A Quadratic Bifactor Hierarchical Model for Jointly Modelling the Nonlinear Relationship Between Response Accuracy and Response Time in Cognitive Testing

In computerized assessments, response accuracy and response time together reflect how examinees engage with test items. Existing joint models typically assume a linear relationship between these two outcomes, yet empirical observations often reveal more complex patterns. This study proposes a Quadratic Bifactor Hierarchical Model (QBi-HM) that captures nonlinear speed–accuracy dependencies through a shared general factor with quadratic terms, while preserving separate ability and speed components. Simulation results across sample sizes (N = 1000, 1500, 3000) and test lengths (m = 30, 60) demonstrated that QBi-HM recovered parameters accurately under both linear and nonlinear conditions, whereas the conventional linear bifactor model produced biased estimates when nonlinearity was present. An empirical application to PISA 2012 computer-based mathematics data (N = 1527 examinees from four economies, m = 10 items) showed that QBi-HM achieved superior model fit (AIC = 41,825.190, BIC = 42,251.675, SABIC = 41,997.535) compared with Bi-HM (AIC = 41,969.750, BIC = 42,289.613, SABIC = 42,099.099) and revealed an inverted U-shaped speed-accuracy pattern. These findings may suggest that modeling nonlinear dependencies can improve the precision of ability estimation in large-scale assessments and inform item design by identifying task features that elicit distinct response processes. The QBi-HM framework offers a flexible tool for researchers and practitioners seeking to leverage response time data to better understand test-taking behavior.