Optimal Sample Size

Optimal Sample Size

The phrase “optimal sample size” needs an explicit objective. In a conventional trial, investigators often seek the smallest feasible sample meeting a prespecified power, type I error and precision requirement for a meaningful target difference. In decision-focused health economic research, “optimal” can instead mean the sample size whose expected information benefit, across the population affected by a future decision, is greatest after the study's costs and delays. These objectives can produce different answers; neither defines a universal number of participants.

Start with the decision and estimand

Specify the target population, comparator, intervention, primary outcome or estimand, unit of randomisation, allocation ratio, analysis method, follow-up and intended use of the results. A superiority trial powered for a clinical outcome is not automatically powered for an economic outcome, subgroup analysis, adverse event, equity question or an incremental net benefit. State whether the aim is hypothesis testing, interval precision, reliable estimation in strata, or a health system decision about further research.

For a frequentist design, prespecify the two-sided significance level $\alpha$, desired power $1-\beta$, clinically or decision-relevant target difference $\delta$, and nuisance quantities such as outcome variability. Target differences should be justified with patients, clinicians and the decision setting rather than selected solely to make recruitment convenient. If the economic analysis is central, specify the willingness-to-pay threshold, uncertainty in costs and effects, and whether net benefit, QALYs or a clinical outcome drives sample size.

A transparent two-arm calculation

Consider independently randomised, equally sized groups, a continuous primary outcome with common standard deviation $\sigma$, a difference in means $\delta>0$, complete observations, and an approximately normal large-sample test. A common planning approximation for participants per arm is

$$n_{\mathrm{arm}}\approx\frac{2\sigma^2(z_{1-\alpha/2}+z_{1-\beta})^2}{\delta^2},$$

where $z_p$ is the standard normal $p$ quantile. This is a planning approximation, not an exact finite-sample calculation, and it does not apply unchanged to binary, count or time-to-event outcomes. For a two-sided $\alpha=0.05$ test with 80% power, $z_{0.975}\approx1.96$ and $z_{0.80}\approx0.84$.

Suppose $\sigma=10$ outcome units and $\delta=5$ units. Substitution gives $n_{\mathrm{arm}}\approx2(10)^2(1.96+0.84)^2/(5)^2=62.72$, rounded up to 63 analysable people in each arm, or 126 total. If 20% of randomised people are expected not to provide an analysable primary outcome and that loss is approximately independent of allocation and outcome after planned handling, recruitment becomes $\lceil63/(1-0.20)\rceil=79$ per arm, or 158 total. The attrition adjustment is an illustrative gross-up, not a cure for informative missingness; prevention and an appropriate missing-data strategy are essential.

If the same standard deviation is assumed and the target difference is halved to 2.5 units, the approximation requires four times as many analysable participants per arm, about 251 after rounding the unrounded formula. This inverse-square relationship illustrates why the target difference matters; it does not show that the smaller effect is clinically important or feasible to study. Recompute with an exact method, realistic event rates and the actual analysis design when planning a real trial.

How design changes the required sample

Equal allocation is often statistically efficient when per-person costs and variances are similar, but ethical, recruitment and cost constraints can favor another ratio. Unequal allocation must be reflected in the variance and power calculation; “total sample size” alone is insufficient. Cluster randomisation requires allowance for intracluster correlation and unequal cluster sizes; under equal clusters of size $m$, a rough design effect is $1+(m-1)\rho$, where $\rho$ is the intracluster correlation. The number of clusters and finite-cluster degrees of freedom also matter, so multiplying a person-randomised sample by this effect alone can be inadequate.

Survival outcomes are often planned from the number of observed events rather than only enrolled participants. Recruitment rates, follow-up, censoring and expected hazard patterns determine how many people are needed to achieve that event count. Binary outcomes require plausible risks in both arms; repeated measures require a covariance structure; equivalence and non-inferiority require justified margins and different decision criteria. Multiple co-primary endpoints, interim looks and subgroup claims may require multiplicity and operating-characteristic adjustments.

State whether the assumed standard deviation, event rate, attrition rate and correlation come from suitable data, a pilot, or explicit judgment. Examine plausible alternatives because an underestimated variance or event shortfall can substantially reduce actual power. Simulation is especially valuable when allocation, clustering, missing data, adaptive rules, non-normal costs or complex estimands make a simple closed-form approximation unreliable.

Precision as the objective

When estimation matters more than a rejection decision, plan sample size against an interval width or margin of error. For an independent sample mean under a large-sample normal approximation, a two-sided $(1-\alpha)$ confidence interval has approximate half-width $h=z_{1-\alpha/2}\sigma/\sqrt{n}$, giving $n\approx(z_{1-\alpha/2}\sigma/h)^2$. Specify the quantity being estimated, its target precision, coverage and expected variance. A confidence interval calculated after the study should report actual uncertainty even if achieved width differs from the planning goal.

The appropriate precision formula changes with binary proportions, correlated observations, ratios and model-based estimands. For cost-effectiveness studies, the variance of incremental net monetary benefit depends on both cost and effect variances and their covariance. At threshold $\lambda$, if $N=\lambda E-C$, then $\operatorname{Var}(N)=\lambda^2\operatorname{Var}(E)+\operatorname{Var}(C)-2\lambda\operatorname{Cov}(E,C)$. Ignoring this covariance can distort a sample size based on net benefit; estimate it from suitable paired patient-level data where possible.

Decision-theoretic optimum in health economics

A value-of-information design asks how much a proposed study is expected to improve future decisions, not simply whether a test has 80% power. For candidate design $d$ and sample size $n$, a stylised objective is

$$\mathrm{ENBS}(n,d)=\mathrm{EVSI}(n,d)-C_{\mathrm{study}}(n,d)-C_{\mathrm{delay}}(n,d),$$

where EVSI is the population-level expected value of sample information over the relevant decision lifetime, and the two costs capture research resources and the opportunity cost of delayed implementation when applicable. Define the population horizon, adoption or implementation assumptions, perspective and whether trial costs already include any delayed benefits. Choose the feasible $n,d$ maximizing expected net benefit of sampling; if all candidates have negative ENBS, conducting no new study may be preferred under the stated objective. Avoid double counting the same opportunity cost in EVSI and the delay term.

EVSI requires a probabilistic model linking study data to uncertain decision inputs and a rule for revising the decision after observing data. It normally rises with informative sample size but with diminishing returns, while study costs and sometimes delay rise. Recruitment feasibility, data quality, ethics, regulatory requirements and irreducible structural uncertainty constrain the answer. Value-of-information methods can allocate research to the parameters or outcomes most likely to change an actual decision; they are sensitive to the prior evidence, model structure and population affected.

Do not call a conventional power calculation the economic optimum merely because it reaches a power target. Conversely, an ENBS-maximising design may not satisfy a regulator's required operating characteristics or a trial's primary clinical objective. Present both objectives when both apply, and explain the final choice under the binding constraints.

Reporting and checks

Record the design objective, estimand, all numerical assumptions and their evidence, formula or simulation method, allocation, rounding, losses, cluster or event assumptions, planned analysis and sensitivity analyses. A compact table of sample sizes under alternative target effects, variances and recruitment rates makes fragility visible. Keep “per arm,” “total randomised,” “expected analysable” and “number of events” distinct; report a rounded-up feasible integer and never quietly round down.

The CONSORT 2025 statement and explanation call for transparent reporting of how the sample size was determined and the assumptions used. The NIHR DELTA2 guidance discusses choosing and justifying target differences and alternative sample-size approaches. For complex adaptive designs, FDA guidance on adaptive designs discusses evaluation of operating characteristics. Bénard and colleagues' value-of-information sample-size study illustrates a decision-theoretic approach in cost-effectiveness research. Apply the guidance for the actual study design and jurisdiction rather than treating this teaching approximation as a protocol-ready calculation.