Table 1 Summary of simulation study scenarios

Total number of clusters

4

8, 12, 20, 30

Mean Cluster size (\(\overline{{\varvec{m}} }\))

3

10, 50, 1000

Coefficient of variation (CV) of cluster size

3

\(CV=\frac{s}{\overline{m} }= 0, 0.5, 0.8\) Where s is the standard deviation in cluster sizes and\(\overline{m }\)is the mean cluster size, Cluster size

\({m}_{ij}\)is sampled from a negative binomial distribution as follows:

\(\delta \sim Negbin\left(no of fails=\frac{{(\overline{m }-2)}^{2}}{{s}^{2}-(\overline{m }-2)},p of fail=\frac{\overline{m }-2}{{s}^{2}}\right)\)

\({m}_{ij}=2+\delta\)  

Control cluster prevalence

2

10%, 30%

Intervention effect

2

No effect, or odds ratio between 1.12 and 11.49 selected for each scenario to achieve 80% power

ICC

4

0.001, 0.01, 0.05, 0.1

Cluster effect distribution

3

Normal:

\({u}_{ij}\sim N(0,{\sigma }_{b}^{2})\)

Gamma

\({u}_{ij}=\frac{{\sigma }_{b}\left({a}_{ij}-2\right)}{\sqrt{2}}\)where

\({a}_{ij}\sim Gamma\left(\mathrm{2,1}\right)\)

Uniform

\({u}_{ij}\sim Uniform\left(-\sqrt{3{\sigma }_{b}^{2}},\sqrt{3{\sigma }_{b}^{2}}\right)\)

Distributions are defined to give the specified between cluster variability set by the ICC