obitools4/autodoc/docmd/pkg/obistats/udist.md

Mann-Whitney U Distribution Implementation in obistats

The obistats package provides efficient computation of the Mann-Whitney U distribution, used in nonparametric hypothesis testing to compare two independent samples.

Core Types

• UDist: Represents the discrete probability distribution of the U statistic for sample sizes N1, N2. It optionally handles ties via a tie-count vector T.

Key Features

• ✅ Exact distribution computation, both with and without ties.

  • No ties: Uses dynamic programming (Mann–Whitney recurrence) in O(N1·N2·U) time.
  • With ties: Implements the linked-list-based algorithm from Cheung & Klotz (1997) via memoization (makeUmemo).

• ✅ PMF & CDF evaluation:

  • PMF(U) returns the probability mass at U.
  • CDF(U) computes cumulative probabilities using symmetry to minimize computation.

• ✅ Support for tied ranks:

  • T encodes tie multiplicities per rank; if nil, no ties are assumed.

• ✅ Optimized recurrence:

  • Exploits symmetry (p_{n,m} = p_{m,n}) and incremental DP to reduce memory/time.

• ✅ Boundary handling:

  • Bounds() returns support [0, N1·N2].
  • Step() = 0.5, reflecting U’s discrete unit in tied cases.

Algorithm Notes

• p(U) uses a 2D DP table (rows = n, columns = U), computing only necessary states.
• makeUmemo builds a 3D memoization table (k, n1, 2U) for tied distributions.
• Performance bottlenecks noted in comments (e.g., map overhead) suggest future optimization paths.

Use Case

Enables exact p-value calculation for the Mann-Whitney U test, especially valuable when:

• Sample sizes are small-to-moderate (exact methods needed).
• Data contain ties.