Updating Cluster-Trial Enrollment Through The Design Effect

Why the Design Effect Enters a Blinded Update

In a cluster-randomized trial, groups such as clinics or schools—not individual participants—are assigned to treatment. Participants within a cluster tend to have correlated outcomes. As a result, enrolling more people in the same cluster usually provides less information than enrolling the same number of people in independent clusters. A sample-size target derived for individual randomization therefore cannot be used unchanged.

A blinded re-estimation can update the target using an estimate of the intraclass correlation (ICC) while keeping treatment assignments concealed. The calculation is a conversion: start with the required number of individuals under an individual-randomized design, inflate it for within-cluster correlation, and translate the inflated total into a whole number of clusters. The calculation does not itself estimate a treatment effect or guarantee that the assumptions used to plan enrollment remain accurate.

Definition: For two distinct individuals in the same cluster, the intraclass correlation $\rho$ is their outcome correlation under the specified model. If the marginal outcome variance is $\sigma^2$, the covariance for two such individuals is $\rho\sigma^2$. The design effect is the ratio of the variance under cluster sampling to the variance under independent individual sampling for the same number of individuals. With equal cluster size $m$, it is $DE=1+(m-1)\rho$.

For the usual two-arm calculation, let $n_{0,a}$ be the individually randomized target in arm $a$, where $a$ denotes either treatment arm. This target must come from a sample-size calculation for the intended outcome, effect, significance level, and power. The design-effect step adjusts that target for clustering; it does not replace the underlying calculation.

Model, Assumptions, and the Design Effect

For the standard equal-size derivation, suppose each cluster contributes $m$ individuals and the outcome has the same marginal variance $\sigma^2$ for every individual. Within a cluster, each pair of distinct individuals has the same correlation $\rho$; outcomes from different clusters are independent. Assume a common ICC across treatment arms for planning and that the estimate inserted into the calculation applies to the outcome and population being studied. The standard formula is exact for the variance of the mean under this exchangeable covariance structure and equal cluster sizes.

The effect of clustering follows by summing variances and covariances. For a cluster with outcomes $Y_1,\ldots,Y_m$, the variance of its sum contains $m$ individual variances and $m(m-1)$ ordered pairwise covariance terms. Dividing by $m^2$ gives the variance of that cluster’s mean.

\operatorname{Var}\left(\frac{1}{m}\sum_{j=1}^{m}Y_j\right) =\frac{m\sigma^2+m(m-1)\rho\sigma^2}{m^2} =\frac{\sigma^2}{m}\left[1+(m-1)\rho\right].

With $K$ independent clusters, each of size $m$, the overall mean has variance $\sigma^2[1+(m-1)\rho]/(Km)$. Under individual independence, the variance for the same $N=Km$ individuals is $\sigma^2/N$. Their ratio is therefore $1+(m-1)\rho$. This argument explains both the formula and why adding participants within a cluster has diminishing returns when $\rho>0$.

Key result: If an individual-randomized calculation requires $n_{0,a}$ individuals in arm $a$, the equal-size cluster design requires approximately $n_{0,a}DE$ individuals in that arm before rounding, where $DE=1+(m-1)\hat{\rho}$ uses the blinded ICC estimate $\hat{\rho}$. With equal planned cluster size $m$, the number of clusters in arm $a$ is $$ K_a=\left\lceil\frac{n_{0,a}DE}{m}\right\rceil. $$ The ceiling function rounds upward to the next whole cluster.

Rounding should be performed by arm when the trial requires a specified allocation of clusters to each arm. The resulting number of individuals is $K_a m$ per arm under equal cluster sizes. It can exceed the unrounded inflated target; it should not be rounded down merely to make the total closer to the original target.

Worked Conversions

Worked Example: Updating a Two-Arm Trial With Equal Cluster Sizes

Suppose a hypothetical two-arm trial’s individual-randomized calculation requires $n_{0,a}=120$ participants per arm. The planned size is $m=20$ people per cluster, and a valid blinded interim estimate is $\hat{\rho}=0.04$. Assume equal cluster sizes, a common ICC across arms, and independent outcomes across clusters.

First calculate the design effect:

DE=1+(m-1)\hat{\rho} =1+(20-1)(0.04) =1+0.76 =1.76.

The inflated target is $120(1.76)=211.2$ people per arm. Converting to clusters gives

K_a=\left\lceil\frac{211.2}{20}\right\rceil =\lceil 10.56\rceil =11\text{ clusters per arm}.

Thus the planned enrollment is $11(20)=220$ people per arm, or 22 clusters and 440 people in total. The upward rounding means the design exceeds the unrounded target of 211.2 people per arm. The 1.76 design effect describes variance inflation relative to independent individuals; it does not mean the trial needs 1.76 times as many clusters.

Worked Example: The Effect of a Revised Blinded ICC Estimate

In a second hypothetical trial, the individual-randomized target is 90 people per arm. Clusters are planned to contain $m=15$ participants. An earlier planning value was $\hat{\rho}=0.01$; a blinded interim update instead supplies $\hat{\rho}=0.06$. Assume the same equal-size, common-ICC conditions as in the preceding calculation.

Using the earlier value would give $DE=1+(15-1)(0.01)=1.14$. The inflated target would be $90(1.14)=102.6$ people per arm, requiring $\lceil102.6/15\rceil=7$ clusters per arm, or 105 people per arm.

Now use the updated estimate:

DE=1+(15-1)(0.06)=1+0.84=1.84, \qquad 90(1.84)=165.6.

The updated cluster count is $\lceil165.6/15\rceil=\lceil11.04\rceil=12$ clusters per arm. That corresponds to 180 people per arm, or 24 clusters and 360 people total. Under the stated assumptions, the larger ICC increases the calculated target because it implies greater within-cluster redundancy. This is a nuisance-parameter update, not evidence that the treatment effect has changed.

Worked Example: Allowing for Unequal Cluster Sizes

Suppose a hypothetical trial plans for average cluster size $\bar m=18$, with coefficient of variation $CV=0.25$ for cluster size, and uses a blinded ICC estimate $\hat{\rho}=0.03$. Assume the sizes are not extremely unequal, the ICC is common, and this approximate unequal-size design effect is prespecified:

DE_{\text{unequal}}\approx 1+\left[(1+CV^2)\bar m-1\right]\hat{\rho}.

The size adjustment is $(1+0.25^2)(18)=1.0625(18)=19.125$. Therefore,

DE_{\text{unequal}}\approx 1+(19.125-1)(0.03) =1+18.125(0.03) =1.54375.

If the individual-randomized target is 100 people per arm, the approximate inflated target is $100(1.54375)=154.375$ people per arm. Using the planned average size gives $\lceil154.375/18\rceil=\lceil8.5764\rceil=9$ clusters per arm. At the planned average size, that is about $9(18)=162$ people per arm in expectation.

This is an approximate planning conversion, not a promise that nine realized clusters will contain exactly 162 people. Actual enrollment depends on the realized cluster sizes. The unequal-size formula should not be substituted automatically for the equal-size formula: it relies on a size distribution adequately represented by the stated mean and coefficient of variation, and on the model assumptions behind the approximation.

Using the Update Responsibly

A blinded ICC estimate can be inserted only if it is suitable for the planned outcome and analysis. In particular, the re-estimation procedure should preserve masking and should specify how the pooled information is used without relying on treatment-specific effect estimates. The ICC should correspond to the outcome scale and population relevant to the sample-size calculation. A value estimated from a materially different outcome or cluster structure may not be a useful substitute.

Several checks help expose a calculation that is going wrong:

Check units and inputs. Cluster size $m$, the individual target, and the ICC must refer to the same design and outcome. An ICC is dimensionless; $m$ and the sample-size targets are counts.
Check the direction. For $m>1$ and a nonnegative ICC, $DE\geq1$. A larger ICC or larger equal cluster size cannot reduce this design effect.
Check the allocation. Convert the target within each arm, then round clusters upward as required by the randomization plan. Rounding a combined total can leave one arm short.
Check size variability. If cluster sizes differ materially, treating every cluster as exactly average-sized can understate the variance inflation. Use a prespecified, justified approximation or a more appropriate design calculation.
Check stability and relevance. An ICC estimated from few clusters can be uncertain, and an interim value may differ from the value at the final analysis. Report the estimate and examine how plausible values affect the target; a point estimate is not certainty about the true ICC.

A method-of-moments estimate can sometimes be negative because of sampling variation, even when the working random-intercept model restricts the population ICC to be nonnegative. Do not silently insert such a value or replace it with zero without a prespecified rule and a compatible model. The estimate’s construction and the handling of boundary cases belong in the re-estimation plan.

The design-effect conversion is attractive because it is transparent and connects directly to an individual-randomized target. It can be a reasonable approximation when its covariance assumptions and cluster-size assumptions hold. However, it is not a universal substitute for a sample-size calculation that directly represents the planned cluster-level analysis. In particular, simply multiplying by the design effect does not resolve issues such as very few randomized clusters, unequal cluster sizes beyond the approximation’s range, or differences in cluster behavior across arms.

Nor should the formula be confused with the alternative of ignoring clustering. Using the individual-randomized target without inflation assumes, in effect, that within-cluster dependence does not reduce information; when $\rho>0$ and $m>1$, the variance derivation shows why that assumption is inappropriate. Conversely, a design-effect inflation can be unnecessarily large if the chosen ICC is not relevant or if the planned cluster sizes are misstated. The result is conditional on the inputs and assumptions, not a guarantee of final power.

A Practical Calculation Sequence

Fix the individual-level target.
Record the required $n_{0,a}$ for each arm from the appropriate outcome-specific sample-size calculation before applying any cluster adjustment.

Check the cluster assumptions.
Specify the planned cluster size or size distribution, the ICC model, and whether a common ICC is being used across arms.

Insert the blinded estimate.
Use the prespecified estimate $\hat{\rho}$ and calculate the equal-size design effect, or a justified unequal-size approximation when applicable.

Inflate and round by arm.
Calculate the inflated individual target and divide by the planned cluster size. Round the number of clusters upward in each arm, then report the implied enrollment.

Common Pitfall

A common error is to calculate $DE$, multiply the target by it, and then report the inflated number of individuals as though it were the final enrollment. For example, an inflated target of 211.2 people per arm with 20 people per cluster is not 211.2 clusters, nor can it be enrolled as a fraction of a cluster. The correction is to divide the inflated individual target by the planned cluster size and round upward to whole clusters in the required allocation. In that example, the answer is 11 clusters per arm, corresponding to 220 people per arm if cluster sizes are exactly 20.

Key takeaway: Under equal cluster sizes and an exchangeable within-cluster correlation model, $DE=1+(m-1)\hat{\rho}$. Apply it to the individual-randomized target, then convert the inflated target into whole clusters per arm. The estimate is useful only to the extent that the ICC, cluster-size assumptions, and planning model are appropriate.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A hypothetical trial needs 80 individuals per arm under individual randomization. With $m=10$ and $\hat{\rho}=0.05$, find the design effect, clusters per arm, and resulting enrollment per arm.
For fixed $m>1$, what happens to the design effect as the ICC increases? Explain using the formula.
A calculation gives an inflated target of 143 people per arm and a planned cluster size of 12. How many clusters per arm are needed, and what is the implied enrollment under equal sizes?
Why can the equal-size formula be unsuitable when cluster sizes vary substantially?

Solutions. (1) $DE=1+(10-1)(0.05)=1.45$. The inflated target is $80(1.45)=116$ people per arm. Thus $\lceil116/10\rceil=12$ clusters are needed, giving 120 people per arm. (2) Since $DE=1+(m-1)\hat{\rho}$, its slope in $\hat{\rho}$ is $m-1>0$; a larger ICC increases the design effect. (3) $\lceil143/12\rceil=\lceil11.9167\rceil=12$ clusters per arm, or $12(12)=144$ people per arm. (4) The equal-size derivation assumes every cluster has size $m$; variable sizes change the contribution of within-cluster covariance and can increase variance inflation. A justified unequal-size method or a more suitable calculation is needed.