Bürkner 2021 - IRT Model Classes 1PL 2PL and Extensions

Summary

The Rasch (1PL), 2PL and 3PL models are the same logistic model with progressively more item-level parameters: easiness only; plus discrimination ${\alpha }_i$; plus guessing ${\gamma }_i$. Discrimination makes the predictor non-linear (a product of parameters), which brms handles through non-linear formulas.

Overview

Notation: person $p$ with ability ${\theta }_p$; item $i$ with easiness ${\xi }_i$ (brms convention; classical difficulty is $-{\xi }_i$), discrimination ${\alpha }_i$ and guessing ${\gamma }_i$. The distributional parameter is ${\psi }_n=f({\theta }_{p_n},{\xi }_{i_n})$.

Main Content

Rasch / 1PL

$\psi =\text{logistic}({\theta }_p+{\xi }_i)=\frac{\exp ({\theta }_p+{\xi }_i)}{1+\exp ({\theta }_p+{\xi }_i)}$. Higher ${\theta }_p$ raises success probability on every item; higher ${\xi }_i$ raises it for every person. Equal discriminations weight all items equally, so persons with the same total score get the same estimate.

2PL

$\psi =\text{logistic}({\alpha }_i({\theta }_p+{\xi }_i))=\text{logistic}({\alpha }_i{\theta }_p+{\alpha }_i{\xi }_i)$. The argument is a product of parameters, so no longer a linear predictor (a generalized non-linear mixed model).

3PL

$\psi ={\gamma }_i+(1-{\gamma }_i)g({\alpha }_i({\theta }_p+{\xi }_i))$ with $g$ a link inverse such as the logistic; ${\gamma }_i$ is the probability of a correct answer from guessing (e.g. 0.25 for four options). A 4PL adds an upper lapse bound.

Scale non-identifiability of the 2PL

The 2PL likelihood is unchanged under ${\theta }_p\to c{\theta }_p$, ${\xi }_i\to c{\xi }_i$, ${\alpha }_i\to {\alpha }_i/c$. Hence the person standard deviation must be fixed (usually ${\sigma }_{\theta }=1$), otherwise the scale of the person distribution is absorbed by the discriminations. A sign switch of ${\alpha }_i$ with sign switch of ${\theta }_p+{\xi }_i$ is also a symmetry; impose ${\alpha }_i>0$ by modelling $\log {\alpha }_i$.

Proof. ${\alpha }_i({\theta }_p+{\xi }_i)=({\alpha }_i/c)(c{\theta }_p+c{\xi }_i)$; the likelihood depends only on this product. (Checked numerically below.)

Examples

Likelihood invariance in a simulated 2PL

Setup. 200 persons, 20 items, $\theta \sim N(0,1)$, $\xi \sim N(0,1)$, $\log \alpha \sim N(0.2,0.2^2)$; responses simulated from the 2PL. Solution. Log-likelihood at the truth is $-2015.48$; rescaling with $c=2$ ($2\theta$, $2\xi$, $\alpha /2$) gives exactly the same value. Interpretation. The data cannot distinguish scale; choose ${\sigma }_{\theta }=1$ (or an informative prior). A 3PL with $\gamma =0.25$ gives $P(y=1)\approx 0.2519$ at ability $-5$.

import numpy as np
rng = np.random.default_rng(0)
P, I = 200, 20
theta = rng.normal(0, 1, P); xi = rng.normal(0, 1, I); alpha = np.exp(rng.normal(0.2, 0.2, I))
y = rng.binomial(1, 1/(1+np.exp(-alpha[None, :]*(theta[:, None]+xi[None, :]))))
def ll(theta, xi, alpha):
    q = 1/(1+np.exp(-alpha[None, :]*(theta[:, None]+xi[None, :])))
    return (y*np.log(q)+(1-y)*np.log(1-q)).sum()
print(ll(theta, xi, alpha), ll(2*theta, 2*xi, alpha/2))   # identical

PyMC-style 2PL sketch (untested; the E116 model is of this form with marginalisation over $\theta$):

with pm.Model():
    sd_xi = pm.HalfNormal("sd_xi", 3)
    xi = pm.Normal("xi", 0, sd_xi, shape=I)               # item easiness
    log_alpha = pm.Normal("log_alpha", 0, 1, shape=I)     # discrimination > 0
    theta = pm.Normal("theta", 0, 1, shape=P)             # sd fixed to 1: identification
    eta = pm.math.exp(log_alpha[item]) * (theta[person] + xi[item])
    pm.Bernoulli("y", logit_p=eta, observed=y)

Connections

• Equivalent to a one-factor model for binary indicators (Confirmatory Factor Analysis and SEM, Factor Analysis and PPCA).
• Rasch is the logit-linear “ability minus difficulty” structure also used in annotation models (Paun 2018 - Partially Pooled Models Hierarchical D&S ItemDiff LogRndEff) and in Bradley-Terry (Caron 2012 - Bradley-Terry Likelihood and Latent Exponential Augmentation: BT is a Rasch model where “items” are opponents).

See Also

Bürkner 2021 - Priors Hierarchical Pooling and Identification · Bürkner 2021 - Worked Examples Binary Models VerbAgg