Sample Size Calculation

Sample Size Calculation in Trials and Economic Evaluation

A sample size calculation turns four planning choices into a number of participants, or of events, that a study has to analyse. The choices are the target difference between groups, the variability or baseline risk of the outcome, the significance level and the statistical power. In health economics it matters twice over, because randomised trials are a key source of cost-effectiveness data yet are normally sized on their primary clinical objective. A trial large enough for its clinical question can leave the economic question badly underpowered, which later shows up as wide uncertainty around net benefit.

Inputs that fix the number

The DELTA2 guidance, funded through the UK MRC and NIHR methodology programme, describes the conventional approach. The significance level and power are set first, typically a two-sided 5% level with 80% or 90% power, and the sample size then follows from the target difference. The ICH E9 guideline on statistical principles for clinical trials also asks for the primary variable, the test statistic, the hypotheses and the approach to withdrawals and protocol violations. It notes that the type II error is conventionally set at 10% to 20%.

The target difference carries most of the weight. DELTA2 says it should be viewed as important by at least one key stakeholder group, such as patients, clinicians, regulators or healthcare funders, and observes that trial reports often suggest it was set on convenience or the research budget. ICH E9 allows either the minimal effect of clinical relevance, often a minimal clinically important difference, or the anticipated effect where that is larger. The target is paired with a nuisance parameter, which DELTA2 defines as the quantity that, together with the target difference, fixes the difference on the original outcome scale, such as the control group event proportion for a binary outcome or the standard deviation for a continuous one.

Formulae for binary and time-to-event outcomes

The formula for comparing two means is worked through on the statistical power page, and DELTA2 notes that for a continuous outcome halving the target difference quadruples the sample size. For a binary outcome compared between two equal groups, a standard large-sample approximation gives the number per arm.

$$n = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2\left[p_1(1-p_1) + p_2(1-p_2)\right]}{(p_1 - p_2)^2}$$

where $n$ is the number of analysable participants per arm, $p_1$ and $p_2$ are the expected proportions with the outcome in the control and intervention arms, $\alpha$ is the two-sided significance level, $\beta$ is the type II error probability, so power is one minus $\beta$, and $z_q$ is the $q$ quantile of the standard normal distribution. Whitley and Ball tabulate the squared sum of the two quantiles as 7.9 for 80% power and 10.5 for 90% power at a 5% significance level.

Time-to-event trials are sized by events. ICH E9 notes that where the event rate is of primary interest, assumptions are needed to turn the required number of events into a sample size, and DELTA2 asks for the number of events to be stated. Under proportional hazards and equal allocation, the formula now known as Schoenfeld's gives the total number of events.

$$E = \frac{4(z_{1-\alpha/2} + z_{1-\beta})^2}{(\ln \mathrm{HR})^2}, \qquad N = \frac{E}{p_{\mathrm{event}}}$$

where $E$ is the total number of events across both arms, $\mathrm{HR}$ is the target hazard ratio, $N$ is the number of participants to recruit and $p_{\mathrm{event}}$ is the expected probability that a participant has the event during follow-up. Cortés Martínez and colleagues note that the number of patients needed to observe the required events depends on the probability of observing the event during follow-up, so recruitment and follow-up periods change $N$ while leaving $E$ unchanged.

Worked example with a readmission trial and its economic endpoint

All numbers are illustrative. A trial of a pharmacist-led medicines review after hospital discharge plans to detect a fall in 90-day emergency readmission, with an expected 20% in the control arm and a target of 15% in the intervention arm. The trial uses a two-sided 5% significance level, 90% power and equal allocation, and a cost-utility analysis will run alongside it.

1. Multiplier. With quantiles of 1.960 and 1.282, the multiplier is (1.960 + 1.282) squared, or 3.242 × 3.242 = 10.51.

2. Variance term. The sum 0.20 × 0.80 + 0.15 × 0.85 = 0.2875, and the squared difference is 0.05 × 0.05 = 0.0025, so the ratio is 0.2875 / 0.0025 = 115.

3. Analysable sample. The product 115 × 10.51 = 1,208.65 rounds up to 1,209 per arm, or 2,418 analysable participants.

4. How the target is expressed. If the control risk were 15% instead, keeping the same 25% relative reduction (a target of 11.25%) gives a variance sum of 0.2273 and a squared difference of 0.00140625, or about 1,700 per arm. Keeping the same 5 point absolute reduction (a target of 10%) gives 87 × 10.51 = 914.37, or 915 per arm. This is why DELTA2 asks a binary target to say whether the absolute or the relative difference takes primacy.

5. Losses. With 10% expected to have no primary outcome, recruitment becomes 1,209 / 0.9 = 1,343.3, or 1,344 per arm.

6. Economic endpoint. The review is expected to add 0.01 QALYs and save GBP 100 per patient, with patient-level standard deviations of 0.15 QALYs and GBP 2,500 and a correlation of minus 0.3 between costs and QALYs. At GBP 25,000 per QALY the incremental net monetary benefit is 25,000 × 0.01 + 100 = 350. The per-patient variance of net benefit adds 25,000 squared times 0.15 squared (14,062,500), 2,500 squared (6,250,000) and, because the correlation is negative, 2 × 25,000 × 0.3 × 0.15 × 2,500 = 5,625,000, giving 25,937,500 and a standard deviation of about GBP 5,093. Powering the economic analysis on its own terms needs 2 × 10.51 × 25,937,500 / 122,500 = 4,450.7, or 4,451 per arm, where 122,500 is 350 squared.

7. Economic power at the clinical size. With 1,209 per arm the standard error of incremental net benefit is the square root of 2 × 25,937,500 / 1,209, about GBP 207, so the expected test statistic is 350 / 207 = 1.69. Power is the normal probability below 1.69 minus 1.96, about 39%.

8. Interpretation. The trial is well sized for the readmission difference but has roughly a two in five chance of showing a significantly positive net benefit. The economic endpoint needs more than three times the clinical sample, so the protocol has to say which question is primary and either accept an imprecise economic estimate or justify a larger trial.

Sizing for the cost-effectiveness analysis

Briggs and Gray noted in 1998 that many commentators wanted trial sample sizes to meet the needs of the economic analysis as well as the clinical one. Willan and O'Brien proposed a size large enough for the confidence interval to separate the cost-effective and not cost-effective regions of the cost-effectiveness plane. Willan then showed that recasting the analysis as incremental net benefit allows standard statistical procedures for power and sample size.

$$\sigma_{NB}^2 = \lambda^2\sigma_E^2 + \sigma_C^2 - 2\lambda\rho\,\sigma_E\sigma_C, \qquad n = \frac{2(z_{1-\alpha/2} + z_{1-\beta})^2\,\sigma_{NB}^2}{(\lambda\Delta E - \Delta C)^2}$$

where $\sigma_{NB}^2$ is the patient-level variance of net benefit, assumed equal in both arms, $\lambda$ is the value placed on a unit of health such as a QALY, $\sigma_E$ and $\sigma_C$ are the standard deviations of effects and costs, $\rho$ is their correlation, $\Delta E$ and $\Delta C$ are the expected incremental effect and cost, and $n$ is the number per arm. Glick lists these inputs together with the two error rates and maximum willingness to pay, and shows that power need not rise steadily as willingness to pay increases. A negative correlation between costs and QALYs inflates the variance of net benefit, as step 6 shows. The NICE manual (PMG36, section 4.2.16) asks for net health benefits to be presented at GBP 25,000 and GBP 35,000 per QALY, which gives natural values of $\lambda$ for UK sensitivity runs.

DELTA2 lists this health economic route among its methods for setting the target difference, alongside a value of information analysis that compares the value of an extra observation with its cost and so needs no target difference. That decision-theoretic optimum, built on the expected value of sample information and the expected net benefit of sampling, is the subject of the optimal sample size page. ICH E9 accepts sizing on another basis, such as important secondary objectives needing larger numbers, if this is made clear and justified.

Design features and assumptions that move the answer

ICH E9 states that the calculation should refer to the number needed for the primary analysis, and that if this is the full analysis set the assumed effect may need reducing for dilution from withdrawal and poor compliance. Inflating for attrition protects power only and does nothing about bias from informative missing data. Cluster allocation multiplies the requirement by the design effect.

For a non-inferiority trial or an equivalence trial, ICH E9 bases the size on a confidence interval showing that the treatments differ by at most a clinically acceptable margin. For a non-inferiority trial it warns that assessing power at a zero difference underestimates the sample needed if the investigational product is less effective than the active control. In a group sequential design the final size cannot be fixed in advance, so planning works with expected and maximum sample sizes.

Every input is an estimate. ICH E9 asks for the basis of each estimate and for the sensitivity of the sample size to deviations from it. DELTA2 recommends sensitivity calculations and the use of pilot data for nuisance parameters, and warns that a small change in one assumption can change the sample size substantially, as step 4 shows.

Reporting the calculation

The CONSORT 2025 statement asks, in item 16a, how sample size was determined, including all assumptions supporting the calculation. Its explanation and elaboration paper asks for the anticipated values in each group with their provenance, the type I error and sidedness, the power, the size per group, any inflation for attrition or non-adherence and the formulas or software used. DELTA2 gives a matching checklist of core items for protocols and results papers so that the calculation can be replicated. For an economic evaluation alongside a trial, the clinical calculation alone leaves readers unable to tell whether an imprecise cost-effectiveness estimate reflects a small effect or a trial never designed to detect one, and stating the power for incremental net benefit closes that gap.

Boundaries with power, precision and optimal sample size

Power is a property of a design under a stated alternative, and the calculation chooses enough participants to reach a target power. A precision-based calculation instead aims at a confidence interval of a desired width, one of the alternatives DELTA2 lists. Effective sample size describes how much independent information a clustered or weighted sample holds. The optimal sample size in the value of information sense trades expected gains in decision quality against research costs and can favour no further study.

Common errors in sample size work

Most errors come from treating the result as more exact than its inputs. Each input is uncertain.

1. Fitting the target difference to the budget. DELTA2 notes that trial reports often suggest this.
2. Leaving the nuisance parameter unexamined. An underestimated variance or a misjudged control event rate can leave the trial underpowered.
3. Assuming the clinical size covers the economic analysis. Net benefit can need several times as many participants.
4. Reading a non-significant result as no effect. Whitley and Ball recall that absence of evidence is not evidence of absence.

Sources

• Briggs AH, Gray AM. Power and sample size calculations for stochastic cost-effectiveness analysis. Medical Decision Making. 1998;18(2 Suppl):S81-S92. https://doi.org/10.1177/0272989X98018002S10
• Cook JA, Julious SA, Sones W, 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
• Cortés Martínez J, Geskus RB, Kim K, Melis GG. Using the geometric average hazard ratio in sample size calculation for time-to-event data with composite endpoints. BMC Medical Research Methodology. 2021;21:99. https://doi.org/10.1186/s12874-021-01286-x
• Glick HA. Sample size and power for cost-effectiveness analysis (part 2): the effect of maximum willingness to pay. PharmacoEconomics. 2011;29(4):287-296. https://doi.org/10.2165/11585080-000000000-00000
• Hopewell S, Chan AW, Collins GS, et al. CONSORT 2025 statement: updated guideline for reporting randomised trials. BMJ. 2025;389:e081123. https://doi.org/10.1136/bmj-2024-081123
• Hopewell S, Chan AW, Collins GS, et al. CONSORT 2025 explanation and elaboration: updated guideline for reporting randomised trials. BMJ. 2025;389:e081124. https://doi.org/10.1136/bmj-2024-081124
• International Conference on Harmonisation. ICH E9 Statistical principles for clinical trials. Step 4 version, 5 February 1998. Section 3.5. https://database.ich.org/sites/default/files/E9_Guideline.pdf
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Section 4.2.16. https://www.nice.org.uk/process/pmg36
• Whitley E, Ball J. Statistics review 4: sample size calculations. Critical Care. 2002;6(4):335-341. https://doi.org/10.1186/cc1521
• Willan AR. Analysis, sample size, and power for estimating incremental net health benefit from clinical trial data. Controlled Clinical Trials. 2001;22(3):228-237. https://doi.org/10.1016/S0197-2456(01)00110-6
• Willan AR, O'Brien BJ. Sample size and power issues in estimating incremental cost-effectiveness ratios from clinical trials data. Health Economics. 1999;8(3):203-211. https://doi.org/10.1002/(SICI)1099-1050(199905)8:3<203::AID-HEC413>3.0.CO;2-7