# Exploratory analysis of the accuracy of age-based maximal heart rate equations across cardiorespiratory fitness levels

## TL;DR

This study evaluates how well age-based formulas predict maximal heart rate across different fitness levels, finding limited influence of fitness on prediction accuracy.

## Contribution

The study is novel in examining how cardiorespiratory fitness affects the accuracy of age-based maximal heart rate prediction equations.

## Key findings

- CRF had limited influence on MHR prediction error, with small associations observed in males but not females.
- The Fox equation showed the most stable performance across MHR ranges without proportional bias.
- None of the equations demonstrated high individual-level accuracy for MHR prediction.

## Abstract

Maximal heart rate (MHR) is a key measure for cardiorespiratory exercise prescription yet is often estimated using age-based prediction equations. The accuracy of these equations may vary by individual characteristics, including cardiorespiratory fitness (CRF), but limited research has examined predictive accuracy across CRF levels. Therefore, we evaluated the accuracy of seven commonly used MHR prediction equations in adults with varying CRF to assess whether prediction error differs by fitness level.

Data from 230 healthy adults (76% male, mean age 38.5 ± 12.3 years) who completed maximal graded exercise tests between 2019 and 2024 were analyzed retrospectively. Predicted MHR values were calculated using the Fox, Tanaka, Gellish, Arena, Åstrand, Nes, and Fairbairn equations. Linear mixed-effects models (LMM) tested the influence of VO₂max and its interaction with prediction equation on error, with sex included as a covariate. Estimated marginal means and slopes were extracted, with pairwise contrasts adjusted by the Tukey method. Prediction equation accuracy was evaluated by comparing predicted and measured MHR using Bland-Altman analyses, and metrics including mean absolute error (MAE), root mean square error (RMSE), and intraclass correlation coefficients (ICC).

LMM indicated a significant main effect of prediction equation on error (p < 0.001) and a significant equation × VO₂max interaction (p = 0.015), though neither sex (p = 0.49) nor VO₂max (p = 0.18) alone influenced error. The conditional R2 for the LME model was 0.70, with a marginal R2 of 0.02. Post-hoc linear regressions showed higher VO₂max was associated with greater prediction error for several equations in males, but not females, with a small amount of variance explained (R2 ≤ 0.06). Agreement analyses indicated small mean biases across equations (–3 to +6 bpm) but wide limits of agreement (~±18–24 bpm). Arena, Tanaka and Gellish equations showed the lowest MAE and RMSE. Among the equations, Fox showed the most stable performance across MHR ranges, being the only formula without proportional bias across the sample.

The findings indicate that CRF had only a limited influence on MHR prediction error, with small associations observed in males but not females, reinforcing age as the primary determinant of MHR. Although some equations (e.g., Tanaka, Gellish, Arena, Fox) performed better than others across agreement metrics, none demonstrated high individual level accuracy, which highlights a lack of precision when estimating MHR for exercise prescription and monitoring purposes. Future work should explore more individualized modeling approaches, though adjusting for CRF alone may not substantially improve prediction accuracy in healthy adults.