TABLE 3.

Design of a confirmatory cluster randomized trial (CRT) accounting for missing outcome data based on a single fully observed baseline binary variable

Estimation of parameters from the pilot CRT data
$X_{ki}$
1 community	${\hat{\pi }}_1=\hat{P}\left(X_{ki}=1\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}1\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}I\left(X_{ki}=1\right)\right\}$	0.67
2 elsewhere	${\hat{\pi }}_2=\hat{P}\left(X_{ki}=2\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}1\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}I\left(X_{ki}=2\right)\right\}$	0.33
$R_{ki}$, $X_{ki}$, $Z_k$
1 observed, 1 community, 1 direct delivery	$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}({\hat{\beta }}_{11})=\hat{P}\left(R_{ki}=1|X_{ki}=1,Z_k=1\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}I\left(X_{ki}=1\right)Z_k\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)Z_k\right\}$	0.61
1 observed, 2 elsewhere, 1 direct delivery	$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}({\hat{\beta }}_{21})=\hat{P}\left(R_{ki}=1|X_{ki}=2,Z_k=1\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}I\left(X_{ki}=2\right)Z_k\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=2\right)Z_k\right\}$	0.96
1 observed, 1 community, 0 standard testing	$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}({\hat{\beta }}_{10})=\hat{P}\left(R_{ki}=1|X_{ki}=1,Z_k=0\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^{m}I\left(X_{ki}=1\right)\left(1-Z_k\right)\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)\left(1-Z_k\right)\right\}$	0.57
1 observed, 2 elsewhere, 0 standard testing	$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{i}\mathrm{t}({\hat{\beta }}_{20})=\hat{P}\left(R_{ki}=1|X_{ki}=2,Z_k=0\right)=\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{I\left(X_{ki}=2\right)\left(1-Z_k\right)}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I(X_{ki}=2)(1-Z_k)\right\}\right.$	0.97
$Y_{ki}$, $X_{ki}$, $Z_k$
1 HIV test, 1 community, 1 direct delivery	${\hat{\mu }}_{11}=\hat{E}\left(Y_{ki}|X_{ki}=1,Z_k=1\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)Z_k\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)Z_k{Y}_i\right\}$	0.94
1 HIV test, 2 elsewhere, 1 direct delivery	${\hat{\mu }}_{21}=\hat{E}\left(Y_{ki}|X_{ki}=2,Z_k=1\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=2\right)Z_k\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=2\right)Z_k{Y}_i\right\}$	0.98
1 HIV test, 1 community, 0 standard testing	${\hat{\mu }}_{10}=\hat{E}\left(Y_{ki}|X_{ki}=1,Z_k=0\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)\left(1-Z_k\right)\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=1\right)\left(1-Z_k\right)Y_i\right\}$	0.85
1 HIV test, 2 elsewhere, 0 standard testing	${\hat{\mu }}_{20}=\hat{E}\left(Y_{ki}|X_{ki}=2,Z_k=0\right)={\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=2\right)\left(1-Z_k\right)\right\}}^{-1}\left\{{\mathrm{\Sigma }}_{k=1}^K{\mathrm{\Sigma }}_{i=1}^m{R}_{ki}I\left(X_{ki}=2\right)\left(1-Z_k\right)Y_i\right\}$	0.94
$Y_{ki}$, $Z_k$
1 HIV test, 1 direct delivery	${\widehat{\mu }}_1=\widehat{E}\left(Y_{ki}\mid Z_k=1\right)={\left\{\sum _{k=1}^K\sum _{i=1}^m{Z}_k{R}_{ki}{\widehat{e}}_{1ki}^{-1}\right\}}^{-1}\left\{\sum _{k=1}^K\sum _{i=1}^m{Z}_k{R}_{ki}{\widehat{e}}_{1ki}^{-1}{Y}_{ki}\right\}$	0.95
1 HIV test, 0 standard testing	${\widehat{\mu }}_0=\widehat{E}\left(Y_{ki}\mid Z_k=0\right)={\left\{\sum _{k=1}^K\sum _{i=1}^m\left(1-Z_k\right)R_{ki}{\widehat{e}}_{0ki}^{-1}\right\}}^{-1}\left\{\sum _{k=1}^K\sum _{i=1}^m\left(1-Z_k\right)R_{ki}{e}_{0ki}^{-1}{Y}_{ki}\right\}$	0.88
$\delta$
intercluster correlation	$\begin{array}{l}\widehat{\delta }=\mathrm{c}\widehat{\mathrm{o}}\mathrm{r}\mathrm{r}\left(Y_{ki},Y_{kj}\right)={\left[\sum _{k=1}^K\sum _{i\ne j}{R}_{ki}{R}_{kj}(Z_k{e}_{1ki}^{-1}{e}_{1kj}^{-1}+\left(1-Z_k\right)e_{0ki}^{-1}{e}_{0kj}^{-1}\}\right]}^{-1}\\ \left[\sum _{k=1}^K\sum _{i\ne j}{R}_{ki}{R}_{kj}\left\{\frac{Z_k{e}_{1ki}^{-1}{e}_{1kj}^{-1}\left(Y_{ki}-{\widehat{\mu }}_1\right)\left(Y_{kj}-{\widehat{\mu }}_1\right)}{{\widehat{\mu }}_1\left(1-{\widehat{\mu }}_1\right)}+\frac{\left(1-Z_k\right)e_{0ki}^{-1}{e}_{0kj}^{-1}\left(Y_{ki}-{\widehat{\mu }}_0\right)\left(Y_{kj}-{\widehat{\mu }}_0\right)}{{\widehat{\mu }}_0\left(1-{\widehat{\mu }}_0\right)}\right\}\right]\end{array}$	0.36
Sample size calculation for a confirmatory CRT
IPRW design component	$\begin{array}{l}{\tau }_{\text{C}-\text{IPRW}}=\sum _{c=1}^2{\kappa }^{-1}\frac{{\widehat{\pi }}_c}{{\left\{{\widehat{\mu }}_1\left(1-{\widehat{\mu }}_1\right)\right\}}^2}\left\{\frac{{\widehat{\mu }}_{c1}\left(1-{\widehat{\mu }}_{c1}\right)}{\mathrm{expit}({\widehat{\beta }}_{c1})}+{\left({\widehat{\mu }}_{c1}-{\widehat{\mu }}_1\right)}^2\right\}\\ +\sum _{c=1}^2{\kappa }^{-1}\frac{{\widehat{\pi }}_c}{{\left\{{\widehat{\mu }}_0\left(1-{\widehat{\mu }}_0\right)\right\}}^2}\left\{\frac{{\widehat{\mu }}_{c0}\left(1-{\widehat{\mu }}_{c0}\right)}{\mathrm{expit}({\widehat{\beta }}_{c0})}+{\left({\widehat{\mu }}_{c0}-{\widehat{\mu }}_0\right)}^2\right\}\\ +(m-1)\widehat{\delta }\left[\frac{{\left\{{\widehat{\mu }}_1\left(1-{\widehat{\mu }}_1\right)\right\}}^{-1}+{\left\{{\widehat{\mu }}_0\left(1-{\widehat{\mu }}_0\right)\right\}}^{-1}}{2}\right]\end{array}$	214.6
number of individuals	$n_{\mathrm{C}-\mathrm{I}\mathrm{P}\mathrm{R}\mathrm{W}}={\tau }_{\mathrm{C}-\mathrm{P}\mathrm{P}\mathrm{W}}{\left(z_{0.9}+z_{0.975}\right)}^2/{\left\{\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\hat{\mu }}_1\right)-\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{t}\left({\hat{\mu }}_0\right)\right\}}^2$	2150
number of clusters	$K_{\mathrm{C}-\mathrm{I}\mathrm{P}\mathrm{R}\mathrm{W}}=2⌈\left(n_{\mathrm{C}-\mathrm{I}\mathrm{P}\mathrm{R}\mathrm{W}}/2\right)/m⌉$	360
$I(.)$ is the indictor function. $X_{ki}$ is the covariate ‘Where do you get your healthcare?’ with answer choices 1 ‘community clinic’ or 2 ‘elsewhere’. $Z_k$ is the intervention group: 1 direct delivery of HIV self-tests to sex workers or 0 refer sex workers to standard HIV testing. $R_{ki}$ is the observed indicator: 1 sex worker’s HIV testing status is observed or 0 sex worker’s HIV testing status is missing. $Y_{ki}$ is the outcome: 1 sex worker had a HIV test and 0 sex worker did not have a HIV test.