Sample Size for Cluster Randomized Trials

Introduction

In a conventional randomized controlled trial, individual participants are randomized independently to treatment groups. In a cluster randomized trial (CRT), the unit of randomization is instead a group, or cluster.

Examples of clusters include hospitals, clinics, physician practices, schools, communities, villages, nursing homes, or geographic regions.

Every participant within a cluster receives the intervention assigned to that cluster. Consequently, participants within the same cluster tend to be more similar to one another than participants from different clusters.

Key idea: The primary sample-size consequence of clustering is that observations are not statistically independent. The resulting loss of information is usually quantified using the intracluster correlation coefficient (ICC) and the design effect.

Why Clustering Changes the Sample Size

Suppose two patients attend the same clinic. They may have similar physicians, clinical practices, socioeconomic characteristics, referral patterns, or baseline risk.

As a result, knowing the outcome of one patient gives some information about the likely outcome of another patient in the same clinic.

This means that 100 patients distributed across 100 independent clinics generally provide more statistical information than 100 patients concentrated in only 5 clinics.

The number of participants alone therefore does not determine the effective amount of information in a cluster randomized trial.

Important distinction: In a CRT, both the number of participants and the number of clusters matter. Adding participants to existing clusters does not provide the same information as adding independent clusters.

The Intracluster Correlation Coefficient

The intracluster correlation coefficient, usually abbreviated as $$ICC$$ or $\rho$ , measures the degree of similarity among participants within the same cluster.

Conceptually, it can be written as the proportion of total outcome variance that occurs between clusters:

ICC = \rho = \frac{\sigma^2_{\text{between}}} {\sigma^2_{\text{between}}+\sigma^2_{\text{within}}}

An ICC of zero means that participants within the same cluster are no more similar than participants from different clusters, whereas larger ICC values indicate greater within-cluster similarity.

ICC	Interpretation
$0$	No additional correlation within clusters
0.01	Small within-cluster correlation
0.05	Moderate clustering in many practical settings
0.10	Substantial within-cluster similarity
1.00	Participants within each cluster are perfectly correlated

The ICC is often small. However, even a seemingly small ICC can have a substantial effect when clusters contain many participants.

The Design Effect

For equal-sized clusters, the most commonly used design-effect formula is:

DE = 1+(m-1)\rho

where:

$$DE$$ = design effect
$$m$$ = average number of participants per cluster
$\rho$ = intracluster correlation coefficient

The design effect quantifies how much the variance is inflated because of clustering relative to an individually randomized design.

Interpretation: If the design effect is 2, the clustered trial requires approximately twice as many participants as the corresponding individually randomized trial, before accounting for other factors such as attrition.

Why Cluster Size Matters

The design effect depends directly on the average cluster size.

For example, if the ICC is 0.05 and the average cluster contains 20 participants:

$$ DE = 1+(20-1)(0.05) = 1+0.95 = 1.95 $$

But if the average cluster contains 50 participants:

$$ DE = 1+(50-1)(0.05) = 1+2.45 = 3.45 $$

The same ICC therefore produces very different sample-size inflation depending on cluster size.

Average Cluster Size	ICC	Design Effect
10	0.05	1.45
20	0.05	1.95
30	0.05	2.45
50	0.05	3.45
100	0.05	5.95

Practical lesson: When the ICC is greater than zero, larger clusters are statistically less efficient than simply adding more independent clusters. This is one reason CRT sample-size planning focuses heavily on the number of clusters.

Basic Sample Size Strategy

A convenient way to understand the calculation is to begin with the sample size that would be required if individuals were randomized independently.

Call this:

N_{\text{individual}}

The clustered design then inflates this number by the design effect:

N_{\text{clustered}} = N_{\text{individual}}\times DE

Substituting the design-effect formula gives:

N_{\text{clustered}} = N_{\text{individual}} \left[1+(m-1)\rho\right]

This gives the approximate number of analyzable participants needed before accounting for dropout or other enrollment losses.

Accounting for Attrition

Suppose the required analyzable sample size is $N_{\text{clustered}}$ and the anticipated proportion of participants lost to follow-up is $$d$$ .

The enrollment target can be inflated as:

N_{\text{enroll}} = \frac{N_{\text{clustered}}}{1-d}

For example, with 15% attrition:

N_{\text{enroll}} = \frac{N_{\text{clustered}}}{0.85}

Do not multiply by $1+d$: For a 15% expected loss, dividing by 0.85 is the appropriate simple retention-based inflation. Multiplying by 1.15 produces a slightly smaller target and does not exactly compensate for 15% loss.

From Participants to Clusters

A CRT ultimately randomizes clusters rather than individuals. Therefore, a participant-level calculation must eventually be converted into a number of clusters.

If the target total enrollment is $N_{\text{enroll}}$ and the planned cluster size is $$m$$ , the approximate number of clusters is:

K = \frac{N_{\text{enroll}}}{m}

Because clusters cannot generally be divided into fractions, the result must be rounded up.

For a two-arm trial with equal allocation:

K_{\text{per arm}} = \left\lceil \frac{K}{2} \right\rceil

The final number of clusters should therefore be an even number when equal allocation is required.

Worked Example: Continuous Outcome

Consider a two-arm cluster randomized trial comparing a new intervention with usual care.

Suppose the primary endpoint is a continuous outcome, and an individually randomized trial would require 120 participants per treatment arm to achieve the desired power and significance level.

The investigators expect:

Parameter	Planning Value
Individual-level sample size per arm	120
Average cluster size	20 participants
ICC	0.05
Expected attrition	10%
Allocation	1:1

The calculation proceeds in several steps.

Step 1: Calculate the Design Effect

The design effect is:

DE = 1+(m-1)\rho

Substitute:

$$ DE = 1+(20-1)(0.05) $$

Therefore:

$$ DE = 1+(19)(0.05) = 1+0.95 = 1.95 $$

Interpretation: The clustering increases the required variance by a factor of 1.95. Consequently, approximately 1.95 times the individual-randomization sample size is required before accounting for attrition.

Step 2: Inflate the Individual-Level Sample Size

The individually randomized design requires 120 participants per arm.

After accounting for clustering:

N_{\text{clustered, arm}} = 120(1.95)

Thus:

N_{\text{clustered, arm}} = 234

So the trial requires approximately 234 analyzable participants per arm before attrition.

Step 3: Inflate for Attrition

The anticipated attrition rate is 10%, so the expected retention rate is:

$$ 1-0.10=0.90 $$

Therefore:

N_{\text{enroll, arm}} = \frac{234}{0.90} = 260

The target is therefore 260 participants per treatment arm.

At this point: The calculation has produced a participant-level enrollment target, but the trial cannot simply enroll 260 individuals per arm. Participants must be enrolled in complete clusters.

Step 4: Convert Participants to Clusters

The planned cluster size is 20 participants.

Therefore:

K_{\text{per arm}} = \frac{260}{20} = 13

Thus, the design requires:

K_{\text{total}} = 13+13 = 26

The proposed design therefore has 26 clusters total, with 13 clusters randomized to each treatment arm.

Calculation Stage	Result
Individual-level sample size	120 per arm
Design effect	1.95
After clustering	234 per arm
After 10% attrition inflation	260 per arm
Average cluster size	20
Clusters per arm	13
Total clusters	26
Nominal total enrollment	520 participants

Step 5: Verify the Final Cluster-Based Design

The resulting design enrolls:

26\times20=520

participants in total, or 260 per arm.

After 10% attrition, the expected number of evaluable participants is:

$$ 520(0.90)=468 $$

or approximately 234 evaluable participants per arm.

This matches the target produced by the design-effect calculation.

Final design: 26 clusters total, randomized 1:1, with approximately 20 participants per cluster and a target enrollment of 520 participants.

The Complete Calculation in One Equation

For this simplified equal-cluster design, the entire participant-level calculation can be written as:

N_{\text{enroll}} = \frac{ N_{\text{individual}} \left[1+(m-1)\rho\right] }{ 1-d }

where:

$N_{\text{individual}}$ is the sample size that would be required under individual randomization.
$$m$$ is the average cluster size.
$\rho$ is the ICC.
$$d$$ is the expected attrition proportion.

The resulting enrollment target is then converted to a whole number of clusters.

What Happens When the ICC Changes?

The ICC is one of the most influential assumptions in CRT sample-size planning.

Using the same average cluster size of 20, consider several possible ICC values.

ICC	Design Effect	Relative Sample Size
0.00	1.00	1.00×
0.01	1.19	1.19×
0.03	1.57	1.57×
0.05	1.95	1.95×
0.10	2.90	2.90×
0.20	4.80	4.80×

The increase can be substantial even when the ICC itself appears numerically small.

Planning implication: A poorly justified ICC assumption can materially affect the number of clusters required. ICC selection should therefore be supported by prior studies, pilot data, registry data, historical trial data, or other relevant evidence whenever possible.

What Happens When Cluster Size Changes?

Now hold the ICC at 0.05 and vary the cluster size.

Cluster Size $m$	Design Effect
5	1.20
10	1.45
20	1.95
30	2.45
50	3.45
100	5.95

This demonstrates why adding more participants to the same clusters has diminishing efficiency when the ICC is positive.

Equal Versus Unequal Cluster Sizes

The simple design-effect formula:

DE=1+(m-1)\rho

assumes approximately equal cluster sizes.

Real clinical trials often have unequal cluster sizes. One clinic might enroll 10 participants while another enrolls 40.

Unequal cluster sizes can reduce statistical efficiency beyond what would be predicted using only the nominal mean cluster size.

Important: When substantial variation in cluster size is expected, a more refined sample-size method should be used rather than relying blindly on the simple equal-cluster formula.

Coefficient of Variation for Cluster Size

One way to characterize cluster-size variability is the coefficient of variation:

CV = \frac{SD(m)}{\bar m}

where $\bar m$ is the mean cluster size and $$SD(m)$$ is its standard deviation.

A larger coefficient of variation indicates greater imbalance in cluster sizes and generally implies some additional loss of efficiency.

In practice, specialized CRT sample-size methods can incorporate this variation directly.

Number of Clusters Versus Number of Participants

A particularly important principle in CRT design is that the number of clusters is often more important than the number of individual participants.

Suppose a study has 20 clusters containing 25 participants each:

20\times25=500

participants.

An alternative design might have 40 clusters containing 12.5 participants each, producing approximately the same number of participants.

40\times12.5=500

With a positive ICC, these designs are not statistically equivalent. The second design generally provides more independent cluster-level information.

Rule of thumb: When choosing between larger clusters and more clusters, increasing the number of independent clusters is often statistically preferable, assuming the intervention, recruitment logistics, cost, and analysis plan permit it.

Cluster-Level Degrees of Freedom

Cluster randomized trials are fundamentally limited by the number of independent randomized units.

For example, a trial with 10 clusters per arm has only 20 randomized clusters, even if each cluster contains many participants.

This matters for hypothesis testing, confidence intervals, small-sample corrections, and the robustness of the statistical analysis.

Consequently, a sample-size calculation should not focus exclusively on achieving a desired number of individual participants.

Individual-Level Versus Cluster-Level Randomization

Feature	Individual RCT	Cluster RCT
Randomization unit	Individual	Cluster
Within-group correlation	Usually not induced by randomization unit	Present within clusters
ICC required for basic planning	No	Yes
Design effect	Typically 1	Usually greater than 1
Primary independent units	Participants	Clusters
Sample-size challenge	Participant count	Participants + clusters

When Is Cluster Randomization Useful?

Cluster randomization is often selected for practical or scientific reasons.

The intervention is delivered at the clinic or hospital level.
Individual randomization would create substantial contamination between treatment groups.
Healthcare providers cannot realistically deliver two different treatment policies simultaneously.
The intervention changes organizational practice rather than individual treatment.
Schools, communities, or practices are the natural units of intervention.

The statistical cost of clustering must then be incorporated into the sample size.

A General CRT Sample-Size Workflow

A practical planning process can be summarized as follows:

Specify the primary endpoint. Determine whether the outcome is continuous, binary, count-based, time-to-event, or another endpoint requiring a specialized CRT method.

Calculate the individual-randomization sample size. Determine the sample size that would be needed if participants were randomized independently.

Specify the ICC. Use appropriate empirical or historical evidence whenever possible.

Specify cluster size. Estimate the expected number of participants per cluster.

Calculate the design effect. For equal cluster sizes, use $DE=1+(m-1)\rho$.

Inflate for clustering. Multiply the individual-level sample size by the design effect.

Inflate for attrition. Account for expected participant loss.

Convert participants to clusters. Round upward to an appropriate whole number of clusters and preserve the planned randomization ratio.

Common Sample-Size Mistakes

1. Ignoring the ICC

Treating all participants as statistically independent will generally underestimate the required sample size when clustering is present.

2. Using the Number of Participants as the Only Design Criterion

A CRT with many participants but very few clusters may have limited statistical information because the number of independent randomized units is small.

3. Choosing an ICC Because It Produces a Convenient Sample Size

The ICC should be scientifically justified. It should not be selected simply because it produces a desirable number of clusters.

4. Forgetting Attrition

Loss of participants reduces the effective cluster size and can compromise power.

5. Rounding the Number of Clusters Down

A calculation that produces 12.2 clusters per arm does not justify 12 clusters. The design should generally round upward.

6. Assuming the Simple Design Effect Always Applies

The formula

DE=1+(m-1)\rho

is a useful starting point, but more complex CRT designs may require specialized calculations.

Binary Outcomes Require Additional Care

For a binary endpoint, the same general concept applies: clustering reduces the effective information and the required sample size must account for the ICC.

However, binary-outcome CRT calculations can require additional considerations depending on the analysis method, event rate, cluster size, allocation ratio, and number of clusters.

For example, methods based on marginal models, generalized estimating equations, mixed-effects logistic regression, or cluster-level analyses can lead to somewhat different sample-size calculations.

Practical recommendation: For a definitive clinical trial with a binary, count, or time-to-event endpoint, use a sample-size method that matches the planned primary analysis rather than applying the simple continuous-outcome design effect without checking its assumptions.

Unequal Allocation

If clusters are randomized in an unequal ratio, the number of clusters per arm must reflect the planned allocation.

For example, if clusters are allocated 2:1 between intervention and control, the total number of clusters cannot simply be divided equally.

Unequal allocation can also affect statistical efficiency and should be incorporated directly into the sample-size calculation.

Individual Attrition Versus Cluster Attrition

Participant dropout and cluster dropout are different problems.

Individual attrition occurs when participants leave the study or become unavailable for the primary analysis.

Cluster attrition occurs when an entire randomized cluster withdraws, closes, fails to recruit, or otherwise becomes unavailable.

Why cluster dropout is serious: Losing one participant reduces the sample size slightly. Losing an entire cluster removes an independent randomized unit and can have a much larger impact on power and balance.

When cluster dropout is plausible, the design may need an additional inflation or contingency strategy beyond ordinary participant-level attrition.

Design Effect Summary

Quantity	Effect on Sample Size
Higher ICC	Increases design effect
Larger cluster size	Increases design effect when ICC > 0
More clusters	Generally increases independent information
Higher participant attrition	Increases enrollment target
Cluster dropout	Can substantially reduce power
Unequal cluster sizes	Can reduce efficiency

Worked Example: Compact Calculation

The worked example can be condensed into the following calculation chain:

N_{\text{individual, arm}}=120

$$ DE=1+(20-1)(0.05)=1.95 $$

N_{\text{clustered, arm}} = 120(1.95) = 234

N_{\text{enroll, arm}} = \frac{234}{0.90} = 260

K_{\text{per arm}} = \frac{260}{20} = 13

K_{\text{total}}=26

Thus the final design has 26 clusters and 520 planned participants.

Implementation Checklist

Before finalizing a CRT sample-size calculation, verify all of the following:

The randomization unit is clearly defined.
The primary endpoint and analysis model are specified.
The individual-randomization sample size is appropriate.
The ICC is supported by relevant evidence.
The anticipated cluster size is realistic.
Potential variation in cluster size has been considered.
The treatment allocation ratio is incorporated.
Participant attrition has been incorporated.
Potential cluster dropout has been considered.
The final number of clusters has been rounded upward.
The final cluster count provides adequate balance between treatment arms.
The resulting number of clusters is adequate for the planned statistical analysis.

Key Takeaways

Clustering reduces statistical efficiency. Participants within the same cluster are correlated rather than independent.

The ICC quantifies within-cluster similarity. Even a relatively small ICC can substantially increase the required sample size when clusters are large.

The design effect translates clustering into sample-size inflation. For equal cluster sizes, $DE=1+(m-1)\rho$.

More participants per cluster are not equivalent to more clusters. With a positive ICC, independent clusters provide particularly valuable information.

Attrition must be handled separately. Participant loss and cluster loss are distinct threats to power.

The final answer is a number of clusters. Participant-level calculations must ultimately be translated into whole randomization units.

References

Eldridge, S.M., Kerry, S. & Bland, J.M. (2006). Design and analysis of randomised trials in health care. BMJ, 332, 1437–1440.
Kerry, S.M. & Bland, J.M. (2001). Sample size in cluster randomisation. BMJ, 323, 673–674.
Kerry, S.M. & Bland, J.M. (2004). The intracluster correlation coefficient in cluster randomisation. BMJ, 328, 104–105.
Eldridge, S.M. & Kerry, S. (2012). A Practical Guide to Cluster Randomised Trials in Health Services Research. Wiley.
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.

ICC	Interpretation
\(0\)	No additional correlation within clusters
0.01	Small within-cluster correlation
0.05	Moderate clustering in many practical settings
0.10	Substantial within-cluster similarity
1.00	Participants within each cluster are perfectly correlated

Sample Size for Cluster Randomized Trials

What You'll Learn

Introduction

Why Clustering Changes the Sample Size

The Intracluster Correlation Coefficient

The Design Effect

Why Cluster Size Matters

Basic Sample Size Strategy

Accounting for Attrition

From Participants to Clusters

Worked Example: Continuous Outcome

Step 1: Calculate the Design Effect

Step 2: Inflate the Individual-Level Sample Size

Step 3: Inflate for Attrition

Step 4: Convert Participants to Clusters

Step 5: Verify the Final Cluster-Based Design

The Complete Calculation in One Equation

What Happens When the ICC Changes?

What Happens When Cluster Size Changes?

Equal Versus Unequal Cluster Sizes

Coefficient of Variation for Cluster Size

Number of Clusters Versus Number of Participants

Cluster-Level Degrees of Freedom

Individual-Level Versus Cluster-Level Randomization

When Is Cluster Randomization Useful?

A General CRT Sample-Size Workflow

Common Sample-Size Mistakes

1. Ignoring the ICC

2. Using the Number of Participants as the Only Design Criterion

3. Choosing an ICC Because It Produces a Convenient Sample Size

4. Forgetting Attrition

5. Rounding the Number of Clusters Down

6. Assuming the Simple Design Effect Always Applies

Binary Outcomes Require Additional Care

Unequal Allocation

Individual Attrition Versus Cluster Attrition

Design Effect Summary

Worked Example: Compact Calculation

Implementation Checklist

Key Takeaways

References

Related Tutorials

Related Tools