13.9
Wilcoxon 秩和检验
Wilcoxon 秩和检验，也称为 Mann-Whitney U 检验，是一种非参数检验，用于确定两个独立样本的分布之间是否存在显著差异。此检验专为两个独立总体设计，具有以下关键要求：
- 样本必须随机抽取。
- 数据应为序数或能够转换为序数标度，以便对值进行排序和排名。
零假设是两个总体分布的中位数相同，…
Wilcoxon 秩和检验，即 Mann-Whitney U 检验，是一种非参数检验方法，用于通过比较两组数据的中位数来判断两个总体之间是否存在差异。
它严格适用于样本量相等或不相等的两个独立简单随机样本。
比较两种不同蜘蛛物种的捕食反应时间。
此处，零假设表明两个物种的中位反应时间相同，而备择假设则表明两者不同。
这两个样本中的数值被排序，视为单一的数据点集合。然而，秩和是独立计算的。
当其中一个样本中主要出现较高或较低的秩次，而另一个样本中较少出现时，这两个样本的中位数可能存在差异。
使用以下公式计算检验统计量 z，以检验该假设。
Wilcoxon 秩和检验为双尾检验，因此检验统计量应与正负临界值进行比较，通常显著性水平为 5%。
由于在本示例中，检验统计量超出了这些临界值的范围，因此两个样本的中位数存在显著差异。
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Q1: What is the Wilcoxon rank-sum test used for?
The Wilcoxon rank-sum test, also called the Mann-Whitney U test, is a nonparametric test that determines whether two independent populations have different medians. It compares the distributions of two independent samples without assuming normal distribution. This test is particularly useful when data are ordinal or when assumptions for parametric tests like the two-sample t-test are violated.
Q2: How are data ranked in the Wilcoxon rank-sum test?
Data from both samples are combined into a single ranked list, with each value assigned a rank from smallest to largest. Tied values receive the average of the ranks for those positions. The ranks are then separated back into their respective groups, and the sum of ranks is calculated for each group independently to determine the test statistic.
Q3: What does it mean if high or low ranks cluster in one sample?
When high or low ranks predominantly fall into one sample rather than being distributed evenly, this suggests the two samples may have different medians. This clustering pattern indicates a systematic difference in the underlying distributions of the two populations, which the test statistic evaluates for statistical significance.
Q4: When should you use a z-score approximation versus critical value tables?
For small sample sizes, typically when n is less than 20, critical values for U are obtained from statistical tables. For larger samples, a z-score approximation is applied, assuming a normal distribution. This approximation becomes more reliable and efficient as sample size increases, making it the preferred approach for larger datasets.
Q5: What are the key requirements for the Wilcoxon rank-sum test?
The test requires two independent samples that are randomly drawn from their populations. Data must be ordinal or capable of being converted to an ordinal scale so values can be ranked. There is no assumption that samples are normally distributed, making this test robust for non-normal data and a suitable alternative to parametric tests.
Q6: Why is the Wilcoxon rank-sum test two-tailed?
The Wilcoxon rank-sum test is two-tailed because the alternative hypothesis states that the distributions of the two populations are different, without specifying direction. The test statistic is compared against both positive and negative critical values, typically at the 5% significance level, to determine if a significant difference exists in either direction.
Q7: What limitations should you consider when using this test?
While the Wilcoxon rank-sum test is robust to outliers, it is prone to higher type-I error when data are biased, heteroscedastic (having different variance), or when sample distributions are extremely far from normal. Despite these limitations, the test remains reliable for most ordinal and non-normally distributed data in practical applications.