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Confidence Interval for the AUC-Equalizing Shift (Hodges-Lehmann Pseudomedian) via Test Inversion

Source: R/wmwAUC_pseudomedian_ci.R
wmwAUC_pseudomedian_ci.Rd

Computes a confidence interval for the pseudomedian by inverting the test \(\mathrm{H_0\colon AUC}(X, Y+\delta) = 0.5\).

Usage

wmwAUC_pseudomedian_ci(
  x,
  y,
  conf.level = 0.95,
  pvalue_method = c("EU", "BC"),
  n_grid = 1000
)

Arguments

x

numeric vector, first sample

y

numeric vector, second sample

conf.level

confidence level (default 0.95)

pvalue_method

character, either 'EU' or 'BC'

n_grid

number of grid points for search (default 1000)

Value

list with conf.int, estimate and conf.level

Details

The pseudomedian \(\delta\) is defined generally, for any pair of distributions \(F, G\), as the shift solving \(P(X < Y + \delta) = 0.5\) — i.e. the value of \(\delta\) that equalizes the AUC between \(X\) and the shifted \(Y + \delta\). This definition does not require a location-shift model relating \(F\) and \(G\). Under location-shift (\(F(t) = G(t - \Delta)\)), this \(\delta\) coincides with the classical Hodges-Lehmann pseudomedian and equals the location difference \(\Delta\); this is a special case, not the definition. Outside location-shift, \(\delta\) remains well-defined as the AUC-equalizing shift, but should not be read as "the median difference" between the two groups.

The confidence interval is obtained by grid search over candidate \(\delta\) values, retaining those for which the AUC=0.5 test applied to \((X, Y+\delta)\) is not rejected at level 1 - conf.level. The search range is set heuristically from \(\pm 3\) times the robust MAD scale of the pairwise differences around the point estimate; if the true interval extends beyond this range, or if the acceptance region is not contiguous (which can occur in small, imbalanced samples), a warning is issued. See Warnings.

EU and BC pseudomedian CIs are computed on independent calls, and may occasionally be reported as numerically identical at coarse n_grid even though the underlying p-value curves genuinely differ — both methods' crossing points can land in the same discrete grid cell.

At small min(length(x), length(y)), BC's near-zero power (see wmwAUC_pvalue_BC) can make its pseudomedian interval very wide and largely uninformative rather than merely conservative, since the grid search rarely rejects anywhere in the searched range. EU is recommended in that regime; see the warning issued below.

When pvalue_method = 'EU' and length(x) + length(y) < 20, each grid point's p-value may come from Monte Carlo permutation (see wmwAUC_pvalue_EU), making this function's result stochastic at small sample sizes. For reproducible results, call set.seed() immediately beforehand.

Warnings

Two diagnostic warnings are issued when the grid search may be unreliable:

  • Non-contiguous acceptance region: the interval is reported as [min(accepted), max(accepted)], which assumes the set of non-rejected \(\delta\) values is a single contiguous block. If it is not, the reported interval may be wider than the true acceptance region and should be inspected directly.

  • Search range too narrow: if the p-value has not dropped below \(\alpha\) at either edge of the search grid, the true endpoint may lie outside the searched range; consider widening the range or increasing n_grid.