Question

Explain the factors that needed to consider in calculating sample size for a cluster randomised controlled trial and how these factors affect sample size. (max. 200 words)

Answer 1

RCT: sample size formulae under individual randomisation

For a trial using individual randomisation, for fixed power (1 - β) and fixed sample size (n), the detectable difference, d _I , with variance var(d _I ) = 2σ ²/n _I is:

(1)

where z _α/2 denotes the upper 100α/2 standard normal centile.

For a trial with n sample size, the power to detect a pre-specified difference of d, is 1 - β _I , such that:

(2)

where Φ is the cumulative standardised Normal distribution.

And, finally the required sample size for a trial at pre-specified power 1 - β to detect a pre-specified difference of d, is n _I , where:

(3)

Using Normal approximations, the above formulae can be used for binary outcomes, by approximating the variance (σ ²) of the proportions π ₁ and π ₂, by:

(4)

for testing the two sided hypothesis H ₀ : π ₁ = π ₂.

CRCTs: standard sample size formulae under cluster randomisation

Suppose, instead of randomising over individuals, the trial will randomise over k clusters each of size m, to provide a total of n _C = mk individuals. Then, by standard results [1], the variance of the difference to be detected d _C is inflated by the Variance Inflation Factor (VIF):

(5)

where ρ is the Intra-Cluster Correlation (ICC) coefficient, which represents how strongly individuals within clusters are related to each other. Where the cluster sizes are unequal this variance inflation factor can be approximated by:

(6)

where cv represents the coefficient of variation of the cluster sizes and is the average cluster size. Thus, the variance of d _C (for fixed cluster sizes) becomes:

(7)

and this is simply extended for varying cluster sizes using equation 6. To determine the required sample size for a CRCT with a pre-specified power 1 - β, to detect the pre-specified difference d, and where there are m individuals within each cluster, then the required sample size n _C = km , follows straightforwardly from equations 3 and 5 and is:

(8)

where n _I is the required sample size using a trial with individual randomization to detect a difference d, and VIF can be modified to allow for variation in cluster sizes (equation 6). This is the standard result, that the required sample size for a CRCT is that required under individual randomisation, inflated by the variance inflation factor [1]. The number of clusters required is then:

(9)

assuming equal cluster sizes. This slight modification of the common formula for the number of required clusters (over that say presented in [2]), has rounded up the total sample size to a multiple of the cluster size (using the ceiling function). For, unequal cluster sizes (using the VIF at equation 6) this becomes:

(10)

again with rounding up to the average cluster size.