Bayesian Modal Estimation of the Four-Parameter Item Response Model in Real, Realistic, and Idealized Data Sets

Abstract

In this study, we explored item and person parameter recovery of the four-parameter model (4PM) in over 24,000 real, realistic, and idealized data sets. In the first analyses, we fit the 4PM and three alternative models to data from three Minnesota Multiphasic Personality Inventory-Adolescent form factor scales using Bayesian modal estimation (BME). Our results indicated that the 4PM fits these scales better than simpler item Response Theory (IRT) models. Next, using the parameter estimates from these real data analyses, we estimated 4PM item parameters in 6,000 realistic data sets to establish minimum sample size requirements for accurate item and person parameter recovery. Using a factorial design that crossed discrete levels of item parameters, sample size, and test length, we also fit the 4PM to an additional 18,000 idealized data sets to extend our parameter recovery findings. Our combined results demonstrated that 4PM item parameters and parameter functions (e.g., item response functions) can be accurately estimated using BME in moderate to large samples (N ⩾ 5, 000) and person parameters can be accurately estimated in smaller samples (N ⩾ 1, 000). In the supplemental files, we report annotated [Formula: see text] code that shows how to estimate 4PM item and person parameters in [Formula: see text] (Chalmers, 2012 ).

Analysing Standard Progressive Matrices (SPM-LS) with Bayesian Item Response Models

“…The most complex model I will consider in this paper is the 4-parameter logistic (4PL) model and all other simpler models result from this model by fixing some item parameters to certain values. In recent years, the 4PL model has received much attention in IRT research due to its flexibility in modeling complex binary response processes (e.g., Culpepper, 2016Culpepper, , 2017Loken & Rulison, 2010;Waller & Feuerstahler, 2017). Under this model, we express P (y ji = 1) via the equation…”

Analysing Standard Progressive Matrices (SPM-LS) with Bayesian Item Response Models

“…The most complex model I consider in this paper is the four-parameter logistic (4PL) model and all other simpler models result from this model by fixing some item parameters to certain values. In recent years, the 4PL model has received much attention in IRT research due to its flexibility in modeling complex binary response processes (e.g., Culpepper 2016Culpepper , 2017Loken and Rulison 2010;Waller and Feuerstahler 2017). Under this model, we express P(y ji = 1) via the equation…”

Marginalized maximum a posteriori estimation for the four‐parameter logistic model under a mixture modelling framework

“…The second simulation was conducted to study the relationship between the d parameter and the properties of the MMAP estimation. The third simulation was performed to compare the performances of the proposed MMAP\EM method with the existing BM estimation procedure implemented in the R package mirt (Waller & Feuerstahler, 2017).…”

“…In fact, BM estimation can be seen as a regulation of MML estimation, while MML estimation is a special case of BM estimation that assumes uniform prior distributions of parameters. Waller and Feuerstahler (2017) recently applied BM estimation as implemented in mirt for the 4PLM.…”