Conditional maximum-likelihood estimation in probability-based multistage designs

Abstract

This article introduces conditional maximum-likelihood (CML) item parameter estimation in multistage designs based on probabilities $p^{[b]}(x_{+}^{[b]})$ for choosing a particular module ${\textbf {m}}^{[b+1]}$ conditional on a raw score $x_{+}^{[b]}$ in a previous module ${\textbf {m}}^{[b]}$. This type of multistage design is applied to ensure a minimum exposure rate for all items, for example, in international large-scale assessments (ILSAs). For the item parameter estimation, various likelihood-based methods are available. While the marginal maximum-likelihood method (MML) provides consistent estimates in multistage designs, the CML method in its original formulation leads to biased item parameter estimates. In this contribution, we will propose a modification of the common CML method for probabilistic routing strategies, based on the approach for deterministic routing strategies (Zwitser & Maris, 2015, Psychometrika), that provides practically unbiased item parameter estimates for the Rasch model. In a simulation study, it is shown that this modified CML estimation method also provides in probabilistic multistage designs, practically unbiased item parameter estimates.