Sample size to estimate a hazard ratio per step across ordered groups

When to use it

A dose–response or exposure-gradient estimate for survival, reported as a hazard ratio per category step.

Not for

Showing that a trend exists. Use the log-rank test for trend under “Detect an effect”.

Notation

$n$: Participants in each group
$g$: Number of ordered groups
$S_1$: Survival in the first group at the end of follow-up
$I$: Expected Cox information about the log hazard ratio per step, per participant
$f$: Precision factor: the interval runs from HR / f to HR × f
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$q$: Non-response rate

Formulas

Participants from the Cox information

n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{g\, I\, (\ln f)^{2}} \right\rceil

Each event contributes the hazard-weighted variance of the group scores among those still at risk; I integrates that over follow-up, with exponential survival censored at its end [1]. At no effect it is the share with an event times the scores’ variance.

Non-response

n_{\text{approach}} = \left\lceil \frac{n}{1 - q} \right\rceil

The number to approach so that n respond. Applied last, after any clustering.

Parameters

Confidence level $1-\alpha$

Values: 90%, 95% or 99%

95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

Precision factor $f$

Values: Above 1

1.25 means the interval runs from HR / 1.25 to HR × 1.25.

Survival in the first group $S_1$

Values: Between 0 and 1

At the end of follow-up.

Number of ordered groups $g$

Values: At least 3

Each is planned the same size.

Expected hazard ratio per step $\mathrm{HR}$ Optional

Values: Above 0; 1 if left blank

It sets how many events each group has.

Group scores $s_j$ Optional

Values: One per group; 0, 1, 2, … by default

The hazard ratio is per unit of score.

Expected non-response rate $q$ Optional

Values: 0% up to, but not including, 100%

The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

Assumptions

Hazards are proportional and log-linear in the scores; survival is exponential and censored only at the end of a common follow-up.
Non-response is unrelated to the outcome.

Worked examples

Three groups, no effect assumed

0.6 event-free, factor 1.25: I = 0.4 × 2/3 = 0.267, so n = 1.96² / (3 × 0.267 × (ln 1.25)²) = 96.4, rounded up to 97 per group.

97 per group · Total: 291

Open this example in the calculator

A falling hazard

Expecting 0.8 per step, the later groups have fewer events: I = 0.221 and 117 per group.

117 per group · Total: 351

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
Cox models on the scores, fitted to simulated samples at the planned sizes, gave median interval factors of 1.2497 against a target of 1.25. An independent implementation in R recomputes these sizes and is compared with the engine’s.

How to cite

StatWisely. Sample size to estimate a hazard ratio per step across ordered groups. Method document, engine version 1.0.0, reviewed 2026-10-10. https://statwisely.com/methods/precision-hr-trend/

Cite the original references for the formulas your design uses as well.

References

1.Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.