Bürkner 2021 - Response Families and Distributional IRT

Summary

IRT models are specified by a response distribution $D$ (the “family”) whose parameters are predicted from person and item parameters. Bürkner reviews the families used for binary, categorical, ordinal, count, proportion, response-time and joint choice-and-time data.

Overview

General form: $y_n\sim D({\psi }_{1n},\dots ,{\psi }_{Kn})$ with data in long format (one row per person-item response), which fits multilevel syntax.

Main Content

Families and their IRT models

Response type	Family	Resulting IRT model
Binary	Bernoulli, ${\psi }^y(1-\psi {)}^{1-y}$	1PL, 2PL, 3PL
Unordered categorical, $C$ levels	Categorical, ${\prod }_c{\psi }_c^{I_c(y)}$	nominal; $C=2$ equals Bernoulli
Ordinal	Cumulative: $P(y=c)=F({\tau }_c-\psi )-F({\tau }_{c-1}-\psi )$, logistic $F$	Graded response model (GRM; Samejima)
Ordinal	Adjacent category: $P(y=c)=\frac{\exp {\sum }_{j<c}(\psi -{\tau }_j)}{{\sum }_r\exp {\sum }_{j<r}(\psi -{\tau }_j)}$, thresholds need not be ordered	Partial credit model (PCM; Rasch; generalised PCM used in PISA)
Count	Poisson, ${\psi }^y{e}^{-\psi }/y!$	Rasch-Poisson-counts model
Proportion of total ($C=2$)	Beta, mean-precision $(\mu ,\varphi )$	comparative judgement
Compositional ($C>2$)	Dirichlet	comparative judgement
Response time	Exgaussian, shifted lognormal	RT-IRT
Choice + response time	Wiener diffusion: drift, boundary separation, non-decision time, bias	drift-diffusion IRT
^def-irt-families

Notes. In the Wiener model the initial bias is typically fixed at 0.5 for IRT, giving the three-parameter drift-diffusion model. Exgaussian lets one parameterize the mean directly (easier interpretation) at some theoretical cost versus the shifted lognormal. Response times vary over persons and items in mean, spread and right-skew, so three parameters are warranted.

Examples

Choosing a family for a survey battery

Setup. A brand-attribute grid collects (a) agree/disagree, (b) 5-point agree scales, (c) seconds to respond. Solution. (a) Bernoulli-logit (1PL/2PL); (b) cumulative-logit (GRM) with item-specific threshold shifts; (c) exgaussian with person and item effects on location, scale, skew. Interpretation. One framework, one sampler, one set of post-processing tools for all three.

Connections

• Ordinal thresholds link to ordered-logit models in Discrete Choice Models; the Wiener process is a sequential-sampling choice model (cf. Reward Modeling from Human Preferences for preference responses).

See Also

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