# Bayesian estimation of category typicality using ordered probit models

Virtual MathPsych/ICCM 2023

## Overview

Humans commonly classify nouns (e.g. chair) as members of superordinate categories (e.g. Furniture).1 2

The degree to which nouns “belong” to a category – referred to as typicality – can be measured using normative data.3

Normative category typicality is an important aspect of linguistic4 and cognitive5 research.

## Overview

Normative typicality is often measured via responses to large-scale surveys.

### Example: Van Overschelde et al. (2003) - free response

Category prompt: A four-legged animal

Responses: cat, dog, horse

Measures: % reported, % reported first

## Overview

Normative typicality is often measured via responses to large-scale surveys.

### Example: Castro, Curley, & Hertzog (2021) - Likert responses

Category prompt: A four-legged animal

Exemplar prompt: cat

Not typical Very typical

1 2 3 4 5 6 7 8 9 10

## Overview

Problem:

• All previous studies employ standard central tendency measures (i.e., averages).
• Response frequencies are NOT normally-distributed!
• e.g., caterpillar vs. ant in the Insect category

## Proposed Solution

Here, we use ordered probit models1 and Bayesian parameter estimation to better approximate response distributions.

In these models, ordinal responses are represented as bounded areas on estimated distributions.

• Probability of giving a specific response calculated using the CDF of a fitted distribution.
• Parameter recovery maximizes fit of estimated and observed response densities.

## Ordered-probit model

### Normal distribution (Liddell & Kruschke, 2018)

$\scriptsize{ p(y=k|\mu,\sigma,\theta_1,\dots,\theta_{K-1}) = \Phi \left( \frac{\theta_k - \mu}{\sigma} \right) - \Phi \left( \frac{\theta_{k-1} - \mu}{\sigma} \right) }$

## Ordered-probit model

### Beta distribution

$\scriptsize{ p(y=k|\alpha,\beta,\theta_1,\dots,\theta_{K-1}) = \left( \frac{B(\theta_k;\alpha,\beta)}{B(\alpha,\beta)} \right) - \left( \frac{B(\theta_{k-1};\alpha,\beta)}{B(\alpha,\beta)} \right) }$

## Example: caterpillar

Fit improvements (LL): -33.01 (Gaussian) vs. -24.45 (Beta)

Gaussian Probit

Beta Probit

## Example: ant

Fit improvements (LL): -36.56 (Gaussian) vs. -21.73 (Beta)

Gaussian Probit

Beta Probit

## Conclusions

Ordered probit models that estimate response probabilities using Beta distributions provide a novel method of estimating category typicality.

Specifically, distributions estimated from responses can be used to estimate the probability that a given exemplar is rated as more “typical” than another.

Future work will compare responses across different normative samples, e.g. hierarchical parameter recovery.