Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Since we do not have data on outcomes of interest in our target population, our power calculations require assumptions. If we want to be able to compare each treatment against the control, as well as compare treatments against one another, then we maximize power by making each group the same size. This means that with a total of 208 candidate villages, we can work with 4 groups (1 control and 3 treatment arms) of 52 villages each. If we want to use the likelihood of purchasing a new variety in any given year as the basis for power analysis, we have no data on this, so we must assume some distributional characteristics.
To begin with, let us assume that 10% of HHs are switchers, or do purchase a new maize variety in any given year. With a fixed number of clusters, we then examine the number of households required for data collection in each village to detect a treatment-induced change from that 10%, assuming that the intra-cluster correlation is 0.2 and standard levels of acceptable type I and II errors (i.e., alpha = 0.05 and power of 80%, using a one-sided test).
We first consider an analysis of spillover effect among non-hosts that compares T1 (the lightest treatment) to the control group. Assuming baseline/control outcome of 10% switchers and MDE of 0.086 or 8.6 percentage points increase in uptake of new varieties among the treatment villages, we need 16 households per village. For the analysis of trial-hosts outcomes, one can assume a higher MDE since they receive the treatment directly (heavier treatment). With 52 clusters and MDE of 0.112 or 11.2 percentage points increase in uptake of new varieties among the hosts in treatment villages, we need 4 host households interviewed per village. These results are shown in Figure 1 below.
For an analysis that compares treatment groups among each other, we assume T1 as the comparison group for T2 and T3, and assume that, after the intervention, 20% of T1 will be switchers. Assuming equal sample size across experimental groups, we shall be able to detect an MDE of 0.106 (10.6 percentage points) with 16 non-host households surveyed and 0.137 (13.7 percentage points) with 4 hosts households surveyed.