Power for PPI++ paired test with binary outcomes

Computes power for a paired test when outcomes are binary and predictions are derived from a classifier with sensitivity and specificity. The paired differences \(D = Y^A - Y^B\) and prediction differences \(f^D = f^A - f^B\) are treated via a CLT approximation using population moments implied by a joint Bernoulli model with within-pair correlation.

Usage

power_ppi_paired_binary(
  delta,
  N,
  n,
  alpha = 0.05,
  p_A,
  p_B,
  rho_within,
  sens,
  spec
)

Arguments

delta

Effect size: mean difference \(E[Y^A - Y^B]\). If NULL, uses \(p_A - p_B\).

N

Number of unlabeled pairs with predictions.

n

Number of labeled pairs.

alpha

Two-sided significance level (default 0.05).

p_A

Marginal probability \(P(Y^A=1)\).

p_B

Marginal probability \(P(Y^B=1)\).

rho_within

Pearson correlation between \(Y^A\) and \(Y^B\).

sens

Classifier sensitivity.

spec

Classifier specificity.

Value

Scalar power in \([0, 1]\).

Details

The joint probability \(P(Y^A=1, Y^B=1)\) is set to $$p_{11} = p_A p_B + \rho_{\text{within}} \sqrt{p_A(1-p_A)p_B(1-p_B)}$$ and clipped to the feasible range. Predictions are generated conditionally on \(Y\) with sensitivity/specificity, and population moments are computed by marginalizing over the joint distribution of \((Y^A, Y^B)\).

Examples

power_ppi_paired_binary(
  delta = 0.08,
  N = 5000,
  n = 500,
  p_A = 0.34,
  p_B = 0.26,
  rho_within = 0.3,
  sens = 0.85,
  spec = 0.85
)
#> [1] 0.9809592