# Bayesian estimation of category typicality using ordered probit models

Virtual MathPsych/ICCM 2023

Taylor Curley

Air Force Research Laboratory

711th Human Performance Wing

Wright-Patterson AFB, OH

## 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 linguistic^{4} and cognitive^{5} 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*

`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 models^{1} 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)

## Example: *caterpillar*

### Bayesian Parameter Recovery

**Gaussian Probit**

**Beta Probit**

## Example: *ant*

Fit improvements (*LL*): `-36.56`

(Gaussian) vs. `-21.73`

(Beta)

## Example: *ant*

### Bayesian Parameter Recovery

**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.