多组重复测量场景下，mixed ANOVA对应的非参数检验是什么？

1. Non-Parametric Equivalent to Mixed ANOVA #

Mixed ANOVA is used for designs that combine between-subjects factors (like multiple cities) and within-subjects factors (like repeated measurements over years). When your data breaks parametric assumptions—think non-normality, uneven variances, or extreme outliers—here are your reliable non-parametric options:

• Scheirer-Ray-Hare Test: This is the direct non-parametric stand-in for mixed ANOVA. It extends the Friedman test (which only handles single-group repeated measures) to include independent groups. The test ranks all observations, then splits variance into three key parts: between-subjects group differences, within-subjects time-based changes, and the interaction between the two. It’s robust to messy data distributions and outliers, making it perfect for your multi-group repeated measures scenario.
• Permutation Tests for Mixed Designs: If you want more flexibility, permutation tests can be customized for mixed ANOVA setups. You’ll permute observations only within valid boundaries—for example, shuffle city labels to test between-group effects, but keep each city’s yearly measurements linked to its original group for within-subjects tests. This method doesn’t rely on any distributional assumptions and shines with small sample sizes.

2. Non-Parametric Alternative for Multi-Group Repeated Measures (When Repeated-Measures ANOVA Fails) #

You’re absolutely right that the Friedman test only works for a single group with repeated measurements. For your use case—multiple independent groups (cities) each with yearly crime rate data—your go-to non-parametric solutions align with the options above:

• Scheirer-Ray-Hare Test: This is your primary choice here. It lets you test three critical hypotheses:
  1. Are there overall crime rate differences between cities?
  2. Do crime rates change significantly across years?
  3. Does the pattern of crime rate change over years vary by city (interaction effect)?
• Post Hoc Follow-Ups: If the Scheirer-Ray-Hare test returns a significant result, you can dig deeper with non-parametric post hoc tests:
  • For comparing cities directly: Mann-Whitney U tests (with multiple comparison corrections like Bonferroni to avoid false positives).
  • For comparing years within a city: Wilcoxon signed-rank tests (again, corrected for multiple comparisons).

A quick tip: Permutation tests are also a strong alternative here, especially if your sample size is small or your design has extra complexities (like covariates). They often outperform rank-based tests when dealing with unusual data distributions.

内容的提问来源于stack exchange，提问作者AchiPapakon