Cluster Random Designs

Derived from this video S4b Sample size and power for cluster randomised trial

I love the way that Dr. Karla Hemming goes through these, so I thought I would follow-along and codify here work for fun!

Cluster Randomized

Question: why do the formula's change when using a cluster randomized trial (CRT) design from this:

Where is the variance of the outcome variable, and : is the total number of individuals.

To this?

Where is the variance of the outcome variable, is the average cluster size (number of individuals per cluster), is the intracluster correlation coefficient (ICC), and is the total number of clusters per arm.

Answer: Subjects within the same cluster may be more similar to each other than to subjects in other clusters. This correlation leads to an increase in the variance compared to simple random sampling.

This: is known as the inflation effect or design effect

What is the difference? Does this matter?

For equivalence, set

^ ICC: intra-cluster correlation coefficient.

Quantifying the effect of clustering

The ICC, or quantifies the effect of clustering. It's a quantity between 0--1.

Where is the between cluster variance and is the within cluster variance.

The total variance is equal to the sum of the between and within cluster variance:

So when we're trying to determine the sample size in a cluster randomized trial, the required number of subjects in each arm is inflated by the design effect.

Becomes:

Where we then divide by to get the number of clusters per arm, for a CRT testing two proportions.

For the case of a binary outcome for superiority, let's walk through the 3 steps needed to size this study.

  1. Calculate needed in non-randomized design. In this case, a CRT testing two proportions. Observations needed per arm, =
  1. Inflate by the design effect, . Total number of subjects needed, .
  1. Estimate number of clusters needed by didviding by : = .
  1. If is 10--15, add an addition cluster per arm.

How do you estimate the ICC?

The ICC is assumed or estimated. When historical data is available, you can obtain an estimate by fitting a mixed model to the data. However, the uncertainty associated with any estimated correlations (particular for small n studies) may be large, rendering these estimates to be more or less uninformative.

When the endpoints are binary, the ICC's put into these calculations need to be put in the proportion scale. (fitted by linear mixed models, not logistic models, to estimate the ICC's).

When there's no information, not all is lost. However, as the ICC is a measure of correlation within the same cluster, it really only make sense to use an ICC value from other similar trials or studies will be appropriate when:

  1. Same (similar) outcome - Outcome type (ICCs for process outcomes higher than for clinical outcomes)
  2. Same (similar) cluster size - Cluster size (ICCs from smaller clusters tend to be higher)
  3. Same (similar) setting - Setting (ICCs in secondary care are typically higher than ICCs in primary care)
  4. Same (similar) prevalence - Prevalence (ICCs from more prevalent outcomes tend to be higher)

Further, in the absence of specific information on likely ICCs, recommendations are to use patterns observed from empirical studies of correlations.

Caveats for RCTs with a small number of clusters

These formulas are generally derived/applied to large cluster studies.

  1. For parallel arm analysis, use critical values from the t-distribution, rather than the normal (z) distribution
  2. Set the number of degrees of freedom to:
  3. If needed at the sample-size design stage, add one cluster per arm (or two, if you're feeling extra conservative).

Qunatifying the ICC is typically done by measuring the degree of clustering. However, an alternative is to use the Coefficient of Cluster Variation (CCV):

where is the true (unknown) cluster mean, proportion, or rate; and is the between-cluster variance.

This can be a useful metric when only cluster-level means are available from any data being used to estimate correlations, or when the outcome is a count or rate rather than a continuous or binary outcome (e.g., gap closure).