Sample size determination

Key inputs

1. Primary outcome
   • Define one primary outcome and its assessment time point before recruitment.
   • It may be continuous, binary, time-to-event, or a count outcome.
2. Target difference
   • The smallest effect that is clinically important and worth detecting.
   • Examples:
     • Continuous: mean difference in pain score.
     • Binary: difference in recovery proportion.
     • Survival: hazard ratio or difference in event-free survival.
3. Type I error, alpha (α)
   • Probability of finding a statistically significant difference when no true difference exists.
   • Usually α = 0.05, two-sided.
4. Power, 1 - beta (1 - β)
   • Probability of detecting the target difference if it truly exists.
   • Usually 80% or 90%.
   • β is the Type II error, failing to detect a real effect.
5. Variability or event rate
   • Continuous outcome: anticipated standard deviation.
   • Binary outcome: expected control-group event proportion.
   • Time-to-event outcome: expected number of events, follow-up duration, and survival assumptions.
6. Allocation ratio
   • Commonly 1:1.
   • Unequal allocation, such as 2:1, may be used for safety data or recruitment preference but normally requires a larger total sample size for the same statistical power.
7. Expected attrition
   • Inflate the calculated sample for loss to follow-up, withdrawal, missing data, and non-adherence.
8. Design effects
   • Cluster randomized trials require adjustment for intracluster correlation: [ \text{Design effect} = 1 + (m-1)\rho ] where (m) is average cluster size and (\rho) is the intracluster correlation coefficient.
   • Repeated measures, crossover, non-inferiority, adaptive, and multiple-comparison designs need design-specific calculations.

Common formulae for two parallel groups

• (\sigma) = assumed common standard deviation
• (\Delta) = target mean difference
• (Z_{\alpha/2}) = standard-normal value for two-sided alpha
• (Z_\beta) = standard-normal value for desired power

Practical workflow

1. State the design and primary endpoint.
2. Choose a clinically meaningful target difference.
3. Obtain plausible SD, event-rate, or survival assumptions from pilot data, prior trials, registries, or literature.
4. Specify α, sidedness, and desired power.
5. Calculate the analyzable sample size.
6. Inflate for attrition and protocol deviations.
7. Have the calculation reviewed by a biostatistician.
8. Prespecify any interim analyses or stopping rules, because they can alter the required sample size.

Randomization

Essential distinction

Methods of randomization

1. Simple randomization
   • Computer-generated random numbers or a random-number table.
   • Suitable for large trials.
   • Can create unequal group sizes in small samples by chance.
2. Permuted block randomization
   • Ensures approximate balance in numbers assigned to each arm throughout recruitment.
   • Use varying, undisclosed block sizes to reduce predictability.
   • Example in a 1:1 trial: randomly ordered blocks of 4 and 6.
3. Stratified randomization
   • Randomizes separately within important prognostic subgroups, such as site, disease severity, or sex.
   • Usually implemented with blocks within strata.
   • Use sparingly because too many strata create small, impractical subgroups.
4. Minimization
   • An adaptive approach that assigns the next participant to minimize imbalance across key covariates.
   • Include a random element rather than making allocation deterministic.
5. Cluster randomization
   • Allocates groups such as hospitals, wards, schools, or communities rather than individuals.
   • Useful when individual-level allocation would cause contamination.
   • Requires sample-size inflation for correlation within clusters.
6. Crossover randomization
   • Participants receive treatments in randomized order, separated by washout when appropriate.
   • Suitable only for stable conditions and interventions without important carryover effects. Miller’s Anesthesia, p. 12596.

Good implementation

• Generate the sequence using validated software, ideally by an independent statistician or data manager.
• Use central web or telephone randomization where possible.
• If physical envelopes are unavoidable, use sequentially numbered, opaque, sealed envelopes prepared independently, with safeguards against tampering.
• Keep recruiters unable to predict the next allocation.
• Record the randomization system, sequence generator, allocation ratio, blocks, strata, and who enrolled and assigned participants.

Avoid

• Allocation by date of birth, day of week, medical-record number, clinic day, or alternation.
• These are quasi-random methods and are predictable, so allocation concealment is difficult and selection bias becomes likely. Miller’s Anesthesia, p. 12596.

Example protocol wording

The trial will use a parallel-group, 1:1 randomized design. The primary endpoint is mean pain score at 24 hours. Based on a clinically important between-group difference of 1.5 points, a common SD of 3.0, two-sided α of 0.05, and 90% power, 85 evaluable participants are required per group. Allowing for 15% attrition, the target enrollment is 200 participants.

An independent statistician will generate a computer-based random allocation sequence using variable permuted blocks of four and six, stratified by study site. Allocation will be concealed through a secure central web-randomization system. Recruiting staff will not have access to the sequence.

Sample size determination and randomization

From Miller’s Anesthesia, 10th ed.

1. Sample size determination

Why sample size matters

• Too small: a true treatment effect may be missed, leading to a false-negative conclusion.
• Too large: wastes resources, delays useful clinical information, and exposes unnecessary participants to research-related risk.
• Therefore, sample-size calculation should be completed during the design phase.
  Miller’s Anesthesia, p. 12613-12614.

Basis of calculation in comparative studies

1. Primary outcome
   • The primary outcome is the central focus of the study and forms the basis for the sample-size calculation.
   • Its definition and measurement time point must be specified clearly.
2. Expected effect size
   • The expected difference in the primary outcome between intervention and control groups.
   • Ideally this represents a minimal clinically important difference, not merely a statistically detectable difference.
3. Variance
   • Required for continuous outcomes.
   • It describes the expected dispersion of the outcome measure.
4. Type I error (α)
   • False-positive finding: concluding that a treatment effect exists when it does not.
   • Conventionally set at 0.05.
5. Type II error (β)
   • False-negative finding: failing to detect a true effect.
   • Statistical power is (1-\beta), commonly 80% or 90%.
6. Number of groups and allocation ratio
   • More groups generally need more participants.
   • Unequal allocation may affect the total sample required.
7. Expected dropout
   • The calculated number should be inflated for anticipated withdrawal, loss to follow-up, missing data, or protocol non-adherence.
8. Planned statistical analyses
   • Analysis of subgroups, repeated outcomes, multiple comparisons, interim analyses, or special trial designs can alter the required sample size.

Sources of assumptions

• published literature,
• pilot data,
• statistical methods, or
• expert opinion regarding the minimum clinically important difference.

Pilot and feasibility studies

• A feasibility study evaluates whether a future larger study can be conducted, including recruitment, willingness to participate, intervention delivery, data collection, outcome suitability, and resources.
• A pilot study is a feasibility study that tests the proposed methods and hypotheses but is not intended to establish treatment efficacy.
• Pilot data may inform a later sample-size calculation, but this can be unreliable. The pilot study’s own sample should be justified by its feasibility objectives rather than necessarily requiring a formal efficacy-based calculation.
  Miller’s Anesthesia, p. 12615-12616.

2. Randomization

Definition

Purpose

• minimizes selection bias,
• balances known and unknown confounders between groups at baseline, particularly in sufficiently large studies,
• makes treatment allocation the main systematic difference between groups,
• thereby supports causal inference about the intervention’s effect.

Types of randomization

1. Simple randomization
   • Allocation by an unpredictable random sequence.
   • Suitable particularly for larger trials.
2. Block randomization
   • Maintains relatively balanced numbers in study groups during recruitment.
   • Especially useful in smaller trials.
3. Stratified randomization
   • Randomizes within important prognostic categories, such as study center or baseline disease severity.
   • Helps balance these specified factors between arms.
4. Covariate-adjusted randomization
   • Allocation takes important participant characteristics into account to reduce imbalance.

Allocation concealment

• centralized web-based randomization, or
• telephone-based randomization.

Allocation concealment vs blinding

Exam summary

Sample size is calculated before study initiation from the primary outcome, expected effect size, variance, α, β or power, number of groups, anticipated dropout, and planned analyses. An underpowered study risks false-negative findings; an excessively large study wastes resources and exposes extra participants to risk.

Randomization assigns participants to intervention or control by chance, minimizing selection bias and balancing confounders. Methods include simple, block, stratified, and covariate-adjusted randomization. Allocation concealment prevents prediction of the next assignment, whereas blinding hides allocated treatment after randomization.