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    Method document

    Sample size to estimate a hazard ratio per standard deviation

    How many participants are needed to estimate the hazard ratio per standard deviation of a numerical predictor from a Cox model within a chosen factor, from the survival expected at the end of follow-up. Allows for other covariates. Covers non-response.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    https://statwisely.com/methods/precision-hr-per-sd/

    When to use it

    • A prognostic cohort reporting the hazard ratio of a biomarker or another measured factor, with its interval.

    Not for

    • Showing that the predictor matters at all. Use Cox regression with a numerical predictor under “Detect an effect”.

    Notation

    nn
    Number of participants
    SS
    Survival at the end of follow-up for someone at the predictor’s mean
    HR\mathrm{HR}
    Expected hazard ratio per SD; 1 if not given
    II
    Expected Cox information about the log hazard ratio per SD, per participant
    RX∣Z2R^{2}_{X|Z}
    Share of the predictor’s variance explained by the other covariates
    ff
    Precision factor: the interval runs from HR / f to HR × f
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Participants from the Cox information

    n=⌈z1−α/22I (1−RX∣Z2) (ln⁡f)2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{I\,(1 - R^{2}_{X|Z})\,(\ln f)^{2}} \right\rceil

    Each event contributes the variance of the covariate among those still at risk, weighted by their hazard; I integrates that over follow-up, with exponential survival censored at its end. At a hazard ratio of 1 it is Var(x) times the share with an event: Schoenfeld’s π₁π₂ for two groups [1] and 1 per SD [2].

    Non-response

    napproach=⌈n1−q⌉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 level1−α1-\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 factorff

    Values: Above 1; 1.25 means the interval runs from HR / 1.25 to HR × 1.25

    A rare event leaves few events to learn from, so it needs many participants for any factor.

    Survival at the predictor’s meanSS

    Values: Between 0 and 1

    The share expected to be event-free at the end of follow-up for someone at the average predictor value.

    Expected hazard ratio per SDHR\mathrm{HR}Optional

    Values: Above 0; 1 if left blank

    A strong effect shifts events towards those with high values, which changes the information a little.

    R² of the predictor on the other covariatesRX∣Z2R^{2}_{X|Z}Optional

    Values: At least 0 and below 1

    0 with no other covariates.

    Expected non-response rateqqOptional

    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

    • The predictor is roughly normal, and the log hazard changes linearly with it.
    • Hazards are proportional; survival is exponential and censored only at the end of a common follow-up period.
    • Non-response is unrelated to the outcome.

    Worked examples

    30% with an event

    0.7 event-free at the mean, factor 1.25: I = 0.30, so n = 1.96² / (0.30 × (ln 1.25)²) = 257.2, rounded up to 258.

    Sample size: 258

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    Planning at the expected effect

    Expecting HR = 1.3 per SD, more events occur among those with high values: I = 0.302, so n = 256.

    Sample size: 256

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    Adjusting for related covariates

    If the other covariates explain 0.2 of the predictor’s variance: 257.2 / 0.8 = 321.5, so n = 322.

    Sample size: 322

    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.
    • At no effect the information equals the share with an event, Hsieh and Lavori’s form: 1.96² / (ln 1.25)² = 77.1 events at 30%. Cox models fitted to simulated samples at the planned sizes gave median interval factors within 0.2% of the target and coverage of 94% to 96%. 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 standard deviation. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/precision-hr-per-sd/

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

    References

    1. 1.Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics. 1983;39(2):499–503.
    2. 2.Hsieh FY, Lavori PW. Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials. 2000;21(6):552–560.