Methodology & Data Sources

Last updated: 2026. Please open a GitHub issue if you find errors.

Not medical advice. This tool is for educational and research purposes only. All estimates carry substantial uncertainty. Consult a physician before making health decisions.

Overview

This calculator estimates how a person's annual all-cause mortality probability changes across fitness levels. The same mathematical framework applies to both supported biomarkers (VO₂ max and grip strength). The approach has three steps:

Anchor to population life tables. The give the average annual probability of death \(q(a, s)\) for the US population.
Apply continuous fitness hazard model. Using normative percentile data and published continuous hazard ratios (see table below), we compute a smooth, monotone fitness hazard multiplier normalized so the population average equals 1.0.
Adjust for the user's risk factors. Comorbidity hazard ratios from large published studies are applied multiplicatively to shift the user's personal estimate.

Biomarker Parameters

The calculator supports multiple fitness biomarkers. The mathematical model is identical; only the data sources, hazard ratio values, and normative populations differ:

Parameter	VO₂ max	Grip strength
Biomarker variable \(B\)	VO₂ max (mL/kg/min)	Grip strength (kg)
Normative data
Published percentiles	10th–90th by decile (9 levels)	5th–95th by decile + 5th/95th (11 levels)
Age bins	Decades: 20–29 through 80–89	5-year bins: 20–24 through 100+
Normative population	US (FRIEND registry)	International (iGRIPS, 2.4M adults, 69 countries)
HR source
\(\mathrm{HR}_{\mathrm{unit}}\)	0.86 per MET Sex-invariant	1/1.16 ≈ 0.862 per 5 kg (men) 1/1.20 ≈ 0.833 per 5 kg (women) Sex-stratified
HR 95% CI	0.85–0.87 per MET	Men: 1.15–1.17 per 5 kg lower Women: 1.17–1.23 per 5 kg lower
\(\mathrm{unit}\)	3.5 mL/kg/min (= 1 MET)	5 kg
Floor (0th percentile)	~10 mL/kg/min ()	0 kg

Mathematical Model

The equations below apply identically to both biomarkers. The specific values of \(\mathrm{HR}_{\mathrm{unit}}\) and \(\mathrm{unit}\) for each biomarker are given in the table above.

Step 1 — Biomarker quantile conversion

Let \(B\) denote the user's biomarker value (e.g., VO₂ max in mL/kg/min or grip strength in kg) and \(q \in [0,1]\) a population quantile. The quantile distribution of \(B\) is conditioned on age and sex via the interpolation scheme described below. The splines are constructed in two stages:

Age direction — quadratic histospline: Published percentile values are treated as bin averages (not point estimates at midpoints). A C¹ piecewise-quadratic is fitted so that the integral over each age bin exactly reproduces the published value.
Quantile direction — monotone cubic Hermite (PCHIP): At each integer age, the quantile values from the age histosplines are extended to the 0th and 100th percentile with physiological bounds (see table above for floor values; 100th percentile mirrors the gap between the two highest published percentiles). A C¹ monotone piecewise-cubic spline (Fritsch–Carlson) is fitted through all knots.
The percentile direction guarantees monotonicity; both guarantee C¹ continuity and exact polynomial integration. Gauss-Legendre quadrature is used for the exponential HR integral over each polynomial piece.

Step 2 — Raw fitness hazard ratio

The raw (unnormalized) hazard ratio as a function of biomarker value is:

\mathrm{HR}_{\mathrm{raw}}(B) = \mathrm{HR}_{\mathrm{unit}}(\mathrm{sex})^{\;B\,/\,\mathrm{unit}}

\(\mathrm{HR}_{\mathrm{unit}}\) is the hazard ratio per standard unit of the biomarker (protective direction, <1), and \(\mathrm{unit}\) is the size of that standard unit. For VO₂ max: \(\mathrm{HR}_{\mathrm{unit}} = 0.86\), \(\mathrm{unit} = 3.5\) mL/kg/min (= 1 MET), sex-invariant. For grip strength: \(\mathrm{HR}_{\mathrm{unit}} = 1/1.16\) (men) or \(1/1.20\) (women), \(\mathrm{unit} = 5\) kg.

Step 3 — Population-mean normalization

To preserve baseline life-table mortality, we normalize so the population-average fitness hazard equals 1.0. Since the quantile distribution of \(B\) is conditioned on age and sex, the expectation is conditional:

\mathrm{HR}_{\mathrm{fitness}} = \frac{\mathrm{HR}_{\mathrm{raw}}(B)}{\mathbb{E}[\mathrm{HR}_{\mathrm{raw}} \mid \mathrm{age}, \mathrm{sex}]}

This guarantees \(\mathbb{E}[\mathrm{HR}_{\mathrm{fitness}} \mid \mathrm{age}, \mathrm{sex}] = 1\) exactly, so population baseline mortality from the life tables is preserved. The expectation is evaluated using 16-point Gauss-Legendre quadrature on each polynomial piece of the quantile spline.

Three separate normalization constants are precomputed per (age, sex): one for the central HR estimate and one for each 95% CI bound, ensuring population-average HR = 1.0 at every CI level.

Sex-stratified HR (grip strength): Because \(\mathrm{HR}_{\mathrm{unit}}\) differs between men and women for grip strength, normalization is computed separately per sex. For VO₂ max, \(\mathrm{HR}_{\mathrm{unit}}\) is sex-invariant (Kokkinos et al. found no significant interactions), so only the quantile distribution differs between sexes.

Step 4 — User risk factor adjustment

Comorbidity hazard ratios from large cohort studies are applied multiplicatively:

\mathrm{HR}_{\mathrm{total}} = \mathrm{HR}_{\mathrm{fitness}} \cdot \prod_i \mathrm{HR}_{\mathrm{condition}_i}

Independence assumption: This assumes comorbidities multiply independently, which is a simplification.

Step 5 — User's annual mortality

Let \(q_{\mathrm{pop}}(\mathrm{age}, \mathrm{sex})\) be the annual mortality probability from the . The user's adjusted mortality is:

q_{\mathrm{user}} = q_{\mathrm{pop}}(\mathrm{age}, \mathrm{sex}) \cdot \mathrm{HR}_{\mathrm{total}}

Step 6 — Plausible range from HR confidence intervals

Each CI bound of the published hazard ratio uses its own normalization constant, so population-average HR = 1.0 is maintained at every CI level:

\mathrm{HR}_{\mathrm{lo}} = \frac{\mathrm{HR}_{\mathrm{raw,lo}}(B)}{\mathbb{E}[\mathrm{HR}_{\mathrm{raw,lo}}]}, \quad \mathrm{HR}_{\mathrm{hi}} = \frac{\mathrm{HR}_{\mathrm{raw,hi}}(B)}{\mathbb{E}[\mathrm{HR}_{\mathrm{raw,hi}}]}

The displayed range is labeled "plausible" rather than "95% CI" because this is a simplified uncertainty propagation.

Step 7 — Life expectancy

Remaining life expectancy from the user's current age:

\mathrm{LE} = \sum_{y=\mathrm{age}}^{118} \prod_{t=\mathrm{age}}^{y} \bigl(1 - \min(\mathrm{HR}_{\mathrm{total}} \cdot q_{\mathrm{pop}}(t, \mathrm{sex}),\; 1)\bigr)

This is the standard actuarial formula. The hazard multiplier is applied uniformly across all future ages, assuming the user's relative fitness rank stays constant. LE estimates should be interpreted as illustrative comparisons, not clinical predictions.

Step 8 — Risk equivalents

Annual excess mortality \(\Delta q = q_{\mathrm{target}} - q_{\mathrm{current}}\) is expressed as equivalent annual risky events:

This is a statistical comparison only — it does not imply the activities are equivalent in any other sense.

Data Sources

Life tables

These values are essentially equivalent to the (typically <2% difference at any age). Public domain — US federal government work.

Normative fitness data

Hazard ratio studies

Risk Factor Hazard Ratios

Applied multiplicatively to the user's baseline mortality estimate. All are all-cause mortality HRs from large prospective studies or meta-analyses.

Condition	HR	95% CI	Source

Risk Equivalents

Activity	Mortality per event	Source

The risk equivalent calculation converts an annual excess mortality fraction Δq into N equivalent single-event risks: N = Δq / (mortality per event).

Limitations

Causality. Observational hazard ratios do not prove causation. Low fitness may partly reflect underlying disease (reverse causation) rather than being a purely modifiable risk factor. Exercise intervention trials show real but smaller mortality benefits than the observational HRs suggest.
Spline interpolation and extrapolation. Normative percentile data are fitted with custom splines. The age direction uses a quadratic histogram-preserving spline (bin-average exact); the percentile direction uses monotone cubic Hermite (PCHIP/Fritsch–Carlson) with physiological extrapolation at the tails. Interpolation between published percentiles is accurate to within ~0.5 units.
Risk factor independence. Multiplying hazard ratios assumes each risk factor's effect is independent. Many conditions interact synergistically (e.g., diabetes + hypertension), so combined risk estimates may be biased in either direction.
Sex-stratified vs. sex-invariant HR. The VO₂ max HR (0.86 per MET) is reported as consistent across age, sex, and racial groups. The grip strength HR differs by sex (1.20 per 5 kg for women vs. 1.16 for men). The calculator handles both cases, but unmeasured effect modifications in other subgroups may exist.
Measurement error. Field estimates of VO₂ max from wearables carry ±5–15 mL/kg/min uncertainty; grip strength varies with dynamometer type and protocol. This propagates into uncertainty in the fitness hazard multiplier.
Relative fitness constancy. The model applies the hazard multiplier from the user's current percentile rank uniformly to all future ages. In reality, individuals may improve or decline relative to their age cohort.
Population generalizability. VO₂ max norms come from a predominantly White US fitness registry; grip norms from an international meta-analysis (iGRIPS, 69 countries). The grip strength HR comes from UK Biobank (ages 40–69); extrapolation outside this range carries additional uncertainty.
Comorbidity data limitations. Risk factor HRs are drawn from diverse studies, often non-contemporaneous and using different adjustment methods.