Scrap power calcs

Summary

This R notebook contains basic formulas for running power calcs, especially for a typical RCT

Formulas

Standard baseline + end line clustered RCT with covariates

This is the most common RCT I encounter. We randomize at the cluster level, collect data for one baseline and one end line, and run an ANCOVA to estimate ATE.

\[ MDE = M_J\sigma_y\sqrt{\frac{2\rho}{J}+\frac{2(1-\rho)(1-R_K^2)}{JK}} \] \[ M_J = F^{-1}_t\left(1-\frac{\alpha}{2},J\right) + F^{-1}_t\left(\beta,J\right) \] Where…

1. \(\sigma_y\) is the variance of the outcome variable.
2. \(\rho\) is ICC
3. J is the number of clusters in treatment or control (not both). Assumed constant across T and C
4. K is the number of units per cluster. Assumed constant across all clusters.
5. \(\alpha\) is significance level (usually .05)
6. \(\beta\) is power (usually .8)
7. \(R_K^2\) is the R squared from a regression of the outcome on the covariates

mj <- function (j, alpha = .05, beta = .8) {
  multiplier <- qt(1-alpha/2, j)+qt(beta,j)
  return(multiplier)
}

mde <- function (j, k, sigmay, icc, rsq, alpha = .05, beta = .8) {
  quant_under_sqrt <- (2*icc/j)+2*(1-icc)*(1-rsq)/(j*k)
  effect <- mj(j, alpha, beta)*sigmay*(quant_under_sqrt^.5)
  return(effect)
}

Replicating the APRIGP power calcs

villages <- 100
hh_per_v <- 10
icc <- 0.05
rsquared <- 0
sd_cons <- 721

mde(villages, hh_per_v, sd_cons, icc, rsquared)

[1] 109.8495