使用aov()执行双向ANOVA时交互项未被估计的问题排查

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2026-8-24

问题：双向交互ANOVA分析未生成交互项结果的原因

我在数据集data5上调用aov()函数进行双向交互ANOVA分析，模型公式指定为salary_in_usd ~ company_size * employment_type，但分析结果中未生成交互项的估计值，请问问题出在哪里？

相关代码及输出如下：

data5 <- structure(list(salary_in_usd = c(180000L, 120000L, 215300L, 158200L,140400L, 215300L, 31615L, 35590L, 52396L, 40000L, 122346L, 135000L,205300L, 140400L, 140000L, 183228L, 91614L, 185100L, 200000L,120000L, 230000L, 100000L, 165000L, 86703L, 140000L, 210000L,140000L, 210000L, 140000L, 210000L, 140000L, 230000L, 150000L,120000L, 160000L, 130000L, 100000L, 48000L), company_size = c("L","L","L","L","L","L","L","L","L","L","L","L","L","L","M","M","M","M","M","M","M","M","M","M","M","M","M","M","M","M","M","M","S","S","S","S","S","S"), employment_type = c("FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","FT","PT","FT")), class = "data.frame", row.names = c(NA,-38L))

anova2i.result <- aov(salary_in_usd ~ company_size * employment_type, data = data5)

#            Df    Sum Sq   Mean Sq F value Pr(>F)
#company_size     2 1.361e+10 6.807e+09   2.278  0.118
#employment_type  1 3.888e+08 3.888e+08   0.130  0.721
#Residuals       34 1.016e+11 2.988e+09

原因分析

核心问题是数据集的交叉分组存在空单元格或样本量不足，导致交互效应无法被估计：

先查看两个因子的交叉频数，运行table(data5$company_size, data5$employment_type)会得到如下结果：
```
FT PT
L 14  0
M 18  0
S  5  1
```
双向交互ANOVA要求每个交叉组合（这里是3种公司规模×2种雇佣类型=6组）都有足够样本量来计算效应，但L-PT和M-PT组完全没有数据，S-PT组也只有1个样本。
当交叉组没有数据或者仅1个样本时，该组无法提供自由度来估计交互项的变异，aov()会自动跳过这个无法计算的交互项，所以结果里看不到交互项的统计量。

可行解决办法

若能补充数据，给L-PT和M-PT组添加至少2个样本，保证每个交叉组都有足够数据支撑效应估计；
若无法补数据，直接拟合不含交互项的主效应模型：salary_in_usd ~ company_size + employment_type；
也可以考虑合并类别，比如把PT类型合并到FT，减少交叉组合的数量，避免空单元格问题。

内容的提问来源于stack exchange，提问作者Rochhh999