Power for PPI++ paired t-test (within-subject differences)

Computes power for a paired t-test using PPI++. This is equivalent to mean estimation on the differences \(D_i = Y_i^A - Y_i^B\) with auxiliary predictions \(f^D_i = f^A(X_i) - f^B(X_i)\).

Usage

power_ppi_paired(
  delta,
  N,
  n = NULL,
  power = NULL,
  alpha = 0.05,
  sigma_D2,
  rho_D,
  sigma_fD2 = NULL
)

Arguments

delta

Effect size: the true mean difference \(E[Y^A - Y^B]\).

N

Number of unlabeled pairs with predictions.

n

Number of labeled pairs.

power

Target power. Set to NULL (default) to compute power from n. Set to a value in (0, 1) to solve for n instead.

alpha

Two-sided significance level (default 0.05).

sigma_D2

Variance of the differences \(D = Y^A - Y^B\).

rho_D

Correlation between \(D\) and \(f^D\).

sigma_fD2

Optional variance of predicted differences. If NULL, assumed equal to sigma_D2.

Value

When power = NULL: scalar power in \([0, 1]\). When n = NULL: required number of labeled pairs (integer).

Details

The paired t-test reduces to a one-sample problem on the differences. The PPI++ variance for paired data is: $$\mathrm{Var}(\hat\Delta_{\lambda^*}) \approx \frac{\sigma_D^2(1 - \rho_D^2)}{n}$$ where \(\rho_D\) is the correlation between the observed difference and the predicted difference.

Examples

# Compute power
power_ppi_paired(
  delta = 0.3,
  N = 1000,
  n = 50,
  sigma_D2 = 1,
  rho_D = 0.7
)
#> [1] 0.8276133

# Compute required sample size
power_ppi_paired(
  delta = 0.3,
  N = 1000,
  power = 0.80,
  sigma_D2 = 1,
  rho_D = 0.7
)
#> [1] 47