Conditional Maximum Likelihood Estimation in Probability-Branched Multistage Designs

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This article introduces conditional maximum likelihood (CML) item parameter estimation in multistage designs based on probabilities $p(j)$ for choosing a particular path conditional on a raw score x₊ in a previous module. 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. Zwitser and Maris (2015, Psychometrika) proposed a modified CML estimation for multistage designs with deterministic routing rules that provide practically unbiased item parameter estimates. First, this approach is illustrated for sequential and cumulative multistage designs. Subsequently, another modification of the CML estimation method is introduced, which is necessary for multistage designs with probabilistic routing.In a simulation study, it is shown that this modified CML estimation method provides also in probabilistic multistage designs practically unbiased item parameter estimates.