<h1>Linking with the Bayesian Item Response Theory Model</h1> <h2>Brandon LeBeau & Xiaoting Zhong</h2> <h3>University of Iowa</h3> # Research Problem + IRT models specified as linear mixed models are not new (e.g., Adams, Wilson, & Wu, 1997; Kamata, 2001; Mellenbergh, 1994; Rijmen et al., 2003). + Some have discussed evaluating DIF in these frameworks too (e.g., Noorgate and De Boeck, 2005). + To our knowledge, these methods have not been explored to which the extent they properly link scores. # Heterotypic Continuity + We approach this research primarily from a psychological measurement lens, where heterotypic continuity is often of concern. + **Heterotypic Continuity** occurs when the same psychological reasons underlie different behaviors at different ages—the same construct *looks different* across development (Petersen, Choe & LeBeau, 2020). + Examples: Externalizing Problems (Petersen & LeBeau, 2020; Petersen & LeBeau, 2021) # Limitations of current linking + Item parameter estimates treated as known parameters for linking. + Problematic in small samples + Does not account for any clustering in the data collection design + Current software for linking is cumbersome to use. + Limits use for applied researchers # Bayesian Generalized Linear Mixed Model Focusing on dichotomous items, we could define a 2pl IRT model within a linear mixed model: $$P(y = 1) = logistic(\alpha_{i}(b_{p} + \gamma_{i}))$$ + $\alpha_{i}$ is item specific discrimination + $b_{p}$ is a random person effect, often called theta or ability + $\gamma_{i}$ is an item easiness term ## Adding Attributes Attributes can be added as item, person, or both item and person attributes. Example: $$P(y = 1) = logistic(\alpha_{i}(b_{p} + \gamma_{i} + X_{jpi} \beta_{j}))$$ + $X_{jpi}$ is a design matrix. + $\beta_{j}$ are a set of $J$ regression coefficients. ## Specific Example - 3 time points Example $$P(y = 1) = logistic(\alpha_{i}(b_{p} + \gamma_{i} + \beta_{1} time_{2} + \beta_{2} time_{3}))$$ + $time_{2}$ is a dummy attribute, 1 = time 2, 0 = otherwise + $time_{3}$ is a dummy attribute, 1 = time 3, 0 = otherwise # Simulation Conditions + Sample size fixed at 1000 + 3 time points + Number of items fixed at 20 for each time point + 10 common items (40 total items, 10 in common, 10 unique across 3 time points) + Change in common easiness: -0.5, 0, 0.5 # Data Generation + Data were generated by randomly generating 1000 person parameters (ability/theta) + Item parameter were randomly generated for all unique items (30 items) + Common item parameters at each time point were shifted based on the change in common easiness. # Fixed Effect Results - Time 2 ![](/figs/time2_posterior.png) # Fixed Effect Results - Time 3 ![](/figs/time3_posterior.png) # Recovering Ability ![](/figs/ability_posterior.png) # Concluding Remarks + Fitting IRT models as mixed models is highly flexible. + Preliminary evidence suggests adequate recovery of parameters. + May work in small samples due to ability to impose prior information # Future Work + Vary more data conditions to evaluate ability to link across different situations + Non-linear change across time + More time points + Smaller sample size + Evaluate more complex model to allow variance of person parameters to change over time # Get in Touch + webpage: [https://brandonlebeau.org/](https://brandonlebeau.org/) + twitter: [@blebeau11](https://twitter.com/blebeau11) + GitHub: [lebebr01](https://https//github.com/lebebr01) + LinkedIn: [blebeau11](https://www.linkedin.com/in/blebeau11/) + slides: [https://brandonlebeau.org/slides/ncme-2023](https://brandonlebeau.org/slides/ncme-2023)