For an outcome vector of length n, the polynomial rank score of a unit at
ascending rank k is (k / (n + 1))^(zeta - 1), with ties at their average
rank. This is the score of Kim, Su, Bowers, and Li (arXiv 2605.08027). At
zeta = 2 it is the rank divided by n + 1, which gives the Wilcoxon rank-sum
test. Larger zeta puts more of the block's total score on its top ranks:
treating k / (n + 1) as uniform on (0, 1), the score-weighted average
distance from the top of the block is about n / (zeta + 1) units.
Usage
riposte_poly_scores(y, zeta, tail = c("upper", "lower"))Details
Unlike riposte_stephenson_scores(), the score lies in [0, 1) whatever the
block size, so blocks of different sizes enter on the same scale.
With tail = "lower" the score weights the bottom of the block instead: the
unit at ascending rank k gets -((n + 1 - k) / (n + 1))^(zeta - 1), which
lies in (-1, 0]. The minus sign makes the score rise with the outcome, as
the upper-tail score does, so that with alternative = "less" in
[riposte_test()] both tails ask whether treated outcomes are lower. Without
ties, the lower-tail score is -riposte_poly_scores(-y, zeta).
[0, 1)` whatever the block size, so blocks of different sizes enter on the same scale.
With tail = "lower" the score weights the bottom of the block instead: the
unit at ascending rank k gets -((n + 1 - k) / (n + 1))^(zeta - 1), which
lies in `(-1, 0]: R:0,%201)%60%20whatever%20the%0Ablock%20size,%20so%20blocks%20of%20different%20sizes%20enter%20on%20the%20same%20scale.%0A%0AWith%20%60tail%20=%20%22lower%22%60%20the%20score%20weights%20the%20bottom%20of%20the%20block%20instead:%20the%0Aunit%20at%20ascending%20rank%20%60k%60%20gets%20%60-((n%20+%201%20-%20k)%20/%20(n%20+%201))%5E(zeta%20-%201)%60,%20which%0Alies%20in%20%60(-1,%200
[riposte_test()]: R:riposte_test()