Bayesian Inference for IRT Models with Non-Normal Latent Trait Distributions

Abstract

Normality of latent traits is a common assumption made when estimating parameters for item response theory (IRT) models, but this assumption may be violated. The purpose of this research was to present a new Markov chain Monte Carlo (MCMC) method for ordinal items with flexible latent trait distributions (i.e., skewed and bimodal). Specifically, the Davidian curve (DC) was used to approximate the distribution of latent traits. The performance of the proposed MCMC algorithm with DCs was evaluated via a simulation study and compared with an EM method using DCs that is available in the “mirt” package (Chalmers, 2012). The manipulated factors included the number of response categories, sample size, and the shape of the latent trait distribution. The Hanna-Quinn (HQ) criterion was used to choose the best DC order. Results indicated that when informative priors were used, the MCMC algorithm with DCs could fit a flexible distribution well and the method provided good parameter estimates which, under some circumstances, had lower bias and RMSE than the EM method.

Keywords:

• item response theory
• MCMC
• Davidian curve
• flexible distribution

Notes

1 If the second approach was chosen to restrict the model identification (e.g., for a four-category test, fix a₁ = 1, b_1,2 = 0), the standardization procedure should be omitted. In other words, under this situation, f(θ|Φ) in Equation 14 has the same expression as Equation 3.

2 Throughout this article, mean and variance are the location and scale parameter, respectively.

3 The minimum acceptable number of replication (Cohen et al., 2001), which was calculated under different manipulated conditions, was 18. The detailed computation method is described by Zhang et al. (2019).

4 The results for the other LT distributions were similar and hence they were omitted to save space.

5 When model identification was restricted by fixing item parameters of one item (e.g., for a four-category test, fix a₁ = 1, b_1,2 = 0), the simulation results were similar.

6 The mean $m$ and variance $v$ of a lognormal random variable are functions of the lognormal distribution parameters $\mu$ and ${\sigma }^2:$ $m=\exp (\mu +{\sigma }^2/2)$ and $v=[\exp ({\sigma }^2)-1]\exp \left(2\mu +{\sigma }^2\right).$