The article proposes a novel item response theory model to handle continuous responses and sparse polytomous responses in psychological and educational measurement. The model extends the traditional two-parameter logistic model by incorporating a precision parameter, which, along with a beta distribution, forms an error component that accounts for the response continuity. Furthermore, transforming ordinal responses to a continuous scale enables the fitting of polytomous item responses while consistently applying three parameters per item for model parsimony. The model’s accuracy, stability, and computational efficiency in parameter estimation were examined. An empirical application demonstrated the model’s effectiveness in representing the characteristics of continuous item responses. Additionally, the model’s applicability to sparse polytomous data was supported by cross-validation results from another empirical dataset, which indicates that the model’s parsimony can enhance model-data fit compared to existing polytomous models.

The key assumption of conditional independence of item responses given latent ability in item response theory (IRT) models is addressed for multistage adaptive testing (MST) designs. Routing decisions in MST designs can cause patterns in the data that are not accounted for by the IRT model. This phenomenon relates to quasi-independence in log-linear models for incomplete contingency tables and impacts certain types of statistical inference based on assumptions on observed and missing data. We demonstrate that generalized residuals for item pair frequencies under IRT models as discussed by Haberman and Sinharay (J Am Stat Assoc 108:1435–1444, 2013. https://doi.org/10.1080/01621459.2013.835660) are inappropriate for MST data without adjustments. The adjustments are dependent on the MST design, and can quickly become nontrivial as the complexity of the routing increases. However, the adjusted residuals are found to have satisfactory Type I errors in a simulation and illustrated by an application to real MST data from the Programme for International Student Assessment (PISA). Implications and suggestions for statistical inference with MST designs are discussed.