P-value for the Bias-Corrected (BC) Test of AUC

Tests \(H_0\colon \mathrm{AUC} = A_0\) vs the specified alternative, using a bias-corrected finite-sample variance estimator with the mid-rank kernel.

Usage

wmwAUC_pvalue_BC(
  x,
  y,
  alternative = "two.sided",
  A0 = 0.5,
  min_n_warn_threshold = 10
)

Arguments

x

Numeric vector of cases (group 1) values.

y

Numeric vector of reference/control (group 2) values.

alternative

Character: "two.sided", "greater", or "less".

A0

Numeric null value of \(\mathrm{AUC} = P(X < Y)\). Defaults to 0.5.

min_n_warn_threshold

Integer; if min(length(x), length(y)) is below this threshold, a warning is issued that power may be very low at this sample size. Default 10.

Value

Numeric p-value.

Details

BC estimates \(\mathrm{Var}(\hat A)\) by correcting each placement-variance component for its \(O(1/n)\) upward bias, using a plug-in estimate of the bias subtracted from the naive placement variance; each corrected component is floored independently at a small \(\epsilon > 0\) if it would otherwise go negative. The mid-rank kernel \(h(x,y) = 1\{x<y\} + \frac{1}{2} 1\{x=y\}\) is used throughout, for both the point estimate and the variance components.

Uses one-tier approach with \(\hat\sigma^2_{\mathrm{adj}}\).

BC is a conservative test: observed size stays below nominal across a wide range of sample sizes, heteroskedasticity, and tie proportions, at a real cost in power for small or imbalanced samples. See min_n_warn_threshold and its warning text; the EU method (wmwAUC_pvalue_EU) is recommended when min(n1, n2) is small.

x is taken to represent cases and y the reference/control group, matching the convention of wilcox.test(). Internally, the test statistic and variance components are computed in the \(P(X<Y)\) framework.