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40 lines
1.7 KiB
Markdown
40 lines
1.7 KiB
Markdown
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# Mann-Whitney U Distribution Implementation in `obistats`
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The `obistats` package provides efficient computation of the **Mann-Whitney U distribution**, used in nonparametric hypothesis testing to compare two independent samples.
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## Core Types
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- **`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`.
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## Key Features
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- ✅ **Exact distribution computation**, both with and without ties.
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- *No ties*: Uses dynamic programming (Mann–Whitney recurrence) in `O(N1·N2·U)` time.
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- *With ties*: Implements the linked-list-based algorithm from Cheung & Klotz (1997) via memoization (`makeUmemo`).
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- ✅ **PMF & CDF evaluation**:
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- `PMF(U)` returns the probability mass at U.
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- `CDF(U)` computes cumulative probabilities using symmetry to minimize computation.
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- ✅ **Support for tied ranks**:
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- `T` encodes tie multiplicities per rank; if nil, no ties are assumed.
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- ✅ **Optimized recurrence**:
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- Exploits symmetry (`p_{n,m} = p_{m,n}`) and incremental DP to reduce memory/time.
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- ✅ **Boundary handling**:
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- `Bounds()` returns support `[0, N1·N2]`.
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- `Step() = 0.5`, reflecting U’s discrete unit in tied cases.
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## Algorithm Notes
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- `p(U)` uses a 2D DP table (rows = *n*, columns = U), computing only necessary states.
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- `makeUmemo` builds a 3D memoization table (`k`, `n1`, `2U`) for tied distributions.
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- Performance bottlenecks noted in comments (e.g., map overhead) suggest future optimization paths.
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## Use Case
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Enables exact *p*-value calculation for the **Mann-Whitney U test**, especially valuable when:
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- Sample sizes are small-to-moderate (exact methods needed).
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- Data contain ties.
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