How to parameterize and identify an IRT 2-PL model? (pymc-bot test)

Questions v5
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To parameterize and identify an IRT 2-PL model in PyMC, you can follow these steps:

  1. Model Specification:

    • Define the latent traits (e.g., student ability) and item parameters (difficulty and discriminativeness).
    • Use hierarchical priors for these parameters to account for variability across students and items.

  2. Dealing with Non-Identifiability:

    • Additive Non-Identifiability: You can enforce sum-to-zero constraints on the ability parameters to address this.
    • Multiplicative Non-Identifiability: Pin one of the discriminativeness parameters to a fixed value (e.g., 1) to resolve this issue.

  3. Parameterization of the Hierarchical Model:

    • Use a non-centered parameterization for hierarchical models to improve sampling efficiency, especially when the data is sparse or the model is complex.

  4. Constraining Discriminativeness:

    • Use a positive prior distribution (e.g., HalfNormal) for the discriminativeness parameters to ensure they remain positive.

  5. Python Code for PyMC Model:

    import pymc as pm
import numpy as np

# Simulate data
n_students = 100
n_items = 20
true_ability = np.random.normal(0, 1, n_students)
true_difficulty = np.random.normal(0, 1, n_items)
true_discriminativeness = np.abs(np.random.normal(1, 0.5, n_items))

# PyMC model
with pm.Model() as model:
    ability = pm.Normal('ability', mu=0, sigma=1, shape=n_students)
    difficulty = pm.Normal('difficulty', mu=0, sigma=1, shape=n_items)
    discriminativeness = pm.HalfNormal('discriminativeness', sigma=1, shape=n_items)

    # IRT 2PL model
    theta = ability[:, None] - difficulty[None, :]
    p = pm.math.sigmoid(discriminativeness[None, :] * theta)
    observed = pm.Bernoulli('observed', p=p, observed=simulated_data)

    # Sampling
    trace = pm.sample(1000, return_inferencedata=True)

# Fit and test the fit
pm.plot_trace(trace)

  6. Simulation-Based Calibration:

    • You can extend this model to perform simulation-based calibration by generating posterior predictive samples and comparing them to the observed data using plots or statistical tests.

This code provides a basic framework for setting up an IRT 2-PL model in PyMC. You can adjust the priors and constraints based on your specific needs and data characteristics.

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