# Estimating pain visual analogue scale from health assessment questionnaire for rheumatoid arthritis with beta mixture models

## TL;DR

This study uses beta mixture models to estimate pain levels from disability scores in rheumatoid arthritis patients, improving decision-making for treatment.

## Contribution

The novel use of beta mixture models to map HAQ scores to pain VAS in rheumatoid arthritis patients.

## Key findings

- A two-component beta mixture model provided accurate non-linear estimates of pain VAS from HAQ scores.
- The model showed good fit and performance in both internal and external validation datasets.
- The approach supports evidence synthesis and resource allocation decisions for rheumatoid arthritis care.

## Abstract

To map from the health assessment questionnaire disability index (HAQ) to the pain visual analogue scale (VAS) for people with rheumatoid arthritis. The estimation sample comprised adults with rheumatoid arthritis and inadequate response to tumour necrosis factor-α inhibitors in a multicentre phase 4 randomised controlled trial. Beta mixture models were estimated with combinations of HAQ and its square, age and sex as independent variables. Bayesian Information Criteria informed the number of components. Model performance (root mean squared error; mean absolute error; pseudo-R2) was estimated by k-fold cross validation. Graphs illustrated mean observed and predicted pain VAS, and cumulative distribution of observed and simulated pain VAS values. For face validity, a probabilistic analysis simulated 5000 pain VAS values at four HAQ scores. For external validation, the performance of the preferred specification was assessed using the Rheumatoid Arthritis Medication Study cohort. There were 1055 observations from 158 participants in the estimation sample (mean age: 55.8; 81% female; mean HAQ: 1.72). The preferred specification was a two-component beta mixture model (probability variables: HAQ, age, sex; main regression variable: HAQ). Visual plots illustrated good fit across the HAQ distribution, and a similar cumulative distribution of observed and predicted pain VAS values. Probabilistic analysis demonstrated that the preferred specification handled uncertainty appropriately. External validation demonstrated that the preferred specification performed well in an independent dataset. Beta mixture models provide accurate non-linear estimates of pain VAS from HAQ scores to support evidence synthesis and resource allocation decision-making for people with rheumatoid arthritis.

The online version contains supplementary material available at 10.1007/s00296-025-05897-1.