Concept Architecture
Concept
Theoretically, Sample Size Calculation is a statistical procedure used to determine the number of observations required for a study to estimate a parameter with specified precision or detect a prespecified effect with an acceptable probability of statistical error. It is founded on sampling theory, hypothesis testing, estimation theory, and the relationship between effect size, outcome variability, significance level, and statistical power. The calculation exists to ensure that studies are sufficiently informative while avoiding unnecessary recruitment, cost, and exposure of participants.
Mathematically, sample size is derived from the sampling distribution of the planned estimator or test statistic. The required number of observations depends on the study design, outcome distribution, target effect size, variance, significance level, desired power, allocation ratio, and any adjustments for clustering, repeated measurements, non-compliance, or attrition. Distinct recognised formulae apply to comparisons of means, proportions, rates, survival outcomes, equivalence studies, non-inferiority studies, and cluster-randomised designs.
In practice, sample size is calculated prospectively using estimates from previous studies, pilot data, registries, expert judgement, or clinically meaningful thresholds. In health economics it is used in clinical trials, observational studies, cost-effectiveness studies, resource-use analyses, and value-of-information research, although economic outcomes may require separate consideration because cost data are often skewed and highly variable.
Purpose
Used to determine the number of participants or observational units required to achieve specified statistical power, estimation precision, and decision relevance while controlling Type I and Type II error in health economic and clinical research.
Mathematical Formulae
Primary Formula
For comparing two independent means with equal group sizes:
n = 2��(z??�?? + z???)� � ?�
where:
- n = required sample size per group
- �� = outcome variance
- ? = minimum detectable difference
- � = Type I error probability
- 1 ? ? = statistical power
- z??�?? = standard Normal critical value for a two-sided significance level
- z??? = standard Normal critical value corresponding to the required power
Supporting Formulae
For estimating a mean with margin of error d:
n = z??�??��� � d�
For estimating a proportion:
n = z??�??�p(1 ? p) � d�
For comparing two independent proportions with equal group sizes:
n = [z??�??�(2p?(1 ? p?)) + z???�(p?(1 ? p?) + p?(1 ? p?))]� � (p? ? p?)�
where:
p? = (p? + p?) � 2
Adjustment for attrition:
n?d? = n � (1 ? a)
where:
- a = anticipated attrition proportion
Design effect for equal-sized clusters:
DE = 1 + (m ? 1)?
Cluster-adjusted sample size:
n? = n ? DE
where:
- m = average cluster size
- ? = intraclass correlation coefficient
Related Mathematical Methods
- Statistical Power
- Effect Size
- Type I Error
- Type II Error
- Confidence Interval
- Hypothesis Testing
- Variance Estimation
- Cluster Design Effect
- Non-Inferiority Margin
- Value of Information Analysis
Example
A randomised controlled trial will compare mean quality-adjusted life years between two treatment groups. Investigators wish to detect a difference of 0.08 QALYs, assume a standard deviation of 0.20, use a two-sided significance level of 5%, and require 90% power.
The relevant critical values are:
z??�?? = 1.96
z??? = 1.282
The required sample size per group is:
n = 2 ? 0.20� ? (1.96 + 1.282)� � 0.08�
n = 2 ? 0.04 ? 10.510 � 0.0064
n = 131.38
The study therefore requires 132 participants per group.
Allowing for 15% attrition:
n?d? = 132 � (1 ? 0.15)
n?d? = 155.29
The required recruitment target is 156 participants per group, or 312 participants in total.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.S.INV | =NORM.S.INV(1-0.05/2) | Obtain the critical value for a two-sided 5% significance level. |
| NORM.S.INV | =NORM.S.INV(0.90) | Obtain the critical value corresponding to 90% statistical power. |
| POWER | =2*POWER(0.20,2)*POWER(NORM.S.INV(1-0.05/2)+NORM.S.INV(0.90),2)/POWER(0.08,2) | Calculate the required sample size per group for comparing two means. |
| ROUNDUP | =ROUNDUP(B2,0) | Round the calculated sample size to the next whole participant. |
| CEILING.MATH | =CEILING.MATH(B2/(1-0.15),1) | Inflate the sample size for anticipated attrition. |
VBA (Optional)
Automate sample size calculations across alternative effect sizes, power levels, attrition assumptions, and cluster design parameters.
Sources
- Chow SC, Shao J, Wang H, Lokhnygina Y. Sample Size Calculations in Clinical Research.
- Julious SA. Sample Sizes for Clinical Trials.
- Machin D, Campbell MJ, Tan SB, Tan SH. Sample Size Tables for Clinical Studies.
- Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
- NICE. Health Technology Evaluations: The Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a sample size calculation?
The process of determining the number of participants a study needs to have an adequate chance of detecting a specified treatment effect.
Source: Friedman LM, Furberg CD, DeMets DL, Reboussin DM, Granger CB. Fundamentals of Clinical Trials. 5th ed. Springer; 2015. doi:10.1007/978-3-319-18539-2.
What does a sample size calculation determine before a study begins?
A sample size calculation determines how many participants a study needs to stand a good chance of detecting a treatment effect of a specified size, if that effect is real. It weighs the size of effect worth finding, the variability of the outcome, and the acceptable risks of a false positive and a false negative, and turns these into a required number. Getting it right matters because too small a study may miss a genuine effect, while too large a one wastes resources and exposes extra participants. Sizing a study to detect a real effect is its purpose. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
What factors affect a sample size calculation?
A sample size calculation depends on the desired statistical power, usually eighty or ninety per cent; the significance level, typically five per cent; the size of the effect to be detected, with smaller effects requiring larger samples; and the variability of the outcome, with greater variability requiring more participants. So a sample size calculation is driven by the power, significance level, effect size, and variability, since these determine how many participants are needed to distinguish a true effect from chance, which is why estimating the expected effect and variability accurately is important, as misjudging them leads to a study that is underpowered or larger than necessary.
Source: Friedman, Furberg & DeMets 2015
Why is a sample size calculation important?
A sample size calculation is important because a study with too few participants may be underpowered, likely to miss a real effect and waste resources, while an unnecessarily large study wastes resources and may expose more participants than needed. So a sample size calculation matters for designing a study that can reliably answer its question efficiently and ethically, since adequate power is needed to detect a meaningful effect and avoid a false negative, and enrolling the right number of participants balances the ability to answer the question against the costs and burdens, which is why it is a required part of planning a trial.
Source: Friedman, Furberg & DeMets 2015
How is a sample size calculation performed?
A sample size calculation is performed by specifying the primary outcome and analysis, the desired power and significance level, the smallest effect worth detecting, and the expected variability, then using the appropriate formula or software to compute the required number, often with an allowance for dropout. So a sample size calculation is performed by combining these inputs through the relevant statistical formula for the outcome and test, which yields the number of participants needed to achieve the specified power, and the result is typically inflated to account for anticipated loss to follow-up, ensuring enough participants remain for the analysis.
Source: Friedman, Furberg & DeMets 2015
What are the challenges of a sample size calculation?
The challenges of a sample size calculation include estimating the expected effect size and variability in advance, since these are often uncertain and errors affect the calculation; choosing a meaningful effect to detect; and accounting for dropout and other design features. So a sample size calculation is challenging because it relies on assumptions about quantities not yet known, which is why estimates are drawn from previous studies or pilot data and sensitivity to the assumptions is considered, since an inaccurate assumption can leave a study underpowered or oversized, and this uncertainty is a key difficulty in planning an adequately powered study.
Source: Friedman, Furberg & DeMets 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Dec 2025
Content version: 1.0.0
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