feat(analyze): add the gehan generalized wilcoxon test

The rank-sum carried over to right-censored samples: every pair of runs is
scored by which one outlived the other, and a pair censoring cannot order
counts as half rather than as a difference neither run supports. The effect
size and the p-value are the same statistic, and with nothing censored both
are exactly what the rank-sum reports.

cmd/internal-tools/analyze/gehan.go

@@ -0,0 +1,97 @@
+package main
+
+import "math"
+
+// gehanResult is one pairwise comparison of two arms of right-censored runs: an
+// effect size, the run pairs that have no order between them, and the test of
+// the same statistic against the null of equal hazards.
+type gehanResult struct {
+	FirstSize  int
+	SecondSize int
+	Statistic  float64
+	A12        float64
+	Unordered  int
+	PValue     float64
+}
+
+// outlives orders two runs the only way right-censoring allows. A run censored
+// at step t violated at no step up to t and stopped for a reason of its own, so
+// it outlives a violation at or before t and nothing orders it against a
+// violation after t or against another censored run. Comparing the two step
+// counts as plain numbers instead reads a run the wall clock stopped at step 12
+// as one that violated at step 12.
+func outlives(left, right observation) int {
+	switch {
+	case left.Event && right.Event:
+		switch {
+		case left.Steps > right.Steps:
+			return 1
+		case left.Steps < right.Steps:
+			return -1
+		}
+	case left.Event:
+		if right.Steps >= left.Steps {
+			return -1
+		}
+	case right.Event:
+		if left.Steps >= right.Steps {
+			return 1
+		}
+	}
+	return 0
+}
+
+// atRiskWeight is Gehan's weight: an event counts for as many runs as were still
+// at risk when it happened. It is what makes the weighted log-rank statistic the
+// same quantity as the pairwise count below, so the effect size and the p-value
+// are one statistic rather than two that can disagree.
+func atRiskWeight(atRisk float64) float64 { return atRisk }
+
+// gehanTest is the Gehan-Breslow generalized Wilcoxon test: the rank-sum
+// carried over to right-censored samples by scoring every pair of runs by which
+// one outlived the other and leaving the pairs censoring cannot order out of the
+// count. Gehan (1965), "A Generalized Wilcoxon Test for Comparing Arbitrarily
+// Singly-Censored Samples", Biometrika 52(1-2), 203-223; Breslow (1970).
+//
+// Statistic is that count, U, and A12 is it over the number of pairs: the share
+// of run pairs in which the first arm survived longer, an unordered pair
+// counting as half. With nothing censored the two are exactly the Mann-Whitney U
+// and the Vargha-Delaney A12 the uncensored rank-sum reports. Where censoring
+// leaves a pair unordered, the half it contributes is the null value, so an
+// unordered pair can only pull the effect size toward 0.5 and can never
+// manufacture a direction.
+//
+// The p-value is the same statistic standardized: the weighted log-rank with
+// Gehan's weight has this U for its statistic, and its variance is the
+// conditional hypergeometric one summed over event times, which is what keeps
+// the test honest when the arms censor on different schedules. The permutation
+// variance Gehan originally paired with the statistic does not.
+func gehanTest(first, second []observation) gehanResult {
+	result := gehanResult{
+		FirstSize:  len(first),
+		SecondSize: len(second),
+		Statistic:  math.NaN(),
+		A12:        math.NaN(),
+		PValue:     math.NaN(),
+	}
+	if len(first) == 0 || len(second) == 0 {
+		return result
+	}
+	outlived := 0.0
+	for _, left := range first {
+		for _, right := range second {
+			switch outlives(left, right) {
+			case 1:
+				outlived++
+			case 0:
+				outlived += 0.5
+				result.Unordered++
+			}
+		}
+	}
+	result.Statistic = outlived
+	result.A12 = outlived / float64(len(first)*len(second))
+	test := weightedLogRank([]string{"first", "second"}, [][]observation{first, second}, atRiskWeight)
+	result.PValue = test.PValue
+	return result
+}