====== 费里德曼双向方差分析 Freedman's two-way ANOVA ======
===== 1.使用背景 Context of Use =====
费里德曼双向方差分析用于分析**多个顺序型相关样本(Multiple sequential correlation samples)**。

===== 2.检测统计量 Test Statistic =====
{{ ::费里德曼双向方差分析的检测统计量.png?400 |}}\\
其中，n为各样本的样本容量，k为样本个数，R为各个样本的秩次和。\\
n is the sample size of each sample, k is the number of samples, R is the rank sum of each sample.

===== 3.假设检验 Hypothesis Test =====
**Step 1: 给出虚无假设和备择假设 Give null hypotheses and alternative hypotheses**
  * **H0**：无显著差异 No significant difference
  * **H1**：有显著差异 Significant difference

**Step 2: 将每个样本在不同条件下得到的结果进行排序 Sort the results obtained for each sample under different conditions**

**Step 3: 求出每种条件下各个样本的秩次和 Find the rank sum of each sample in each condition**

**Step 4: 代入检测统计量公式求出统计量观测值，根据题目比较观测值与临界值 Substitute the formula for the test statistic to find the observed value of the statistic, and compare the observed value with the critical value according to the topic**
  * 只有当✘<sup>2</sup><sub>r</sub>的观测值大于临界值时才能拒绝虚无假设H<sub>0</sub>。
  * The null hypothesis H0 can be rejected only if the observed value of ✘2r is greater than the critical value.

===== 4.大样本情况下的费里德曼双向方差分析 The case of Large Samples =====
与克-瓦氏单向方差分析一样，当样本数和样本容量较大时费里德曼双向方差分析的✘<sup>2</sup><sub>r</sub>的取样分布近似于自由度为k-1的✘<sup>2</sup>分布。
  * As with the Kerr-Watt one-way ANOVA, the sampling distribution of the Freedman two-way ANOVA approximates the chi-square distribution with k-1 degrees of freedom when the number of samples and the sample size are large.