Principles of Sample Size Planning

Overview

Sample-size planning determines how much information a study needs to answer its primary question with acceptable uncertainty. It is not a ritual of entering numbers into a formula. The correct rationale depends on the objective, estimand, design, outcome, analysis and expected information loss.

Different studies may justify sample size through:

• precision of an estimate;
• power for a prespecified comparison;
• number of outcome events;
• prediction-model complexity;
• finite population or census constraints;
• feasibility and pilot objectives;
• qualitative information power; or
• a combination of design and resource considerations.

This lecture focuses on methodological planning and transparent justification. Detailed formulae and calculator implementation belong in the linked Biostatistics lectures.

Why this matters

An unjustified sample can be:

• too small to produce clinically useful precision;
• too small to detect an important effect with reasonable probability;
• larger than necessary, wasting participant time and resources;
• based on an implausible effect or event rate;
• inflated incorrectly for attrition;
• calculated for the wrong outcome or test;
• incompatible with clustering or repeated observations; or
• presented with false precision despite uncertain assumptions.

Sample size also has ethical implications. An under-informative study exposes participants to burden without a realistic prospect of answering the question. An unnecessarily large study may use resources and participants that could support other research.

Key concepts

Begin with the primary objective

A study may have several objectives, but sample-size planning usually centres on the primary objective. The plan should specify:

• target population;
• primary outcome;
• exposure or intervention;
• comparison where applicable;
• estimand;
• design;
• principal analysis;
• acceptable precision or target effect; and
• expected missing information.

Precision-based planning

Precision-based planning is appropriate when the aim is to estimate a quantity such as:

• prevalence or proportion;
• mean;
• median;
• incidence rate;
• sensitivity or specificity;
• agreement parameter; or
• regression coefficient.

The main design question is the desired confidence-interval width or margin of error. A narrower interval requires more information.

For a proportion, the required sample depends on the anticipated proportion and acceptable absolute precision. When the anticipated value is uncertain, the team may use conservative or scenario-based assumptions, but the choice should be explained.

Precision should be clinically interpretable. “A 95% confidence interval no wider than 10 percentage points” is more meaningful than “using a precision of 0.05” without stating whether 0.05 is an absolute margin, relative precision or standardised value.

Power-based planning

Power-based planning is common for testing a prespecified comparison or effect. It requires:

• null and alternative values;
• clinically meaningful target difference;
• variability or baseline event rate;
• type I error;
• desired power;
• one- or two-sided procedure;
• allocation ratio;
• design features; and
• intended statistical method.

The target effect should not be chosen solely from the largest effect in a small prior study. Published estimates can be exaggerated, and the smallest clinically important difference may not be the same as the expected effect.

Power is the probability that the planned analysis will meet its decision criterion under the assumed alternative. It is not the probability that the study hypothesis is true.

Event-based planning

For survival, logistic, Poisson and prediction models, the amount of information may depend more on the number of outcome events than the total sample. If the event is rare, a large cohort may still contain too few events.

Planning should address:

• anticipated event proportion or rate;
• number and form of parameters;
• censoring or competing events;
• overfitting control;
• model objective—explanation, causal estimation or prediction;
• validation strategy; and
• expected shrinkage or precision.

Rules such as “10 events per variable” are not universal sample-size solutions. Modern prediction-model planning considers outcome frequency, candidate parameters, anticipated model performance and overfitting.

Feasibility-based and fixed sample sizes

Sometimes the available population, budget, time or biological material fixes the maximum sample. A study can still be methodologically transparent by:

1. stating the constraint;
2. calculating the precision or detectable effect under plausible scenarios;
3. judging whether that information is useful;
4. simplifying objectives if necessary; and
5. avoiding a post hoc claim that the fixed number was “adequately powered” without evidence.

A census of a small accessible population does not remove uncertainty about measurement, non-response or generalisation.

Pilot and feasibility studies

Pilot and feasibility studies should be sized for their own objectives, such as:

• estimating recruitment and retention rates;
• evaluating procedure completion;
• assessing acceptability;
• estimating variability for future planning;
• testing data systems; or
• identifying implementation failures.

They are usually not designed to provide a definitive treatment-effect test. An imprecise pilot effect estimate should not be used uncritically as the sole input for a main trial.

Qualitative studies

Qualitative sample adequacy is justified through the aim, sample specificity, diversity, theory, quality of dialogue and analytical approach. It is not calculated using statistical power. The protocol may state an initial range and an iterative stopping process based on information power or a methodology-specific adequacy concept.

DECISION GUIDE

Choose the primary sample-size rationale

• Use precision when the primary aim is estimation.
• Use power when the primary aim is a prespecified hypothesis test or comparison.
• Use event-based planning when information depends on outcome events or model complexity.
• Use prediction-model criteria for model development or validation.
• Use feasibility precision for pilot and process objectives.
• Use information power or methodological adequacy for qualitative inquiry.
• Use a fixed-sample sensitivity analysis when resources or population size are constrained.
• Do not combine rationales indiscriminately; identify which one governs the final target.

Sample-size planning table

illustrative

Table 1. Mapping objectives to information requirements

Objective	Primary quantity	Possible rationale	Key assumptions
Estimate testing completion	Proportion	Confidence-interval precision	Expected completion, margin of error
Compare reminder groups	Risk difference or ratio	Power for target contrast	Baseline risk, effect, allocation
Develop prediction model	Predicted probability	Model-development criteria	Event rate, parameters, anticipated performance
Interview women about barriers	Themes and explanatory account	Information power	Aim, specificity, diversity, dialogue
Assess feasibility	Recruitment and retention rates	Precision around feasibility parameters	Expected rates and decision thresholds

Note: The table is illustrative. The study should declare one primary objective.

Inputs that require justification

Target effect or precision

The target should be connected to clinical or policy decisions. For an intervention, the smallest effect worth detecting may be informed by patient priorities, costs, harms and existing care. For estimation, interval width should determine whether stakeholders can distinguish acceptable from unacceptable performance.

Nuisance parameters

Standard deviation, baseline risk, intracluster correlation, correlation between repeated measures and event rate affect the calculation. Use estimates from populations, settings and definitions similar to the planned study. When uncertain, show scenarios.

Type I error and power

Two-sided 5% type I error and 80% or 90% power are common conventions, not natural laws. Deviations require rationale. Multiple primary comparisons, interim analyses or multiplicity may alter error allocation.

Allocation ratio

Equal allocation often maximises power for a fixed total when group costs and variability are similar. Unequal allocation may be justified by ethics, availability, cost, expected attrition or the need for more safety information.

Design structure

Clustering, unequal weights, repeated measures, matching, crossover and stratification affect information. The calculation should match the intended analysis rather than treating all observations as independent.

Adjustments after the core calculation

Non-response and attrition

If $n_\text{analysable}$ participants are required and the expected retained proportion is $r$, a common inflation is:

$$n_\text{recruit} = \frac{n_\text{analysable}}{r}$$

For 20% attrition, $r=0.80$; the required recruitment is divided by 0.80, not increased by only 20%. If 300 analysable participants are needed, $300/0.80=375$ must be recruited.

This arithmetic addresses numbers, not attrition bias.

Design effect

For cluster designs, the independent-sample requirement may be multiplied by a justified design effect. The number of clusters must also be adequate. A very large number of participants in a few clusters may not support reliable cluster-level inference.

Unusable or ineligible records

Retrospective and laboratory studies may need allowances for missing records, inadequate specimens or failed measurements. These losses should be estimated separately from participant attrition because their mechanisms and prevention differ.

Multiple objectives

Calculate information needs for important objectives, then identify which governs the design. Do not inflate mechanically for every secondary objective. A study cannot always answer all ambitions within one feasible sample.

PROTOCOL CHECKLIST

Sample-size justification to report

• primary objective and estimand;
• outcome and comparison;
• rationale type;
• formula, method, software or calculator;
• confidence level, type I error and power where relevant;
• target effect or margin of error;
• standard deviation, baseline risk or event rate;
• allocation ratio;
• clustering, matching or repeated-measure assumptions;
• design effect and number of clusters;
• non-response, attrition and unusable-data allowances;
• source and uncertainty of every input;
• resulting analysable and recruitment targets; and
• scenario or sensitivity calculations.

Ethical and governance considerations

Interface with statistical analysis

The calculation and analysis must target the same quantity and data structure. Examples of mismatches include:

• calculating for an independent t-test but analysing repeated measurements;
• planning for a simple proportion but using clustered facility sampling;
• calculating from a risk ratio while the primary estimand is a risk difference;
• using the number of predictors rather than candidate parameters for a nonlinear model;
• calculating for complete follow-up while the primary strategy handles intercurrent events differently; and
• applying a cross-sectional formula to a time-to-event endpoint with censoring.

Detailed implementation belongs in the Biostatistics sample-size lectures and calculators. The research-methodology responsibility is to define the question, design and justified inputs.

Why observed post hoc power is unhelpfulMMed extension

After a study is completed, “observed power” calculated by inserting the observed effect into the original formula is largely a transformation of the p-value and adds little information. The estimate and confidence interval directly show the effect and its uncertainty. A non-significant result should be interpreted through its compatible effect range, not rescued or dismissed by post hoc power.

Worked example

WORKED EXAMPLE

Planning for prevalence precision and attrition

Primary objective: estimate the proportion of women completing postpartum testing within 12 weeks.

Planning assumptions: expected completion 60%, 95% confidence level and absolute margin of error of 5 percentage points.

The appropriate Biostatistics method gives an independent-sample requirement based on those inputs. Suppose the calculated analysable sample is 369.

Attrition allowance: Expected outcome ascertainment is 85%.

$$n_\text{recruit} = \frac{369}{0.85}=434.1$$

The recruitment target is rounded up to 435.

Stratification: If the study oversamples a small hospital, the allocation and weighting plan must be incorporated. If cluster sampling is used instead, the calculation must account for the design effect and number of clusters.

Sensitivity: Because expected completion is uncertain, the team should compare sample sizes for plausible proportions and margins of error.

Interpretation: Recruiting 435 addresses the anticipated number of unavailable outcomes. It does not remove bias if loss is related to testing completion.

Suggested protocol or reporting wording

Evidence basis and critical appraisal

The WHO practical manual establishes the classical precision- and power-based foundations for health-study sample sizes [1]. DELTA2 guidance emphasises transparent choice of the target difference in randomised trials rather than selecting a convenient or optimistic effect [2]. Broader sample-size guidance recognises that justification may be based on precision, power, resource constraints or other information criteria [3].

Observational reporting guidance expects sample-size methods and assumptions to be disclosed [4]. Prediction-model research now uses criteria that address overfitting, shrinkage and expected model performance rather than a single events-per-variable rule [5]. Updated randomised-trial reporting guidance likewise requires transparent sample-size determination and design changes [6].

A critical reader should be able to reconstruct the calculation from the report. The presence of a software name or final number is insufficient. Inputs should be clinically interpretable, sourced and consistent with the final primary analysis. Scenario analyses are particularly important when event rates, variability, intracluster correlation or attrition are uncertain.

Checking numerical consistency

The reported analysable target, attrition inflation and final recruitment target should reconcile arithmetically. Rounding should occur at the end and always upward when a minimum is required. For unequal group allocation, site caps or cluster sizes, the final number may need further rounding to preserve complete blocks or clusters. These operational adjustments should be reported separately from the statistical requirement.

Common mistakes

Knowledge check

What should be defined before selecting a sample-size formula?

Answer and reasoning

The primary objective, estimand, outcome, comparison, design and intended analysis. These determine which information criterion and inputs are relevant.

If 300 analysable participants are required and 20% attrition is expected, how many should be recruited?

Answer and reasoning

Divide by the expected retained proportion: $300/0.80=375$. Adding 20% would give only 360 and would leave 288 after 20% attrition.

Does increasing sample size remove selection or measurement bias?

Answer and reasoning

No. It reduces random uncertainty under the design and model. Systematic errors require prevention, design changes, adjustment or sensitivity analysis.

How should a fixed maximum sample be justified?

Answer and reasoning

State the resource or population constraint and show the precision, detectable effect or expected information under plausible scenarios. Do not invent assumptions merely to label the fixed sample adequately powered.

Why should a qualitative interview study not use a conventional power calculation?

Answer and reasoning

Its adequacy concerns conceptual relevance, information richness and analytical depth rather than estimation of a population parameter under random sampling.

Key take-home messages

SUMMARY

Key take-home messages

• Base sample size on the primary objective, estimand and design.
• Choose a precision-, power-, event-, feasibility- or information-based rationale deliberately.
• Justify every numerical assumption and examine uncertainty through scenarios.
• Match the calculation to clustering, repeated observations, allocation and analysis.
• Inflate for expected losses by using the retained proportion, while addressing bias separately.
• Report analysable and recruitment targets and link detailed calculations to the correct Biostatistics method.

References

1. Lwanga SK, Lemeshow S. Sample size determination in health studies: a practical manual. Geneva: World Health Organization; 1991. https://iris.who.int/handle/10665/40062
2. Cook JA, Julious SA, Sones W, Hampson LV, Hewitt C, Berlin JA, et al. DELTA2 guidance on choosing the target difference and undertaking and reporting the sample size calculation for a randomised controlled trial. BMJ. 2018;363:k3750. https://doi.org/10.1136/bmj.k3750
3. Lakens D. Sample size justification. Collabra Psychol. 2022;8(1):33267. https://doi.org/10.1525/collabra.33267
4. Vandenbroucke JP, von Elm E, Altman DG, Gøtzsche PC, Mulrow CD, Pocock SJ, et al. Strengthening the Reporting of Observational Studies in Epidemiology (STROBE): explanation and elaboration. PLoS Med. 2007;4(10):e297. https://doi.org/10.1371/journal.pmed.0040297
5. Riley RD, Ensor J, Snell KIE, Harrell FE Jr, Martin GP, Reitsma JB, et al. Calculating the sample size required for developing a clinical prediction model. BMJ. 2020;368:m441. https://doi.org/10.1136/bmj.m441
6. Hopewell S, Chan AW, Collins GS, Hróbjartsson A, Moher D, Schulz KF, et al. CONSORT 2025 statement: updated guideline for reporting randomised trials. JAMA. 2025;333(22):1998–2005. https://doi.org/10.1001/jama.2025.4347