrate our smoothness-constrained NIRT model efficiently. We conducted computational experiments to assess the effectiveness of our smoothness-constrained model in comparison with the common two-parameter logistic model and the monotone-homogeneity model. The computational results demonstrate that our model obtained more accurate estimation results than did the two-parameter logistic model when the latent abilities of examinees for some test items followed bimodal distributions. Moreover, our model outperformed the monotone-homogeneity model because of the effect of the smoothness constraints.