Internal calculation: stepped wedge power (Hussey & Hughes 2007 Eq 8)

WLS-based Wald power for the treatment effect in a full CS stepped wedge design. Requires (I, T, cluster_size m, tau2, sigma2_ind) - the variance of the treatment effect estimate is computed from the design matrix quadratic forms U/W/V, then power = Phi(|theta|/se - z_(1-a/2)).

Usage

calculate_stepped_wedge(
  theta,
  tau2,
  sigma2_ind,
  m,
  n_clusters,
  clusters_per_step,
  alpha,
  outcome = "mean"
)

Details

Eq 8 as printed (Hussey & Hughes 2007, p. 187):

  Var(theta_hat) = I * sigma^2 * (sigma^2 + T * tau^2) /
    [ (IU - W) * sigma^2 + (U^2 + ITU - TW - IV) * tau^2 ]

where sigma^2 is the variance of the cluster-level mean response (sigma^2_ind / m) and tau^2 is the between-cluster variance.

Anchor: H&H EPT example - I = 24 counties, T = 5 (baseline + 4 steps of 6), m = 100, binary outcome p = 0.05 (sigma2_ind = p(1-p) = 0.0475), CV = 0.3 -> tau2 = (0.3*0.05)^2 = 2.25e-4. Eq 8 returns power = 0.774 at RR = 0.64, matching the paper's "about 80% power to detect a decrease in prevalence of roughly 36% (from 0.05 to 0.032)".