Race Prediction Formulas Compared: Riegel vs Cameron vs Vickers

Four major formulas exist for predicting race performance across distances: Riegel (1981), Cameron (1998), Purdy (1974), and the Vickers-Vertosick model (2016). They use different mathematical approaches, were derived from different datasets, and produce noticeably different predictions — especially for the half-marathon to marathon projection.

This guide implements all four approaches conceptually, compares their outputs for a typical recreational runner, and explains which to use and when.

Why the formulas differ

Race prediction formulas are empirical — they're fitted to data, not derived from first principles. Different datasets, different runner populations, and different mathematical forms produce different coefficients, and therefore different predictions.

The fundamental question each formula is answering: if a runner produces time T₁ over distance D₁, what time T₂ can they produce over D₂? This requires a model of how performance degrades as distance increases.

Power law models (Riegel, Cameron): assume T = a × D^b. The exponent b governs how much each doubling of distance costs. Riegel's b = 1.06; Cameron's modified form is closer to 1.07 for most pairs.

Oxygen-based models (Purdy, Daniels): use the relationship between velocity and oxygen cost explicitly, accounting for the fact that the fraction of VO₂ max sustainable decreases with duration. These models are physiologically grounded rather than purely empirical.

Regression from race data (Vickers-Vertosick): use large-scale observational data from real runners to find empirical multipliers for specific distance pairs.

The formulas in detail

Riegel (1981)

T₂ = T₁ × (D₂/D₁)^1.06

Derived from world record progressions across distances from 1 mile to 100 miles. The exponent 1.06 assumes a homogeneous population of elite competitive runners. Simple to apply and explain.

For half-to-marathon prediction, the multiplier is 2^1.06 = 2.085 for exact doubling of distance. For exact marathon from half (21.0975 to 42.195 km): 2^1.06 = 2.085 (since the distances are exactly doubled). A 1:45:00 half marathon → 3:38:54 marathon.

Cameron (1998)

Cameron's formula uses a modified logarithmic form:

T₂ = T₁ × [ln(D₂ × 2π) / ln(D₁ × 2π)]^c

Where c ≈ 1.07. The natural logarithm structure changes how the formula behaves at extreme distances — it tends to predict more fatigue at very long distances than Riegel.

For the same 1:45:00 half marathon → approximately 3:42:00 marathon using Cameron (about 3 minutes slower than Riegel).

Purdy (1974)

George Purdy's model uses a points-based system derived from the oxygen cost equation and world-record calibration. Purdy points assign a score (on a 0–1300 scale) to any race performance, allowing comparison across distances. A runner's "Purdy points" from a 10K can be used to look up equivalent performances at other distances.

The Purdy model is conceptually similar to Daniels' VDOT — both use the relationship between velocity and VO₂ to derive equivalent performances. The Purdy points scale is not in common use today, but the underlying physiology model is sound.

Vickers-Vertosick (2016)

The most empirically robust approach: Vickers and Vertosick analysed finish times of approximately 2.5 million runners from the New York City Marathon who had prior race records. They fitted regression models for specific distance pair predictions.

Key finding for the half-to-marathon projection: Marathon ≈ 2.14 × half marathon time (not 2.11 as Riegel suggests)

For a 1:45:00 half marathon: 2.14 × 1:45:00 = 3:44:30

The difference between Riegel (3:38:54) and Vickers-Vertosick (3:44:30) is 5.5 minutes — meaningful for race planning. Their model consistently found that recreational runners' actual marathon times exceeded Riegel predictions, particularly at slower overall paces.

Comparison for a typical recreational runner

Runner: 1:45:00 half marathon (5:00/km pace). Predictions for marathon:

The spread is about 6 minutes. For a runner targeting 3:45, the Riegel prediction (3:39) might lead to starting too fast and positive-splitting, while the Vickers-Vertosick prediction (3:44:30) suggests targeting 3:45 is appropriate.

Which should you use?

For adjacent standard distances (5K to 10K, 10K to half marathon): all formulas agree closely. Use Riegel.

For half marathon to marathon: Vickers-Vertosick's 2.14 multiplier is better supported for recreational runners. Consider adding 5–10 minutes to your Riegel prediction if you haven't done extensive marathon-specific training.

For extreme distance gaps (5K to marathon, or shorter to ultramarathon): all formulas are unreliable. Use VDOT tables for within-standard-distance predictions and treat very long extrapolations as rough orders of magnitude only.

The calculator

The calculator implements the Riegel formula. For marathon prediction from a half marathon, consider adding the 2–6 minute buffer suggested by Vickers-Vertosick's data for recreational runners.

Why recreational runners usually underperform Riegel

The Riegel formula was derived from world record data — the performance of elite competitive runners across distances. Several systematic factors cause recreational runners to run marathons slower than Riegel predicts from their half marathon:

Pacing: Elite runners pace nearly perfectly. Recreational runners frequently go out too fast and positive-split. Riegel implicitly assumes optimal pacing.

Fuelling: Glycogen depletion at marathon distances is a limiting factor for recreational runners that doesn't appear at half marathon distances. Riegel's exponent doesn't model glycogen as a distinct variable.

Training specificity: A half marathon in September doesn't mean you've done adequate marathon-specific long run preparation. The aerobic fitness captured in the half marathon time doesn't fully predict marathon capability without the long-run adaptations.

The 21 km wall: Some recreational runners who run confident halves fall apart after 30 km in a way that isn't apparent from half marathon times. This is primarily a pacing and fuelling issue, not a fitness gap.