Cluster Randomized Trial Design: A Worked Example

Introduction

In a conventional individually randomized clinical trial, each participant is randomized independently to treatment or control.

That structure is not always practical. Sometimes the intervention is delivered to an entire practice, hospital, school, community, clinic, nursing home, physician, or other naturally occurring group. Randomizing individuals within the same group may also create substantial risk of treatment contamination.

A cluster randomized trial addresses this problem by randomizing groups of participants rather than individual participants.

The group is called a cluster, and the individuals within that cluster are the participants.

What Is a Cluster?

A cluster is a naturally occurring group whose members are expected to be more similar to one another than participants selected from different groups.

Examples include:

• Patients treated within the same medical practice
• Patients receiving care from the same physician
• Patients treated at the same hospital
• Residents of the same nursing home
• Students attending the same school
• Children attending the same daycare center
• Communities within a public-health intervention
• Households in a community intervention

The important statistical feature is not the name of the group. It is that observations within the group tend to be correlated.

Individual Randomization vs. Cluster Randomization

Consider a trial involving 600 patients and 30 clinics.

Under individual randomization, all 600 patients could theoretically be randomized independently.

Under cluster randomization, the 30 clinics might be randomized, with 15 clinics assigned to treatment and 15 assigned to control.

Why Randomize Clusters?

There are several practical reasons to randomize clusters.

Intervention Is Delivered at the Cluster Level

Suppose an intervention changes the prescribing behavior of an entire clinic. It may be impossible to give the intervention to one physician's patients while withholding it from another patient of the same physician without affecting clinical practice.

Randomizing the clinic avoids this problem.

Preventing Contamination

Participants assigned to different treatment groups may interact with one another.

For example, if healthcare workers receive a training intervention, staff members assigned to different arms within the same facility may share the intervention.

Randomizing the facility can reduce this contamination.

Operational Convenience

Cluster randomization can also simplify implementation. Rather than training hundreds of individuals separately, a trial might train entire healthcare practices.

The Statistical Cost of Clustering

The practical advantages of cluster randomization come with a statistical cost.

Participants within the same cluster tend to resemble one another. For example, patients from the same clinic may have similar:

• Physicians
• Referral patterns
• Local demographics
• Access to healthcare
• Clinical practice patterns
• Socioeconomic characteristics

Consequently, their outcomes are not statistically independent.

The degree of within-cluster similarity is commonly represented by the intracluster correlation coefficient, or ICC.

The Intracluster Correlation Coefficient

The ICC is commonly denoted by:

For a continuous outcome under a simple random-intercept model, the ICC can be expressed as:

where:

• \(\sigma^2_{\text{cluster}}\) is the between-cluster variance.
• \(\sigma^2_{\text{individual}}\) is the within-cluster individual-level variance.

The ICC therefore describes the proportion of total outcome variance attributable to between-cluster differences under this variance-components formulation.

Interpreting the ICC

An ICC of zero means that, under the specified model, individuals within the same cluster are no more correlated than individuals from different clusters.

An ICC approaching one means that members of the same cluster are highly similar.

These values should not be interpreted as universal categories. Whether an ICC is "small" or "large" depends heavily on the outcome, population, cluster structure, and design.

Why a Small ICC Can Matter

Suppose a cluster contains:

participants and the ICC is only:

The effect of clustering is not determined by the ICC alone. It depends on both the ICC and the cluster size.

The standard approximation is the design effect.

The Design Effect

For equal cluster sizes, the classic design effect is:

where:

• \(m\) = number of participants per cluster
• \(\rho\) = ICC

The design effect quantifies the inflation in variance caused by clustering relative to an individually randomized design with the same number of participants.

Worked Design Effect Calculation

Suppose:

Then:

Thus, the variance is approximately 1.95 times the variance that would be obtained from 400 independent observations.

Equivalently, the effective sample size is approximately:

For 400 participants:

So 400 clustered observations provide information roughly comparable to 205 independently observed participants under this simple approximation.

The Most Important Sample-Size Formula

Suppose an individually randomized trial would require:

participants.

Under equal cluster sizes, the approximate clustered sample size is:

The quantity in brackets is the design effect.

A Complete Worked Example

Suppose investigators are evaluating a new intervention designed to reduce hospital readmission. The intervention is delivered at the hospital level, so hospitals rather than individual patients will be randomized.

Suppose the individually randomized calculation indicates that the trial would require 400 evaluable participants.

Assume:

We will calculate the required number of participants and clusters.

Step 1: Calculate the Design Effect

Using:

we obtain:

Step 2: Inflate the Individual-Level Sample Size

The individually randomized requirement is:

Therefore:

Thus, approximately 780 evaluable participants are required after accounting for clustering.

Step 3: Account for Dropout

Suppose 10% of enrolled participants are expected to be unevaluable. The inflation factor for 10% dropout is:

Therefore:

We therefore need approximately 867 enrolled participants before accounting for the practical requirement that enrollment occur in whole clusters.

Step 4: Convert Participants to Clusters

If the expected cluster size is 20:

Because we cannot randomize a fraction of a hospital, we must round upward:

With 44 clusters and 20 participants per cluster, the planned enrollment is:

participants.

With equal allocation, that corresponds to:

The Complete Calculation at a Glance

Effective Sample Size

The effective sample size is a useful way to communicate the information loss associated with clustering.

For equal cluster sizes:

For 880 participants with 20 participants per cluster and an ICC of 0.05:

The effective sample size is therefore approximately 451 independent observations under the simple design-effect approximation.

This does not mean that only 451 patients are actually enrolled. It means that the information contributed by the 880 correlated observations is roughly comparable to that of 451 independent observations for the variance calculation under the stated assumptions.

The Number of Clusters Matters More Than It First Appears

A common mistake is to focus almost entirely on the total number of participants.

Suppose a study has:

participants. That sounds large.

But 1,000 participants distributed across 10 clusters means approximately 100 participants per cluster. The randomization still involves only 10 independent units.

By contrast, 1,000 participants distributed across 100 clusters means only about 10 participants per cluster.

Those two designs can have very different statistical properties.

More Participants per Cluster Is Not Always the Best Solution

The design effect is:

As cluster size increases, the design effect increases whenever \(\rho>0\).

This means that adding more participants to existing clusters becomes progressively less efficient than adding new clusters.

For example, with:

the design effect for different cluster sizes is:

This illustrates why cluster size is a fundamental design parameter.

The Square-Root Relationship

For many basic power calculations, the standard error of an estimator is approximately proportional to:

Therefore, a design effect of 1.95 does not mean that the standard error is 1.95 times as large. Instead:

so the standard error is approximately 40% larger under this simple approximation.

Because power depends on the standard error, the sample-size inflation is typically expressed using the full design effect rather than its square root.

Unequal Cluster Sizes

The formula:

Real studies rarely have perfectly equal clusters. A hospital may contribute 40 patients while another contributes 12. A physician may recruit 25 patients while another recruits 8.

Unequal cluster sizes reduce efficiency beyond what would be predicted by using a simple average cluster size.

The Coefficient of Variation of Cluster Size

Let the coefficient of variation of cluster size be:

A commonly used approximation to account for unequal cluster sizes is:

This is often called a coefficient-of-variation adjustment.

Worked Example With Unequal Cluster Sizes

Suppose:

Then:

Since:

we obtain:

The equal-size calculation gave a design effect of 1.95, whereas the coefficient-of-variation adjustment gives approximately 2.20.

That difference can materially affect the required number of participants and clusters.

ICC Assumptions Are Critical

The required sample size is highly sensitive to the assumed ICC.

Suppose:

Then:

A planning ICC that is too small can therefore result in an underpowered cluster randomized trial.

Where Does the ICC Come From?

The ICC should ideally be based on relevant empirical information. Potential sources include:

• Previous studies using the same outcome
• Pilot or feasibility studies
• Historical data from the same healthcare system
• Published estimates from similar populations
• Variance-component estimates from observational datasets

The population, outcome definition, cluster type, and cluster size should be as comparable as possible.

ICC for Binary Outcomes

Cluster randomized trials frequently use binary endpoints such as:

• Response vs. no response
• Hospitalization vs. no hospitalization
• Event vs. no event
• Success vs. failure

For binary outcomes, the interpretation and estimation of the ICC require additional care because the variance depends on the underlying event probability.

A simple design-effect approximation is often used for planning, but more specialized methods may be appropriate for generalized linear mixed models, binary outcomes, varying cluster sizes, and small numbers of clusters.

Randomization Ratio

Suppose clusters are randomized in a 1:1 ratio.

If the trial requires:

clusters, the allocation is:

where \(K_T\) and \(K_C\) are the numbers of treatment and control clusters.

Unequal allocation can be used when scientifically or operationally justified, but it generally changes the efficiency of the design.

Why Cluster-Level Balance Matters

Suppose a trial has 20 treatment clusters and only 6 control clusters. Even if the total number of participants is similar between groups, the small number of control clusters can produce unstable estimates and reduced power.

The effective amount of information is influenced by the number of randomized clusters in each arm, not merely the number of participants.

Degrees of Freedom

In a cluster randomized trial, the number of independent treatment assignments is determined by the number of randomized clusters.

This is especially important when the number of clusters is small.

A trial with:

clusters does not have the same inferential properties as a trial with 120 clusters, even if each cluster contains many participants.

Analysis of Cluster Randomized Trials

The analysis must account for the correlation induced by clustering. Common approaches include:

• Cluster-level analysis
• Mixed-effects models
• Generalized estimating equations (GEE)
• Randomization-based or permutation methods
• Specialized small-sample methods

The appropriate method depends on the endpoint, number of clusters, allocation scheme, covariates, cluster sizes, and estimand.

Cluster-Level Analysis

A simple approach is to calculate a summary measure for each cluster and then compare those cluster-level summaries between randomized arms.

For a binary endpoint, one possible summary is the proportion of participants with the event in each cluster.

If cluster \(j\) contains \(m_j\) participants and \(Y_{ij}\) is a binary outcome, the cluster event rate is:

The analysis then operates on the cluster summaries rather than pretending that all individual observations are independent.

Advantages and Limitations of Cluster-Level Analysis

Mixed-Effects Models

A mixed-effects model explicitly represents between-cluster heterogeneity. For a continuous outcome, a simple random-intercept model is:

where:

• \(Y_{ij}\) is the outcome for participant \(i\) in cluster \(j\).
• \(T_j\) indicates the randomized treatment for cluster \(j\).
• \(u_j\) is a cluster-specific random effect.
• \(\epsilon_{ij}\) is the individual-level residual.

The cluster random effect induces correlation among participants within the same cluster.

Binary Outcomes and Generalized Linear Mixed Models

For a binary endpoint, a generalized linear mixed model may use a logit link:

The treatment effect may then be represented on the odds-ratio scale through the treatment coefficient.

However, interpretation of cluster-specific versus population-averaged effects requires care.

Generalized Estimating Equations

GEE provides a population-averaged framework for correlated observations.

A simplified model for a binary outcome might be:

with a working correlation structure used to account for within-cluster dependence.

Robust variance estimation can provide valid inference under appropriate conditions, although small numbers of clusters require particular care.

Small Numbers of Clusters

One of the most important design considerations is the number of clusters.

When the number of clusters is small, standard asymptotic methods may produce inaccurate standard errors and type I error rates.

Potential approaches include:

• Small-sample corrections for mixed models
• Finite-sample corrections for GEE
• Cluster-level analyses
• Randomization-based inference
• Permutation procedures that respect the actual randomization scheme

Covariate Adjustment

Cluster randomized trials can often gain efficiency through prespecified covariate adjustment. Potential covariates include:

• Baseline cluster characteristics
• Baseline participant characteristics
• Baseline outcome measurements
• Geographic region
• Cluster size

However, the covariate strategy should be prespecified and aligned with the estimand and analysis model.

Stratified Cluster Randomization

Suppose hospitals differ substantially by region. Investigators may stratify the randomization by region so that each treatment arm contains a balanced number of hospitals from each region.

The randomization might therefore occur within strata:

Matched-Pair Cluster Randomization

Another strategy is to form matched pairs of similar clusters and randomize one cluster within each pair to treatment and the other to control.

For example:

Matching can improve balance when the number of clusters is limited, but it also introduces additional considerations for the randomization and analysis.

Cluster Size Variability

Cluster size variability is not merely a logistical issue. It affects statistical efficiency.

Suppose the planned average cluster size is 20, but actual cluster sizes are:

The average alone does not fully describe the information structure.

Large clusters can contribute many observations but also amplify the effect of within-cluster correlation.

Consequently, investigators should consider the expected distribution of cluster sizes during planning.

Variable Cluster Size in Practice

A practical sample-size workflow may therefore include:

1. Estimate the expected mean cluster size.
2. Estimate the variability of cluster size.
3. Calculate the coefficient of variation.
4. Inflate the design effect for unequal cluster sizes.
5. Round the required number of clusters upward.
6. Assess sensitivity to alternative cluster-size distributions.

Dropout Is Different From Cluster Attrition

There are two different forms of loss that should not be conflated.

Individual-Level Loss

A participant may withdraw or become unevaluable.

Cluster-Level Loss

An entire clinic or hospital may withdraw from the study.

Cluster attrition can be particularly damaging because losing one cluster means losing the independent randomization unit as well as all participants within that cluster.

Inflating for Cluster Attrition

Suppose a design requires 44 clusters and investigators anticipate a 10% cluster-level attrition rate. A simple planning calculation is:

Thus, approximately:

clusters would be required under this simple inflation approach, followed by an assessment of whether the resulting allocation remains balanced.

More sophisticated planning may be preferable when attrition is expected to be differential or correlated with cluster characteristics.

Contamination and Cluster Randomization

Cluster randomization is often selected specifically because contamination between treatment arms is a concern.

For example, if all physicians within a practice share a clinical decision support tool, assigning some patients within that practice to treatment and others to control may be difficult.

Randomizing the practice can preserve separation between the intervention and control conditions.

The Tradeoff: Contamination vs. Efficiency

Cluster randomization can reduce contamination but increases the sample size required because of clustering.

The design decision therefore involves a fundamental tradeoff:

Individually Randomized Group Treatment Trials

Not every study involving groups is automatically a cluster randomized trial.

For example, a trial may randomize individuals but deliver some aspects of treatment in a group setting.

The key question is: what unit was actually randomized?

If individual participants are independently randomized, the design is not necessarily a cluster randomized trial merely because participants interact within groups.

Parallel Cluster Randomized Trials

The simplest cluster randomized design is a parallel-group design.

Clusters are randomized once and remain in their assigned treatment group.

Cluster Randomized Crossover Designs

Some trials randomize clusters to sequences rather than permanently assigning them to one treatment.

For example:

Such designs introduce additional considerations, including period effects, carryover, participant turnover, and cluster-period correlation.

The simple design-effect formula should not automatically be applied to a cluster crossover trial.

Cluster-Period Designs

Some interventions are randomized at the cluster-period level, particularly when the intervention is implemented sequentially.

The statistical structure may then involve:

• Cluster effects
• Period effects
• Cluster-by-period variability
• Repeated observations within clusters

These designs generally require more specialized sample-size methods.

Sample Size for a Continuous Outcome

Suppose the individually randomized trial would require:

for a two-arm comparison with equal allocation and a continuous endpoint.

After accounting for clustering:

under the equal-cluster-size approximation.

Sample Size for a Binary Outcome

For a two-arm comparison of independent proportions, an individually randomized sample-size calculation can first be performed using the expected control and treatment event probabilities.

A generic normal-approximation expression is:

where:

The resulting individual-level sample size can then be inflated for clustering as an initial approximation.

Loss to Follow-Up and Design Effect

Loss to follow-up can alter the effective cluster size. Suppose the planned cluster size is:

but 10% of participants are expected to be lost. The expected evaluable cluster size becomes approximately:

The relationship between recruitment cluster size, evaluable cluster size, and the design effect should therefore be considered explicitly rather than inflating blindly at the end.

Recruitment Feasibility

A statistically sufficient design may still be operationally impossible. Suppose the calculation requires:

clinics, each recruiting 20 patients. If the trial network contains only 35 eligible clinics, the mathematical sample-size calculation does not solve the feasibility problem.

Investigators may need to reconsider:

• Effect size assumptions
• Cluster size
• Number of participating sites
• Allocation ratio
• Follow-up duration
• Eligibility criteria
• Study population

The Number of Clusters Can Be the Binding Constraint

This is one of the most important practical differences between cluster and individual randomization.

In an individually randomized trial, recruiting additional participants generally increases the number of independent observations.

In a cluster randomized trial, recruiting additional participants into an already-randomized cluster does not create another independent randomized unit.

Therefore:

Design Effect vs. Effective Sample Size

R Implementation: Basic Design Effect

The basic equal-cluster-size calculation is straightforward in R.

m <- 20
icc <- 0.05

design_effect <- 1 + (m - 1) * icc

design_effect
# 1.95

R Implementation: Cluster-Adjusted Sample Size

n_individual <- 400

n_clustered <- n_individual * design_effect

n_clustered
# 780

R Implementation: Dropout Adjustment

dropout <- 0.10

n_enroll <- n_clustered / (1 - dropout)

n_enroll
# approximately 866.67

R Implementation: Number of Clusters

cluster_size <- 20

n_clusters <- ceiling(
  n_enroll / cluster_size
)

n_clusters
# 44

R Implementation: Unequal Cluster Sizes

Suppose the mean cluster size and coefficient of variation are:

m <- 20
icc <- 0.05
cv <- 0.50

design_effect_unequal <-
  1 + (m * (1 + cv^2) - 1) * icc

design_effect_unequal
# approximately 2.20

The resulting cluster-adjusted sample size is:

n_clustered_unequal <-
  n_individual * design_effect_unequal

n_clustered_unequal
# approximately 880

Sensitivity Analysis for the ICC

Because the ICC is often uncertain, it is good practice to evaluate several plausible values.

m <- 20
icc_values <- c(0.01, 0.03, 0.05, 0.10, 0.20)

data.frame(
  ICC = icc_values,
  DesignEffect =
    1 + (m - 1) * icc_values
)

This produces:

A sensitivity analysis makes the consequences of ICC uncertainty transparent to the clinical and operational teams.

Sensitivity Analysis for Cluster Size

The same approach can be used to examine cluster size.

icc <- 0.05
cluster_sizes <- c(5, 10, 20, 30, 50)

data.frame(
  ClusterSize = cluster_sizes,
  DesignEffect =
    1 + (cluster_sizes - 1) * icc
)

The design effect increases as the cluster size increases.

A More General Planning Function

A reusable R function can combine the major planning components.

cluster_sample_size <- function(
  n_individual,
  mean_cluster_size,
  icc,
  dropout = 0,
  cv = 0
) {

  de <-
    1 +
    (
      mean_cluster_size *
      (1 + cv^2) - 1
    ) * icc

  n_evaluable <-
    n_individual * de

  n_recruit <-
    n_evaluable / (1 - dropout)

  clusters <-
    ceiling(
      n_recruit / mean_cluster_size
    )

  list(
    design_effect = de,
    evaluable_n = n_evaluable,
    recruitment_n = n_recruit,
    clusters = clusters
  )
}

cluster_sample_size(
  n_individual = 400,
  mean_cluster_size = 20,
  icc = 0.05,
  dropout = 0.10,
  cv = 0
)

Interpreting the Function

For the worked example, the function produces approximately:

Because the final number of clusters must be an integer, the final enrollment should be based on the actual planned cluster structure rather than simply rounding the participant count.

Why the Simple Formula Is an Approximation

The classic design-effect formula is extremely useful, but it is not a universal solution to every cluster trial sample-size problem.

It is most naturally interpreted as an approximation under a relatively simple correlation structure and cluster-size assumption.

More complex designs may involve:

• Unequal allocation
• Unequal cluster sizes
• Binary outcomes
• Count outcomes
• Time-to-event outcomes
• Repeated measurements
• Cluster-period correlation
• Variable follow-up
• Covariate adjustment
• Small numbers of clusters
• Cluster-level attrition

In these settings, simulation or specialized analytical methods may be more appropriate.

Simulation-Based Sample Size Planning

Simulation can be particularly useful when the intended analysis is more complicated than the simple design-effect approximation.

A simulation-based approach can:

1. Specify the cluster structure.
2. Generate cluster-level random effects.
3. Generate individual outcomes conditional on those effects.
4. Randomize clusters according to the planned design.
5. Fit the intended analysis model.
6. Record whether the treatment effect is statistically detected.
7. Repeat the process many times.

The proportion of simulated trials achieving the prespecified decision criterion estimates power.

Simulation Workflow

Simulation Can Reveal Unexpected Problems

Simulation can demonstrate how performance changes when:

• Cluster sizes are highly unequal.
• Cluster dropout occurs.
• The ICC differs from the planning value.
• The treatment effect differs from the assumed value.
• Baseline covariates are strongly prognostic.
• The number of clusters is small.
• The analysis model uses small-sample corrections.

This can make simulation an important complement to closed-form sample-size calculations.

Common Mistakes

1. Using an individual-level sample-size formula without clustering inflation. This can substantially understate the required sample size.
2. Counting participants instead of clusters. The number of randomized clusters is fundamental to the design.
3. Assuming ICC = 0. Even modest clustering can substantially increase the required sample size.
4. Ignoring unequal cluster sizes. Cluster-size variability can reduce efficiency.
5. Inflating only for individual dropout. Cluster-level attrition requires separate consideration.
6. Adding participants instead of clusters when power is inadequate. Once clusters are large, additional participants within existing clusters may provide relatively little independent information.
7. Using an analysis that treats participants as independent. The analysis must account for within-cluster correlation.
8. Ignoring the number of clusters when selecting the analysis method. Small-cluster trials require particular attention to finite-sample inference.
9. Assuming the ICC is a fixed universal property. ICC depends on the outcome and population and should be justified for the specific trial.
10. Failing to coordinate design and analysis. The sample-size method should reflect the intended estimand and primary analysis.

A Practical Cluster Randomized Trial Workflow

What Should Be Reported in a Statistical Analysis Plan?

A cluster randomized trial SAP should make the cluster structure and its statistical implications explicit.

At minimum, document:

• Primary estimand
• Primary endpoint
• Unit of randomization
• Unit of observation
• Number of clusters
• Allocation ratio
• Expected cluster size
• Expected cluster-size variability
• ICC assumption
• Individual-level dropout assumption
• Cluster-level attrition assumption
• Sample-size calculation method
• Primary analysis model
• Covariate adjustment
• Handling of missing outcomes
• Small-sample inference methods, where applicable

Worked Example Summary

The complete worked example can be summarized as follows.

The Most Important Concept

The most important conceptual point is that a cluster randomized trial has two levels of sampling information.

There are individual participants, who provide the observed outcomes.

But there are also clusters, which provide the independent randomization units.

Participants within the same cluster are correlated, so adding participants to an existing cluster does not provide the same information as adding entirely new clusters.

The basic design-effect relationship makes this explicit:

and therefore:

But a rigorous cluster randomized trial design goes beyond this one equation. The investigator must also consider cluster-size variability, ICC uncertainty, cluster attrition, number of clusters, randomization strategy, analysis model, and the intended estimand.

References

Eldridge, S.M., Kerry, S. & Bland, J.M. (2006). Are cluster randomized trials feasible? Journal of Health Services Research & Policy, 11(2), 80–83.
Kerry, S.M. & Bland, J.M. (2001). Randomization in clinical trials. BMJ, 322, 213.
Kerry, S.M. & Bland, J.M. (2001). Statistics notes: Sample size in cluster randomisation. BMJ, 323, 1419–1421.
Kerry, S.M. & Bland, J.M. (2004). The intracluster correlation coefficient in cluster randomised trials: a review of definitions. International Statistical Review, 72(3), 381–394.
Kish, L. (1965). Survey Sampling. John Wiley & Sons.
Eldridge, S.M., Ukoumunne, O.C. & Carlin, J.B. (2009). The intra-cluster correlation coefficient in cluster randomized trials: a review of definitions. International Statistical Review, 77(3), 378–394.
Donner, A. & Klar, N. (2000). Design and Analysis of Cluster Randomization Trials in Health Research. Arnold.
Hayes, R.J. & Moulton, L.H. (2017). Cluster Randomised Trials, 2nd ed. CRC Press.
Teerenstra, S., Eldridge, S., Graffy, J., van Veen, F. & Weel, C. van. (2010). A simple sample size formula for analysis of covariance in cluster randomized trials. Statistics in Medicine.