Joint analysis of dispersed count-time data using a bivariate latent factor model

In this study, we explore parameter estimation for a joint count-time data model with a two-factor latent trait structure, representing accuracy and speed. Each count-time variable pair corresponds to a specific item on a measurement instrument, where each item consists of a fixed number of tasks. The count variable represents the number of successfully completed tasks and is modeled using a Beta-binomial distribution to account for potential over-dispersion. The time variable, representing the duration needed to complete the tasks, is modeled using a normal distribution on a logarithmic scale. To characterize the model structure, we derive marginal moments that inform a set of method-of-moments (MOM) estimators, which serve as initial values for maximum likelihood estimation (MLE) via the Monte Carlo Expectation-Maximization (MCEM) algorithm. Standard errors are estimated using both the observed information matrix and bootstrap resampling, with simulation results indicating superior performance of the bootstrap, especially near boundary values of the dispersion parameter. A comprehensive simulation study investigates estimator accuracy and computational efficiency. To demonstrate the methodology, we analyze oral reading fluency (ORF) data, showing substantial variation in item-level dispersion and providing evidence for the improved model fit of the Beta-binomial specification, assessed using standardized root mean square residuals (SRMSR).