Principles of Sample Size Determination

Overview

Sample-size determination is the process of deciding how many participants, observations, events, clusters or measurements a study requires to address its primary objective with an acceptable level of uncertainty.

It is not a stand-alone arithmetic exercise. A defensible sample size follows from a chain of decisions:

clinical question → estimand → study design → outcome → effect measure → planned analysis → planning assumptions → required sample size

If any link changes, the calculation may also need to change.

A sample may be justified through:

• precision, such as a desired confidence-interval width;
• statistical power, such as the probability of detecting a prespecified effect under a hypothesis-testing framework;
• decision-theoretic or Bayesian criteria;
• feasibility objectives, particularly in pilot studies; or
• a transparent evaluation of what an available sample can estimate.

The phrase “minimum sample size” should be used carefully. A formula usually gives the smallest integer meeting stated assumptions, not a universal guarantee that the study will be informative.

Why this matters

A study can be too small to estimate an effect precisely or to detect a clinically meaningful difference with adequate probability. It can also be unnecessarily large, exposing more participants and consuming more resources than required.

However, increasing sample size does not correct:

• selection bias;
• measurement error;
• confounding;
• an invalid comparison group;
• an inappropriate outcome;
• poor adherence to the study protocol; or
• an analysis that does not match the design.

Sample-size calculations in randomised trials require subjective but explicit clinical judgements, especially the effect considered important and the expected event rate or variability [1]. The objective is therefore not to produce an impressive number but to make the assumptions and consequences transparent.

Three common purposes

Precision-based planning

A precision calculation begins with the width of the confidence interval that the study should achieve.

Examples include:

• estimating a prevalence within ±5 percentage points;
• estimating a mean within ±3 mmHg;
• estimating sensitivity within ±5 percentage points; and
• estimating a correlation with a confidence interval no wider than a specified amount.

The primary input is a margin of error, meaning the desired distance from the point estimate to a confidence limit under the planning approximation.

Precision-based planning is especially suitable for descriptive and estimation objectives.

Power-based planning

Power is the probability that a prespecified statistical procedure will reject the null hypothesis under a specified alternative scenario.

A conventional power calculation requires:

• null and alternative hypotheses;
• effect size under the alternative;
• significance level;
• one-sided or two-sided testing;
• outcome variability or event probabilities;
• allocation ratio;
• planned statistical method; and
• target power.

Power is not the probability that the study hypothesis is true. It is a repeated-sampling property under assumed parameter values.

Feasibility-based justification

Sometimes the number of eligible participants, specimens or clusters is constrained.

A feasibility-based justification should not claim that the study has “adequate power” unless this has been demonstrated for a clearly specified effect. Instead, investigators should report what precision or power the feasible sample supports across plausible assumptions [2].

Pilot and feasibility studies should be sized according to their own objectives, such as estimating recruitment, retention or outcome variability, rather than powered routinely for the definitive treatment effect.

Core inputs

Primary outcome and time point

The calculation should be based on the primary outcome unless another objective is explicitly designated to determine the sample.

For a numerical outcome, planning may require a standard deviation. For a binary outcome, it may require event probabilities. For survival analysis, the required number of events is often central.

The time point matters. A 30-day risk and a one-year risk are different planning quantities.

Effect size

The effect size is the difference, ratio, correlation or other quantity the study is designed to estimate or detect.

The effect should be:

• clinically interpretable;
• justified independently of the desired sample size;
• compatible with existing evidence;
• expressed on the analysis scale; and
• accompanied by sensitivity analyses.

A smaller target difference generally requires a larger sample.

Variability or nuisance parameters

A nuisance parameter influences sample size but is not the main target effect.

Examples include:

• standard deviation;
• control-group event risk;
• prevalence;
• intracluster correlation;
• censoring proportion;
• predictor distribution; and
• expected correlation between paired observations.

Uncertain nuisance parameters should be varied rather than treated as known constants.

Significance level

The type I error rate, commonly denoted $\alpha$, is the probability of rejecting the null hypothesis when it is true under the testing framework.

A two-sided significance level of 0.05 divides the error across both directions. A one-sided calculation should be used only when the design and decision genuinely exclude the opposite direction from the inferential claim.

Power

Power is $1-\beta$, where $\beta$ is the type II error probability under the specified alternative.

Common targets are 80% or 90%, but they are conventions rather than universal ethical thresholds. Higher power requires a larger sample.

Allocation ratio

Equal allocation is often statistically efficient for a two-group comparison when per-participant costs and variances are similar.

Unequal allocation may be justified by:

• limited intervention supply;
• differing costs;
• safety information;
• recruitment constraints; or
• a desire for more experience with one treatment.

Unequal allocation usually increases the total sample needed for the same power.

A basic two-means approximation

For two independent groups with equal allocation and a common planning standard deviation, a large-sample approximation is:

This formula is an approximation. Exact or analysis-specific procedures may use a noncentral t distribution, baseline adjustment, unequal variances or simulation.

Worked example

Sensitivity analysis

A sample-size result should be shown across plausible assumptions.

Sensitivity of required sample size to the target effect

simulated data

Table 1. Approximate analysable sample per group

Mean difference to detect	Standard deviation	Power	Required per group
3 mmHg	12 mmHg	90%	337
5 mmHg	12 mmHg	90%	122
8 mmHg	12 mmHg	90%	48

Note: Two-sided $\alpha=0.05$, equal allocation and large-sample normal approximation.

(Calculations for the sensitivity-table sample sizes are not shown here; they use the formula and assumptions presented above.)

The assumed effect has a large influence because the required sample varies approximately with $1/\Delta^2$.

A calculation that selects a large effect merely to obtain a feasible sample is not a defensible clinical justification.

Assumptions to document

Smallest valid analysis unit

The sample-size unit may be:

• participant;
• paired observation;
• eye, tooth or lesion;
• cluster;
• event;
• person-time contribution; or
• specimen.

Several measurements from one participant do not create several independent participants.

In a cluster-randomised trial, both the number of clusters and participants per cluster matter. A very large number of participants within a few clusters may provide limited independent information.

Internal and external pilot information

Pilot data can inform:

• recruitment rate;
• follow-up completeness;
• measurement feasibility;
• standard deviation;
• intracluster correlation; and
• event frequency.

Small pilot estimates are uncertain. Using one observed pilot standard deviation as though it were the true value can produce an unstable main-study sample size.

The source and uncertainty of pilot inputs should therefore be reported, with conservative or assurance-based approaches considered where appropriate.

Power, assurance and expected power MMed extension

A conventional power calculation treats planning parameters as fixed.

Assurance, sometimes called unconditional power or expected power in particular frameworks, averages the probability of study success over uncertainty in planning parameters.

For example, instead of assuming that the standard deviation is exactly 12 mmHg, an assurance calculation can represent uncertainty about plausible standard deviations.

Assurance requires additional assumptions or prior distributions. It is not automatically more objective, but it can make parameter uncertainty explicit.

Suggested reporting

SPIRIT 2025 requires transparent protocol reporting of sample-size assumptions, while CONSORT 2025 requires reporting how the sample size was determined [3], [4].

The final report should also state any difference between planned and achieved recruitment and explain whether assumptions changed before outcome analysis.

International guidance emphasises alignment between study objectives, hypotheses, design and analysis [5]. Design-specific formulae and computational approaches should be selected from established methodological sources rather than improvised from a superficially similar study [6].

Common mistakes

Knowledge check

Why must the planned analysis be identified before calculating sample size?

Answer and reasoning

The sampling distribution, variance and information depend on the analysis. A calculation for independent groups may be inappropriate for paired, clustered or covariate-adjusted data.

Why does detecting a smaller effect require more participants?

Answer and reasoning

Smaller effects are harder to distinguish from sampling variability. In the two-means approximation, sample size increases approximately with the inverse square of the target difference.

What does 90% power mean?

Answer and reasoning

Under the specified alternative effect and all other planning assumptions, the chosen procedure would reject the null hypothesis in approximately 90% of repeated studies. It is not a 90% probability that the intervention works.

Why is the 10% attrition adjustment calculated as 122 divided by 0.90?

Answer and reasoning

Only 90% of recruited participants are expected to remain analysable. Dividing by the retention proportion gives the recruitment number expected to yield 122 analysable participants.

Can a study with adequate power still produce a biased result?

Answer and reasoning

Yes. Power concerns sampling variation under stated assumptions. Selection bias, confounding, measurement error and protocol problems can distort an estimate regardless of sample size.

Key take-home messages

References

1. Schulz KF, Grimes DA. Sample size calculations in randomised trials: mandatory and mystical. Lancet. 2005;365(9467):1348–1353. https://doi.org/10.1016/S0140-6736(05)61034-3
2. Lakens D. Sample size justification. Collabra Psychol. 2022;8(1):33267. https://doi.org/10.1525/collabra.33267
3. Chan AW, Boutron I, Hopewell S, Moher D, Schulz KF, Collins GS, et al. SPIRIT 2025 statement: updated guideline for protocols of randomised trials. BMJ. 2025;389:e081477. https://doi.org/10.1136/bmj-2024-081477
4. Hopewell S, Chan AW, Collins GS, Hróbjartsson A, Moher D, Schulz KF, et al. CONSORT 2025 statement: updated guideline for reporting randomised trials. BMJ. 2025;389:e081123. https://doi.org/10.1136/bmj-2024-081123
5. International Conference on Harmonisation of Technical Requirements for Registration of Pharmaceuticals for Human Use. Statistical principles for clinical trials E9. Geneva: ICH; 1998. https://database.ich.org/sites/default/files/E9_Guideline.pdf
6. Chow SC, Shao J, Wang H, Lokhnygina Y. Sample size calculations in clinical research. 3rd ed. Boca Raton (FL): CRC Press; 2018.