Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
There are three power calculations that must be made (all interlinked). First, is the effect of the filter on indoor air pollution. In this case, the unit of observation is the same as the unit of randomization (the school).
We assume for the calculations that the power (β) is equal to 0.8, the level (α) is 0.05, and that the control variables (e.g., school characteristics, strata fixed effects, and historical pollution levels) explain 30% of the variance in the final result (i.e., in the indoor air pollution).
Under these assumptions, using the formula described above, the MDE for the Intention-To-Treat (ITT) is 0.33SD against the control. The MDE for the ITT comparing treatments is 0.382SD. Assuming that only 70% of schools use the filter consistently (and correctly), the MDE for the treatment-on-the-treated is 0.47SD against the control.
Next, we can study the power to detect the reduced form effect of providing schools with air filters on test scores. As the experiment is randomized at the school level, but the main results are observed at the student level (i.e., scores from standardized tests), it must be taken into account that these results have some correlation with each other. This is done by clustering the standard errors at the school level.
Assuming an intracluster correlation of 0.3, and that we observe 80 students per school, the MDE (comparing against the control) for the ITT is 0.18 SD and for the TOT is 0.26 SD. Comparing across treatments, the ITT MDE is 0.21SD, and the TOT MDE is 0.3SD.