<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 agesâ€”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)