====曼-惠特尼U检验（Mann-Whitney U Test）====
**1. 使用场景（Usage Scenarios）**
  * 用于两个独立样本的检验；
  * 顺序性数据

  * For tests of two independent samples；
  * For sequential data.

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**2. 步骤（Steps）**
  * 对每种处理条件各得到一个独立的样本，以nA表示样本A中的被试数目，以nB表示样本B中的被试数目；
  * 将两个样本合并，将所有被试（nA +nB）排序；
  * 确定来自两个样本的分数在混合排序中是否系统性地聚集在度量的两端
  * 如图：从下到上，对每个字母，分别计算下面有多少个另外的字母（如：等级6的B下面有多少个A），计算结果作为点数U
  * {{:mann-witney.png?400 |}}

  * An independent sample was obtained for each treatment condition, with nA denoting the number of subjects in Sample A and nB denoting the number of subjects in Sample B.
  * Combine the two samples and rank all subjects (nA + nB).
  * Determine whether scores from two samples are systematically clustered at opposite ends of the metric in mixed ordering.
  * As shown: from bottom to top, for each letter, count how many other letters are below it (e.g., how many A's are below B on a scale of 6), and use the result as the number of points U.

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**3. 公式（Formulas）**
  * UA+UB = nA*nB
  * UA= nA*nB+[nA（nA+1）/2]-∑RA
  * UB= nA*nB+[nB（nB+1）/2]-∑RB

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**4. 假设检验（Hypothesis Testing）**

**第一步：给出虚无假设和备择假设**
  * **H0**：两处理之间无系统差异
  * **H1**：两处理之间有系统差异
**第二步：得到分界点Ucrit**
  * 根据nA,nB,查表得到Ucrit
**第三步：计算U**
  * 选择UA、UB中较小点数作为U
**第四步：Mann-Whitney U的结果分析**
  * 若U=0，其中一个样本不得分，两个样本无重叠，有最大的差异
  * U越大，两个样本越接近
  * ∴当Uobs ≤ Ucrit时,才能拒绝H0 （与参数检验正好相反）

**Step 1: Give null hypotheses and alternative hypotheses**
  * **H0**：Without systematic differences between the two treatments.
  * **H1**：With systematic differences between the two treatments.
**Step2：obtain Ucrit of boundary point**
  * According to nA,nB, look up the table to get Ucrit.
**Step 3: Calculate U**
  * Select the smaller of UA and UB as U.
**Step 4: Analysis of results for Mann-Whitney U**
  * If U = 0, one of the samples is not scored . There is no overlap between the two samples and the difference is the largest.
  * The larger U is, the closer the two samples are.
  * ∴Only when Uobs ≤ Ucrit can H0 be rejected (the opposite of a parametric test).

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**5. 大样本情况下的正态近似（Normal approximation in the case of large samples）**
  * 当n>20, Mann-Whitney U统计量接近正态分布
  *  When n>20, Mann-Whitney U statistic is close to normal distribution.

  * μ= nAnB/2
  * σ= sqrt（nAnB （nA+nB+1）/12）
  * {{:u正态近似.png?200 |}}

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**6. 统计前提（Statistical Prerequisites）**
  * 要求观察独立
  * 要求变量是连续的，即较少相同的等级
  * 不要求正态分布
  * 不要求方差同质

  * Requiring independent observation.
  * Requiring variables to be continuous, i.e., less equal in rank.
  * Not requiring normal distribution.
  * Not requiring variance to be homogeneous.