###### Stepped wedge study design

We used the sample size calculations for a stepped wedge design proposed by Woertman et al^{1}. Their calculations use the unadjusted sample size ($ n_{unadjusted} $) required for an individual randomised controlled trial for comparing two proportions and then adjusts this calculation based on the cluster design effect ($ DEFF $) to be expected in a stepped wedge design. Their equation for estimating the design effect of a stepped wedge study is as follows:

$$ DEFF_{\text {stepped wedge}} = \frac {1 + p(ktn + bn – 1)}{1 + p \left(\frac {1}{2}ktn + bn – 1 \right )} \times \frac {3(1-p)}{2t \left (k-\frac{1}{k} \right )} $$

where

$ k = \text {number of steps} $

$ b = \text {number of baseline measurements} $

$ t = \text {number of measurements after each step} $

$ p = \text {intra-cluster correlation coefficient (ICC)} $

$ n = \text {number of subjects per cluster} $

To estimate the unadjusted sample size ($ n_{unadjusted} $) needed for the comparison of two proportions, we use the following formula^{2}:

$$ n_{\text {unadjusted}} = { \left (Z_{\frac {\alpha}{2}} + Z_{\beta} \right )}^2 \times \frac {p_1(1-p_1) + p_2(1-p_2)}{{(p_1 – p_2)}^2} $$

where

$ Z_{\frac {\alpha}{2}} = \text {level of significance (1.96 for 5% significance)} $

$ Z_{\beta} = \text {desired power (0.84 for 80}\% \, \text {power)} $

$ p_1 = \text {proportion for control group} $

$ p_2 = \text {proportion for intervention group} $

We calculated the sample size based on an expected 5% decrease in MAM prevalence (from 20% to 15%) with an 80% power to detect a difference and a 5% level of significance. These parameters provide the following unadjusted sample size:

$$ \begin{align}

n_{\text {unadjusted}} &= {(1.96 + 0.84)}^2 \times \frac {0.20(1 – 0.20) + 0.15(1 – 0.15)}{{(0.20 – 0.15)}^2} \\

\\

&= 7.84 \times \frac {0.20(0.80) + 0.15(0.85)}{{(0.05)}^2} \\

\\

&= 7.84 \times \frac {0.16 + 0.1275}{0.0025} \\

\\

&= 7.84 \times \frac {0.2875}{0.0025} \\

\\

&= 7.84 \times 115 \\

\\

&\approx 902

\end{align}$$

This results in an unadjusted sample size per group of 902 for a total unadjusted sample size of 1804. Using this estimated sample size, we factor in the DEFF estimator specified above using the following parameters:

We used the following parameters to calculate the DEFF to be used to calibrate the computed unadjusted sample size above:

$ k = 3 \ \text {steps} $

$ b = 1 \ \text {baseline measurement} $

$ t = 2 \ \text {measurements after each step} $

$ p = 0.034 \ \text {intra-cluster correlation coefficient} $

$ n = 192 $

These parameters were based on a study design that would conduct 1 baseline measurement where all areas or clusters are at ‘baseline’ status (as defined previously) and that assumes that rollout of intervention will be staged at 4 month intervals over a one year period hence 3 steps. We planned to conduct 2 measurements at each step for each of the clusters or areas. First measurement will be done 2 months after the start of each step (i.e., turning of some clusters into intervention areas) and the second measurement will be done 2 months after the first measurement. This will allow for a measurement to be made at the start phase of the intervention when the organisation, logistics and protocols of the intervention are getting refined and institutionalised and at 2 months thereafter when the intervention has already been well-established and potentially has had an effect. We estimated the intra-cluster correlation factor to be around 0.034^{3} and will aim for a minimum cluster size of 192 which is the minimum sample size to estimate GAM prevalence with the required relative precision of 30%^{4} using a PROBIT estimator^{5}.

Given these parameters and considerations, we arrived at the following calculations sample size calculations for the initial stepped wedge study design:

$$ \begin{align}

n_{\text {stepped wedge}} &= 1804 \times \frac {1 + 0.034(3 \times 2 \times 192 + 1 \times 192 – 1}{1 + 0.034 \left (\frac {1}{2} \times 3 \times 2 \times 192 + 1 \times 192 – 1 \right )} \times \frac {3(1-0.034)}{2 \times 2 \left (3 – \frac {1}{3} \right )} \\

\\

&= 1804 \times \frac {1 + 0.034(1343)}{1 + 0.034(767)} \times \frac {3(0.966)}{4 \left (\frac{8}{3} \right )} \\

\\

&= 1804 \times \frac {46.662}{27.078} \times \frac {2.898}{10.67} \\

\\

&= 1804 \times 1.72324396 \times 0.27160262 \\

\\

&\approx 844

\end{align}$$

We estimated a total sample size requirement of 844 children 6-59 months old for the study^{6}. Given a minimum cluster size of 192 children (as mentioned in the outcome measures section), we would need about 5 clusters. We decided to bring this up to 6 to round off the number of clusters that will switch at each of the 3 steps of the study. With 6 clusters, we can have 2 clusters switching over to being intervention areas at every step (4 month intervals). By the third step, all clusters would then become intervention areas.

###### Incidence sub-study

For the incidence sub-study, we apply sample size calculations in $ y_{\text {person-years}} $ proposed by Hayes and Bennet^{8} for an individually-randomised cluster controlled trial as follows:

$$ y_{\text {person-years}} = \left (Z_{\frac {\alpha}{2}} + Z_\beta \right) ^ 2 \times \frac {\lambda_0 + \lambda_1}{(\lambda_0 – \lambda_1) ^ 2} $$

where

$ \lambda_0 = \text {incidence rate in control group} $

$ \lambda_1 = \text {incidence rate in intervention group} $

We use a value of $ \lambda_0 = 0.32 $ (assuming a prevalence rate of 20% in the control group) and a value of $ \lambda_1 = 0.24 $ (assuming a prevalence rate of 15% in the intervention group). This gives us a sample size for one arm of the incidence study of:

$$ y_{\text {person-years}} = (1.96 + 0.84) ^ 2 \times \frac {0.32 + 0.24}{(0.32 – 0.24) ^ 2} \approx 686 $$

For both arms, we would therefore need 1372 sample size. To calculate the number of clusters needed based on this sample size, we use the following formula:

$$ n_{\text {clusters}} = 1 + \left (Z_{\frac {\alpha}{2}} + Z_\beta \right ) ^ 2 \times \frac {\frac {\lambda_0 \ + \ \lambda_1}{y_{\text {person-years}} \ + \ {k ^ 2}({\lambda_0} ^ 2 \ + \ {\lambda_1} ^ 2)}}{(\lambda_0 \ – \ \lambda_1) ^ 2} $$

where

$ k = \text {intra-cluster correlation coefficient which we set at 0.034} $

The formula gives us:

$$ n_{\text {clusters}} = 1 + (1.96 + 0.84) ^ 2 \times \frac {\frac {0.32 \ + \ 0.24}{1372 \ + \ 0.034 ^ 2 (0.32 ^ 2 \ + \ 0.24 ^ 2)}}{(0.32 \ – \ 0.24) ^ 2} \approx 2 $$

So, we will need 1372 sample (686 per arm) from 2 clusters (one from each study arm).

###### Endnotes

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